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10 gdz geo_b
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10 gdz geo_b Document Transcript

  • 1. PO3Altl L ct4cTEMAr[3Aqlg TAv3ArAnbHEHHg oAKTIB I METOAIB NTAHIMETPII $ 1. Onopni Qaxru nnaniwrerPn- Teruaruqni saBAaHHq E TecroBii OopMi l-lpnmi i ryru 1'n)HaSa6orra4.Ilpauirran.noulurritrepeTzHaloTbcEa6onapa.ne'rrlrri.,flrrqontllx. Tr4HaroTbcg, ro poa6ttnarorr nr,rotrIr4Hy sa 4 qacrunui SKLIIO [apalTeJlsni, ro na 3 untr .T}l]lTI. 2. n) 50". Hexaii ogran xyr .r", rogi 6i:nurufi (r + 80)"' Cyua cymi:*cHzx nyrin 180"' ToAi x +rc * 80 : 180, gsi,qxu 2x : LOO'r : 50. Ornce, nenurufi xyr 50"' s al] b..{ri npami, fiepneuArlxyJr,Epni 4o rperroi upfiMoi, napa'renlni uint co6oxr s) 30'. Hexait a ll a i c - civua, A : b a c, B : a o c' Hexafr IEBA: 60', BC - 6icexrpmca ZEBC. 1 Tofi ZABC = ! ZAne, ZABC : 30"- lcAB : 180" - IEBA (eHytpirrrui onilo"topo""i npfi napaJrerbHr'Ix trpflMnx i ciunifi), zcAB : 180' - 60" lcAB : 120". v LABC ZC : L80 - /ABC - ZBAC, lc: 1a0'- 30" - 120" tc :30"' 5. a) 6' r) 7. 6) 8.6) 9. 6) lO, 6) 6. Hexafi na ogxifi etoponi xyra O poavrirqeni rouxr'r A, B, C D; 'l,l.a 4pyriE * At, Bt, ct i D,, Ae AA,ll B4ll cc,ll oo'; AB : 2, BC : 3, CD : 5, ArDr: 2O. Megia:ra rpraxyruur{a poa6unae fioro na gna pianone.rr Htrlrvt s pinuurtu n.norqatrv), .(la rpzxyrnr.rxa ai crop 2(a2 +b2-c"[poBe.qeHa Ao cropoHn c, ,4oprBHroc ^; = -' -z-- Yci tprz 6iceitrplrcu rpr,rlryruzRa rrepernuarorbcfi n ogu ca inu,eanpon mpurcAmHurca. Inqenrp rprrnyruurca e r s ganufi, Tpr.rryrHuK, Bicercrplrca rprxyrnrrxa no4i,rae nporulerxry iioro crop qifiui nprzleuuM cropoHaM. .{"rra rpz*yrurzxa ai cropon 4abn( o - c npoBe,rreHa 40 cropoHll c, AoprBHrce l: = @ + b), ' . Vci rpr.r Bvcorrr rpl4KyrHHKa neperr.rHarorbcfl B oAniii opmoqerlmpoJtr. npuKAnlHurca. ,(la rpuxyrHzra ia cro flpoBeAeHa Ao cropoHu c, 4opienroc o" = 4p(p - al(p-- t- 1, 6) 40'. Cyua xyrie rpriryrurzrca 180", ro,qi 2x-l 3x + nporopqifinocri rcyrin; 9* : 180, x - 2O; naftuenrulrf 2. 6) 90', Ocni;rr,nu r1afi6irrurfifi auyrpiurnifi xyr rpr4xyr a sram gosui[rsiri xyr 6yae nafimeHrunu. 3. 6) Jd 4nn2. fkoula pinHocroponnroro rpr.rxyrHrrKa t="+, ro.qi s=af ="6 to*,r. 4. a) 3 cu. 3. 4. 3a yaaramrienoro reopeMo$ @a.neca, flxrqo,4-B : BC : CD : 2 : 3 : 5, rogi ArB, t BrC, : CrD, : 2 : 3 : 5. Hexafi r - iix roe@iqient nponopqiftnocti, rogi 2x'l 3x + 5r - 20; r:2.Orxe,BrCr:2 3:6. Bidnoaida: 6. TplrKyrHxKl4 AB BC. CD 48 B,C, CrD Ba Hac,ri4xov B TeopeMr.r cuHycin, ,ACr= = 2R,' sin lB (oJIa, olucaHoro HaBKoJ'Io rpuxyrnzna, ro4i R= 3 =3 (cml. 2 sin 30" 5, 6) 6,5 cnr. 4e.R - 2 iii ,jiXl ,'d' iir irt V npsnroHyrnoMy rprrxyrur4xy uaitmenuroro rue4iaro ue4iana, npoBe,qeHa go rinorenyar. i gopirnioe ii n no.na). V npfiMoxyrnoro rpr{r{yrHnrca s r(areraMrr 5 c roAi nneaiana 6,5 crra, 6. a) 4. Hexafi y A,ABC npone4euo cepe4ni risii MN, NK, MK /ABC: lAMK: ZCNK (nnyrpilrni o4nocroponni n pa,rerbrrrx np,ffMrrx i civnzx). IMBN : ZMKN (npor ni xyru raparrerrorpaMa MBNK. IBMN : IBAC : I (nnyrpinrui o4nocroponni rrpu napaJreJrbrrzx i ciqnrx). ZMA-I(: IMNK (nporu.nencni xyrrz rapa,'rerrorpaMa AM LMBN : MMK : AKNC : ANKM N LABC (aa ABoMa r,ru). Orxe, Eorr{eH B yrBopeur,rx rplrxyrnrznin nogilnuf Maeuo 4 napH no4i6Hux rplrxyrnaxia. 7. r) s. p :;:iiiiili;;iiiiiiiiiii: TEOMETPI9 Eeae F. It. ra ix. t-tpoe Bnacrrsocti gonilrnoro rpr{nyrrrr$a Cyrrna nnyrpimnix xyrie: o + B + y - 180'' Teopema rcocunycie: a2 : b2 I c2-2bc cos a. Teopeua cunvcis: -:- = -!- = -!- = zn.--"-'---- sinc sinP sinY rDopny,rn nnouli rpurcyrnnrca: S = lrtO" = lln, = )rn", S = )aO s:'n l : S - ff : s = rp,se o =tTt s = filp - o)(p -b)(p - ct. Vci rpn me4ianu rpr4rtyrHrlxa neperr4Harorbcff a o4niii rouqi, Rxa rraol{Baerr'cs qeHmpoJw Mac mputymHuna. le:rrrp Mac rpunyrur'rrta no,4i'nae Horxny ue4iauy y siAHouenni 2 : 1, paxyrouu ri4 repurianrz rplrxyrrr4xa' 856 ::::i: rEOMETPIS Eees f. fl. ra ir. flpoQinux14fi piBeHb::i:::i;i:i:i:i:ii:i:i:i::::i:i::
  • 2. 8.") Js. V pinno6egpeHoro fipsMorcyrHolo rpr{KyrHuKa 6icenrpuca. onyrleHa na rinoteHyay' e Me,qiaHoro. Me4iaHa, trpoBeAeua go rinorenysn, 4opinHroe no'loszHi rinoreaysra' ,o6ro 11io Js. 2 9. r) 21 cn'z. IlrorqyrpuryrHtr(aMo)fiHasHafituaaSopuyloro S - aUsiny (y xyr nixqzull 1 croDoHaMu). S-l 6 14 sin30'.21 1cm';.'2 1O.r) 2:3. II;rorIi noAi6irzx rpuxyrnrzxir ni4noCarr,ca sft KBaApaT nignoueuna ix cropis (xsa{- par xoe@iqicHra no4i6eocri). f - t' u 3 9oruplrryrxuxu 1. n) 4. lni oci npoxogsrb qepe3 giaroHali KBaApara' 4ri iHui qepes cepeAuHrr [po- tz.nexcuux croPiH. 2, 6 7 cv. Cepegns ;riuic rpaneqii 4opinuroe niecymi ocnon' 3. r) 3 clr. flinnepuuetp8cu, roAiApyracropona8- 5:3(crt)' 4, e) 4O":140". Cyva nyris pom6a (fiH i naparerorpaMa). nphrer;tux 4o o4uiei cropo- uu, 4opinnroe 180'. flnufo xoeSiqienr nponopqiinocri Hyrie r, ro qi nyru 2x"' 7x.'logi 2x i 7x: 180, ;r : 2O; 2x : 4O, 7x : l4O. 5. a) 1o cu. I Hexafi y npfiMor{yrHuny ABCD CD: 5 cu, ICAD :3O": V LACD (ICDA:90') rarer CD ne:x<url nporu xyra n 30", rorvy AC 2CD, AC - l0 cm. 6. e) 8rE u'?. Hexait y pov6i ABCD BC : AC : 4 v' LABC - pinnocroponnifi, rogi ZABC : 6O". S*"o: AB ' BC ' sinZA-BC, S*,o- 4. 4 .sin 60" - AuE (no'). 7. a) | :2. Hexafi y npsMonyrHrrrcy,A-BCD nponegeuo 6icexrpucy A1{, IBA]<: IDAK. Tofi BK: BA. B Ilpoeegemo KKr L AD. S-u. = S*r.,, 3a yuonoro S*"o: 3S*" I{e rro:r.nrlno, aHulo KrD : -AX, a ornte, AD : 2A-I{t: 2AB. AB:AD:I:2. K1 8, a) 5 cu. flHuqo uorzpnxyrHl{ri e onucaur M HaBrcoJro Itora' ro cyuu fioro npor[Je]fi- 1 Hux cropiH pinHi, or:rxe,.4opinsrororr 1 20 cu: 10 clr. OcHi,rlrcn rpaneqia pieno- 2 6egpena, ro 6iqsa cropoHa 4opienroe I ro "* : 5 cn. 2 858 ;:::::: TEOMETPIq SeBo r. fl. ra ix. rlpoQinunui piBeHb i;:iiiiiiliiiiiiii{lii'i:: 9. 6) 120" a6o 135'. Hafi6i,rrnnu ia rcyrir nparvroxyrHoi rpaneqii e rynufi, y 2 pasu a6o nig npsuoro Hyra, a6o ri4 rynoro rcyra mennruii ni4 npnrroro Kyra, ro niu 4opieHroe 45", a (cyma xyrie, rrpnrerJrrrx 4o 6iuuoi cropoHr rpaneqii ,flrcrqo rocrprfi xyr y 2 pasr uenurrafi ei4 rynoro Hyra ny x", a BerxqtrEy ryfioro Hyra 2x": x I 2x: 180; x nyr gauoi rpaneuii 120' a6b 135". 1O. r) 6 cm. Hexait y porvr6i ABCD ZBAD : 12O", r Ocxialnz n porrn6i 4iaroua.ni e 6icexrpracauz rior D / 4u "1 "n',/ / D K? .zBAC - I .tZO" = OO.. 2 Togi IBAC : ZBCA: /ABC : 60", MBC - pinn AC : AB; AB =1. P^o,,, AB - 7..24 "*- 6 cv; AC - 6 cv. 4*'" 4 Kono i xpyr 1, a) n2 u. flonrxuHd xo,ra C : 2nR, iroro vnepri C'- -. n.2nL"= -='I" (Ml. 2 2, s) 2On cu. flroula rpyra S : nR2 :100n cu2, rnAi -R )rcuHa HoJra C : 2nR, C : 2n . 10 crra : 2On ctt. rii i,li .ii'i ii$ 3, r) 55'. Hexafi y xo.ni s rlenrpou O nponegeuo paAiyc n xinqnx fagiycin CA i CB, ge C' roqlca rreperrrny 3a ymonoro IAOB : L25". 3a n,racrusicrxl AorrqHux ZCAO : ICBO :90'. pr,rryrHrxa ACBO cyva nnyrpimuix nyris 360", I - 360. - (tcAo + zcBo + 44ozl:360" - 180. - t25 4. r) 120". Hexafi y pinnocroponuifi rpzryruzx ABC xo.no a qeHrpom O, Ldenrp nnzcanoro Rora O - roqxa neperr4ny licexr i CO rpuxyurzxa. Ocxi.rrlxu y AABC ZA: lC: 60o, ro - ,locA - 30". IAOC - 780" 2ZOAC - 120". Ocni.nsxra cropoHy AC BuAHo 3 qeuTpa BtrucaHoro nola rom AOC, ro ii nz4no nig rcyrou 120o. 5, r) 4 cm. V npanumHoro [recrurcyrnrzna cropoHa gopin IroJro HEoro lro,Ta. 'foAi nonra 4opieHroro ro !.rncu : o 6.n)4u. Hexaii nola s r{eHTpaMr{ O i O, marort nnyrpinuro y ro'rqi O, OA < O,4. 3a yrraoeoro OA: 7 yr, OrA: 3 "Io4iOOr:7 - 3:4 (rvr). 7,6) I6n m2. Ilron4a 6i.nsnroro orpyra S : n. 52: 25 MeHuoro - S' : tt . 3'? : 9n (u,). flrorqa ni,rrqa S,: S S', S,: 25n- 9n: 16n (ur). DA ffiiiliiiiittilli rEoMETPle 6eee r. n. ra ix. npoe
  • 3. 8. a) 16 cm. f,IHIqo r{BaApar onficaHo HaBKoJao RoJra'. To cropona r{Baaqpara AoprBxro€ 4iaruerpy Ho.na. [on:rcr,rua r(oJra 3 4iarterpou d: C : nd, ssigx.r- d = 9. Togi d, =16n = 16 (crvr), i cropona rcna4para 16 cu. n It 9. s) 9 cNr. Hexaft y Ko,IIo 3 l{errrpoM O rfiuca:.c'o LABC' AB : BC : AC. V LABC 1A: ZB : lC, orxte, Ayrnl' Ha sfii crapalorr'cs qi nnrcaxi Hytu, pisHi. wAB : wBC : wAC- Ocxi.nsxu uAB + u AC : 6cM, ro uBC=1 O=3 (cu), aAoBlr{r4- na xoaa 9 cm, 2 1O. a) 6n crvrz. Il.norqa cer{ropa 3 qenrpaJlbHraM ryrom 60" / ' I Hu 60" = l soo". s"^: n-R'?, roAiAopiBHloe e5*e., ocRlrb 6 , ls^=1.-n.o2=6n(cu'/). 6"p 6 9" 6) 3A: 4r. l{eutp Ho:ra (0; 0), Pinnauna npfiMoi, rqo x-x. u-u.TOqHU: - xz-rr Az-Ut !,pyra rovxa (3; 4), roa; x -o - Y -o x - atr_o=ni,"=2,4x:3s 10, e) y : 3. I{enrp rco.na (0;.2), pa4iyc 3, ro4i 4oruvH neperr.rHac xo.no n 2-x roqirax, Berropu 1.6) (-8; t't. AB(-7 -r:12- (-3)), ,4ii(-8;15). 2.9 a.6 =o. 3. a) (8; -7). 2F = f-+,,21, 34 = Qz';S), m(-4+72i2*9 c. a 2J5. l,i' = [-2fu 4' = J+ r+n to = zJs. 5, r) (4: -16). BianosiaHi xoopgraHaru rco.nineapuax nexr 4 = -q, ro4i no,rineapH uir (4; -16)._16 6. n) 0. nC+ dn+ OA+CD =(BC+ Cil ,OA+ ttf)= 7. 6) 60". (i 6 =lal.lal cos(a, a); cos(ri. a) = ,r:h, cos 8, n) 3. Bexropu [eprreuAnnyJrspni, ariqo ix cra.nnpnra Iogi -2 .I * 6rn: 0; 6m : 18; m : 3. 9. s) -8.5. a = {, x =3'7 =?} = -! = -".u.' -6 7' -6 -6 2 10. r) -b. Benrop BA nporu.terxnr.rfi po en, axulo np npoerqia EA naOy * -b. S 2. Mero,4'r posBtssyBaHHc nraHiMerpnqHhx 3ag Mero4 xoopgznar. Braxopucrann.E MeroAy rcoopAranar cnJr4 1) nnogarr 3pyrrHr.rM cuoco6ovr AeraproBy crrcreMy noopA 2)yuony aagavi ra ii srcuoeox nepexJra,4arort Ea MoBy K 3)AosoA,firB a6o o6qlrc.nrotorr neo6xi4ne Ba AorroMororc e HOrO anapaTy: 4) o4epxanr,rii peoyJ'r brar Sopuylrorcrr, y repm i Hax sa4av Ilpr,r 4rolry BurcoprrcroByrorr rari @arcrra: 1) ,flrcqo sa4ani gni rovru A(xr; y) i B (xr; yr) cnoiMu Koop nigpiaxa AB o6qucnroerrcs Ba OopMyroio: lesl= ,UQ, - 2) .flxrqo ro.rna C no4inae ei4pisox A-B y nignourensi AC Harr,r roqlcn C(x; y) o6wcnrororbcs sa ([oply,rarvrr: r=t:*1*',r=u'-+^u'.To4ixoop4nnarr,rcepeAr4Hrrni4pien 1+1. " l+L aa dopuvraurri x = rrl xz . y = a'!-!z . 2 '" 2 KoopAraxaru sa nnouluxi 1. 6) (4; 1). Itoopgzirarz (x; y) cepegunu nigpiara e ninqauu (xr; a) i (x"; yr) o6wc- r,rrororbcn aa abopMy,laMu, * = {t;:, g = yLj-!' . Tori r = t -7 = 4,, = -3,* 5 = 1. orxe, Koop4r'IHartt cepeA[HI{ ei.qpiaxa (4: 1)' 2"2- 2. e) AC = Ct -* + <z - q,sf = t6;z,u =,ltt,zs, BC =,!@-27" +16-4,5y =,10;z,zs = ^11t,%, rofiAC: BC. 3. s) (2; 5). V rovnu (2; 5) x : 2, y : 5, ro1i 2x + a : 2' 2 + 5 : 9 * 7, ovxe' (2; 5) He HaJrexrrrb nParriii 2x + Y : 7, 4, s) 29 - x: 2. V napalenr,Hnx rrpnMnx xyroei xoeabiuieuru pinni. V np:rmoi x+2 1 I 1. 2g x:2, u=^;!=i*-'mvronufiKoediqicgr r'nxi vnpamoi a= rx+5' 5. e) 135". x+ A:5, A: -x * 5.,{auanpfiMa - 6icercrpuca IIiIV xoop4zHarl{r4x vneprefi, nepeuecena napaJreJrbHo BBAoBT{c Oy na 5 oAr4ur.rrlb yropy. 3 AoAarHr4M Ha- np.fiMxoM O.r yrnoproe rcYr 135". 6. n) (0; 2',1. x2 + (a - 2)', - 8 : 0; (r - 0)' + (y - 2)' - 8 : 0; qeurp (0; 2)' 7. r (6;0). V toqxpt M xooppnuaru (x; O). Toni MA2 : (5 - r)' + (4 - O)'z : 25 - 7Ox't xz * 16 : x'z - IOx * 4L; MB2: (2- x)2 + (l- O)":4- 4x t x2 !l: x2 4x + 5' Ba ymonoro MA: MB, MA2 : MBr; x2 - 10x * 4 : x2 - hc + 5, 6x : 36, x : 6. Or:xe, rovxa M(6: O), B. n) r'? + y" : ?5.1{e'rp xo,ra o - cepesnna Aut ol+t f) . oro, 0), nsaApar pa4iyca n': (4- 0y + (3 - 0)': tO + 9:25. 'fo4i pinnnnHa ronatc2 + A':25' 860 ll{: reOurerplg Eees F. 11. ra in' IlpoQinsxrr piBeHb i:jiilitilfii:li:;i{i*i iil ii I ii:::ii:{ilttiiil:f; TEOMETPI9 Eees F. Il. ra in. ItpoQi
  • 4. 3) 4) PisHssssrpauoi-AJ3: r-rt -a-at . Iz-rt Az Ut PieHssH,E noJra B qeHrpoM (a,'bJ i pa.4iycoM r: (.r - a)2 + (A - b)': P. 1. Tpz npnui, nponegeHi rrepes oAHy ror{Hy, po36r4Barorb trrorrluHy Ha 3 napu Beprrr- KaJrbHr,rx, a oTlrce, piBHax RyriB. Ilpunycrurvro, qo )t(oAeH is 4zx xyrin He MeHrrrrrfi aa 61o, ro6ro Korr(eH i3 HyriB 6i,'rbilufi a6o AopiBHIoe 61'. To,4i cyMa Bcix 6 yrnope- nux ryrin 6i.nr,ura a6o 4opinuroe 61' . 6 : 366", rqo HeMor{rrlBo, ocrci,rrxra ix cyma 360". Orxte, Harxe npr4rryu{enHri HenpaBurbse i o6os'seroso e Kyr, meuurufi ea 61", a, ocrcirsxz 4Jrs Hboro e Beprr.rxaJr,HrzI? (i pinunfi), ro npuHafiuni ABa B yrBopeuux nyrir menrni ea 61'. IIfo fi rpe6a 6yro goeecru. 2. ,flrcqo xoe(pirlieHr rponopqiiiuocri Hyrin r, to niryrpiurui r{yru .r, 2x,3x, a ix cyma 180'. To.qi x * 2x i 3r : 180, t : 30. Hafi6iasnrufi ryt 3.r : 90, a orxe, rprrnyr- nzr npRuoxyrnufi. Bi4noai}t: npsuoxyrnufi . 3. Cyma aoeHilxuix Hyrin rpurcyrnrrxa, B3srux tro oAHoMy npra aeprnuui,4opieuroe 360". Sossiusi xyru 2x, 3x, 4x, Ee r - xoe@iqieut nponopqifinocri. 2x i 3x + 4.r : 360, x : 40, Hafiuesunfi sosuiuHiii xyr (2x: 80") - 80". Haii- 6i,rruuu euyrpiuxiu xyrom 6yge xyr, cyvlir+*urfi s Haiiuesmnu soeuiuruiu. Orrxe, uyranrafi xyr 180'- 80' : 100'. BiOnoaiOu !OO". 4. flxqo npnunfi rcyr 4irnrrcn Brrcorolo i wIe,qiaHoo Ha 3 pisHi qacrhHfi, ro xoxel i: nzx 30o. I{yr uirx xareroM i eacorolo 30'. Kyr uix quu )+(e HareroM i 6icexrpacoro npsMoro *yra45'. Orxe, xyr mix 6icexrpzcoro i sl{cororc: 45o - 3Oo - 15o. Bidnoaidu: 15" . 5. ,{o6ygyerrno qorrrpuryrHnK ABDC. flxt4o y qorupunyrH14na 4iarona;ri neperzHarorb- 6) V LCAM i ACBN: AC: BC, ZC - cninrrt:aft, ZC orxe, LCAM - ACBN 3a cropoHolo i npuaer,rrzun g II{o ri rpe6a 6yro gonecrn. 8, Ilnan nodydoeu. Ilponegeuo: 1) npauy a ra o6epeuo xa airt rourcy If (riHia ocuoez rcyrH rxa): 2) nepneu4nrcy.nap B roqxlr I{ go npauoi a (lirrin en TptrKyrEERa, npone4enoi go ocnonu); 3) eig rcivru -FJ Ha nepren4zny,rapi ni4naa4emo ni4 BH :6 cm; 4) aa gorouororc rlt4pnyJ'rs anaft4euo na nparrifi a 4 roqxuBsa8cu. Y MBC AB : BC :8 cv., BH: 6 cu. 9. Ilnan no6g0oau. flo6y4yeuo: l) npauufi xyr C; 2) ua o4uiir fioro cropoai ri4xaa4euo aigpisox CA: 3) ua inrrouy Hareri aa AolroMotoro rlr4pnyJrfi anafige ei44aleny ni4 rovxra A ua 6 cu; 4) Ha npoueni CO nigxaa4etro ni4 rovna O ei4piaon ni4piexy OC s 4pyroro 6oxy ni4 roqrcu O, y LABC ZC :9O", AC : B cm, AO - vesiaua, AO : 10. Ilnan nodAAozu. 'flo6y4yeuo: 1) e xiHrl;r A nigpiarca ,4-B lpone4er,ro npouinr nig ro crplru nyrorr 9o,43; 2) na npomeni ni4n,ra4euo Z pinnr,rx ei4pisxie 6y4r-axo ,qots)nrrHr AA, : ArA": AAr: AA^: AA": AAu =Mi3) s'egnaeuo ei4pienom ro.rtiu Az i B; 4) uepes rourr-r Au, As, A4, Ai, A,, A, npoee1er,ro npaiv napare:rrHi ArB, axi neperHyrbAB nignoei4no B roql 3a reopeuoro @aleca, A-8, : BrBr: B2BB: B"Bn: B^ cs 1 ToqItolo nepeTtrHy AlJrfiTbcJr HaBnur, To qez qorupr{rcyrHr{}c - napareJrorpatt, orwe, ABDC -naparerorpaM, A-B 1 DC, BD ll AC. a) Pocr.nsueuo LMAO i LNDO: AO - OD, IAOM - - lDON (aepruxa,rsHi), . MAO - INDO (suyr- ^ir ,i ii.-...... pinrni pisHocropouni npn AB ll DC i ciqnifi BC). To,qi LMAO : ANDO sa cropoHoro i npu,rerrrauu 4o nei xyrauu, orxe, MO : OC. 2) Ocxi.nucu LMAO : ANDO, ro AM : DN. 3) Ocximxu LMAO : ANDO, ro ZDNO : 1AMO. IIIo ft rpe6a 6y.no goeecrr.r. 6. 3a yuoeoro AC : BC, AN : NC, BM : MC. C1 To,qi -41r' : BM, NC : MC. / a)YMMBiLBNA:AB -cninr,na, MB:AN,ZNAB: IMBA -- / . (xyrn npu ocuoei pinno6eApeHoro rpuxyrHrxa). "A ' Tor.i LAM B - LBNA 3a ,rBoMa cropoHaMu i rcyrov mixr HNvu. I 7 6) v LCAMiACBN: CA:CB,CM :CN, lC *i"t""ti,-t"^t a Z g LCAM : ACBN oa ABoMa cropoHaur i xyrorvr mirx nrznu. III,o ii rpe6a 6y,ro gonecru, 7. 3a 'ryronorcAC.: BC, ro4i 1A: lB; ZCAM : IBAM : ICBN : .IABN. a Y LAMB i ABNA: lNAB : IMBA, INBA: IMAB, ,43 - cui.rura, rogi AAMB: LBNA 3a cropoaoro i npr.r.ner,rzurz 4o nei nytauu. A A62 i{::: |EOMETPI9 6eB3l. Il. ra in. flpoQinrxri piBeHb iii::::rl;:till:ji{i:l iiiiiiiiiii;iliil:i rEoMETpte Eees F. n. ra im. npoei
  • 5. I a)Y ACNB: CN = rAC = 1,6 (cM); BC : 4,8 cu; ro,4i BN = ,F,o' + l,z" = '125'6 = f,6J10 (cu). 6) V AACB: AC = 3,2 mr; CK = i"" = 2,4 (cr*t); rosi At( = 'ts,r' - r,4'= JG = a (crw). n) V AABC: AB = $,2' + 4,8" = G3,r8 = r,ol[5 (crra). V npnrroryrnoMy rprztyrHrrxy me4iaua, npoBeAeua go rinorenysz, 4opiniroe ii no- 1 ,rogilHi, orn<e, CM = )an = 0,8r43 (cvrt. Bi}noei}u: 0,8'[5 (cu); 1,6{0 cu; 4 cm. 13. Hexafi y napaJrerrorpav,i" ABCD 6icenrpwca BK, ZBAI{ : ICBK, AK: 5 ciir, KD : 15 cm. AD : AK + KD, AD: 20 cru. ICBK : IAI{B (sHyrpiuHi pienocroponni npu BC ll ,4-D i ciuuiia BK). To.qi LABK : IAKB i MBK - pinno6e4pe- suft s ocsogoro BK' AB : A-I{ - 5 c*t. A P*"r: 2(AB + AD), P*"o: 2. (5 + 20) : 50 (cm). Bi}noei1a: 50 cttt. 14. Hexafi y fip,flMoxyrHraxy ABCD 4o giaroHali AC npoae4eno repnen4raxy;rap BfI i sexaia All : CH : 1 : 2. flxulo xoe(fiqienr ix nponopqiftnocri B * (r > 0), ro AiI : x, CH : 3x, AC : 4x. y LABC ZB - 90", BH - wtcora, flpoBeAeHa 4o rinorenyer' A To.4iA3'z : AC . AI{, AB2 : 4x' x : 4x2, 1l roili AB = r, = i.+x = jnC. II{o ii rpe6a 6y.no gonecru. 15. Hexafi y LABC BH LAC, BH:4cv, AB -: 6 cm, BC = 4G c*. ts LABH (zAlIB :90"): ,4-H = J An" - BH'; ilr = ^tE6 - 16 = zJs (crvr). 7a LCBH (LCHB - 9O'): cH = '[Ec" - BH',crl = r,(4.f6f - 16 = aG (cu). TosiAC : AH + HC, AC = 6J5 cu. = 1 aG .4 - Lz,fb @.''.S*. = iAC BH, Sffi, 2 BilnoaiTt : l2n!B cw'z. 16. Hexaii ocuonx rpaleqii a rrn i b u, a > 0, b > 0; cepegna.niuin a=15, b=5. 6 sittlA= BC "inz/.=!. "o"1A= AC . "o". AB 13' AB' Ls, zA = 4. tr za =Y = 2.4.. "AC-a n) sin lC : sin 90' : 1; cos lC : cos 90' : 0; tg lC 18. ,flHruo cnHyc o4Horo nyra 0,5Jd = f. t" BeruqnHa'2 urifr ocNoni). ToAi lcyr npu ueuurift ocnosi 180o - 60 rpaneqii, [prrJ'rerJrux Ao 6i.rnoi croponr4, - 180'. O 'ro Kyrr,r npfi ocHoBax pisni, orme, Kyrn trp[ snxuif uifi - no 120". Bidnoaida: 60', t20'. 60', 120'. 19. Ilo6ygyerrro npanrnfi xyr-4 i Ha fioro croponi ei4n.n Hzr1r, Ha 4pyrifi croponi xyra anafi4er'ro rouxy C, n .nena ni4 ro.rrclr B na ei4cranr 1 o4r.ruraqa. Togi /-B - uryrcauufi, ocxi,rlnra B npflMo(yrr{oMy M (zA:so') costB= #= T=t,t r,Ao'58 Toil /B - 60'; sin60'=;; tc60'= J3. t; Bi1noaida: Y; Ja. 2 17" Hexaii y LA-BC lC : 9O", AC : 5 c*r, BC :'l-2 cttt. Togi AB: 13 cu. a) sinrB = ffi, sinzn = ft, ".o"za = 99, cos lB = tglB= AC BC' tglB= 2O. llo6y4yenro rpxuufi tyr a nepruznoro A i AB : 5 oAr4glrrlb i AC : 1 oArrHr4qfi. Tovy a AABC (ZA : g0") ts ZC = 4, E zC =- AC' " i lC - uryxanrfi. 3a reopeuoro fligaropa, BC2 : AB2 * AC2i BC2:25+7:26; BC=Jr6. 51 Toni sin lC = -;: , cos ZC = - =-.'126 v26 51lJlOnOALdb: -i -. 126 .126 21, Ilnaa no6g0oau. a) Ha npamift a o6epemo roury 11 i nporegelro a nei KyJrfip Ao o. 6) Ha nprmifi a nig rovxu 11 no pisHi 6oxu ei4r.na4eu HA=Hc=l o-z tr*).n iiiiiittil{i:litt::.:il rEoMETPle Eees r. !-1. ra is. Ilpo 28 vti raP. lo s.. c. I 5 - 12 i3' B D K na iioro c 5_ = 1 a+b 2 3a ymonoro {a-b=10' la-b=lo. l2a'Bo, 1-' =ro: la+ 6=2st zd =to, Bilnolilb:15ui5u. i) 15 964 iiil: rEOMETPIS Seas F. n. ra ir. IlpoQinuxilfi piBeHb i:i:i:i:::i:i:li:::'ii:i:i:::::
  • 6. s) 3Hafi4euo Ha rreptrenArrnyJrspi roqry B, siAAareHy ni4 rounn A Ha 6 cM' LABC - ruynauuii: AB : 6 c.u: BC, BH. Bilcora. coszA=ffi, "o"za=?=I 22. Hexafi cosZA: O,8; coslD : O,6. a) lA+ ZB:18O", lB:18O" - 14; coslB : cos(180' - ZA) : -cos.1A: -0'8; sitr lB - ^tf *", n = fi o,aa = o,o; sinzB 0,6 3 tp llJLa " cos, B -0.8 4' 6)ZD+ lC:18O"; ZC:1'8O'- lD;coslC: cos(180" -. D): coslD:-O'6; - 0,8 4 sinzB- Vl-( 0,6)'=Jl-0,36 0.8: tg-B= O.O=-a 23. 3a yuonoro si4omi,qni cropoHr{ i xyr mirx Hurrrz. ,{.ns 3gaxoA}r{egHs Heni4ouoi cro- poHu BlrKoptrcraeMo reopeMy xoczHyci s. a) AC2 : AB2 + BC2 - 2AB ' BC ' cos lB; AC2 : 32 + 82 2 3 8' cos60' : 9 + 64 - 24: 49:' AC : 7 cvr. 6) BCz : AB2 + AC2 2AB AC . cos 1A; nc' = (arE)" + 42 - 2' arE'' q' cos 45" = 7 2 + L6 - 48 - 4o; BC = 2l[b cu' Binoaidu: a) 7 cm;6) 2J10 cm. 24, Hexafi r xoedriqieHr nponopqifinocri cropiir, ro4i cropouu tpuxyrr;axa 7x,8x i 13r i sa yMocoro 7x l8x + 13r: 56, x :2.,{onxuHr'r cropin tprznyrnnxa 14 cm, 16 cm, 26 cu. Haii6i;rr,urufi Hyr o JreE{r.rTb nporz nafi6i:rl)]Iroi croponz TpltlcyrHl{rca. 3a reopemoro xocuHycie: 262 : 74' + 162 2 ' 14 ' 16 cos q, ., _ 142 +16'_ 26'z = _ l - rooi a : 120..3BrAHu cosd = 2.w.rc 2 Bidnoair)a: 72O", 25. Hexafi y raparrerrorpami ABCD BD : 12 cr'tt, AC : 32 cv, AB :'14 cm, To,ui BO : 6 cm, AO : 16 cnr, 4e O - rorrga rreperuHy Aiaro- 27, [:ze. puc. 4o aagaui 26. }{exaft y pou6i ABCD AC an-rp^"-^, AB=! 6,8=l,z (cv). Ao=!o", 4 *'" 4 2 3 LA2B (ZAB : so") Bo = 'EE - ,qo"; Bo .EJ' 4.8'- 6.r .z,r = 0,8.5 = l,b (cv). so",, . 1 AC.BD = AC.Bo, s*., = l.6 l,b = 2.4 2 Bidnoaida: 2,4 cv2. 28. Hexafi cropoHu naparerorpaMa aib,rocrpuir xy P:2(a + b):52, a + b:26,b:26 - a. S: ab 6O: a(26 a).sin 3O', 6O-a(26-14 +, 26a -2' Seigxu ar:6 a6o ar:2O, ro4i br:2O a6obr:6. a-6cv,b-2Ocw. flrorqanaparerrorpaMa S:a.h.:b.ho,aeipxu h- f, ," = !9 = s t"*r."h'20 Bi)noai}u: qropoHu 6 cm, 20 cM; Errcorr4 10 cu i 3 29. Hexa:ii v rpaneqii -A-BCD A-B : CD, BC : 8 cu, AD AD: 18 cm, BH LAD, BH eucora rpaneqii, Rxa 4opinnroe 4iamerpy BtrrrcaHoro ItoJra. Ocni.nrnr e rpaneqiro Mor+{Ha Btrucarn xono, roAB + + cD : BC + AD, AB + CD : I + 18 : 26 (cu), aos IBOA 2BO.AO 2.6't6 2' A lt.lii::iil: IBOA:60' (ryr uirx giarouanaMu traparlerrorpanta). 3 ABOC maeno: :..:iiii:ii:!:i: BC, : BO2 + AC2 2BO .OC . cos tBoc, :!!i.1:;ii:i:i BC2:62 +162 - 2.6 .16.cos(180'- tBoA):36 + 256 2'6't6'(-cos tBoA): tii;iiiii : zsz +e6 : 888; BC =2Js7; P*"o =2(74+2"ffi)- 4(7+"i67) (cM). ,:l:,iiiiii Bidnoeida:60'; n(7 - "67) "*.:i'i:i::li:l:: B -i;,iiii,ii 26. Hexait y pov6i ABCD roq$a neperhHy.u.iaroHa;reii o. AC ' l2 cvt' n BD -t6cv. s*, ,=Ior'BD, sM.D=) tr 16=96 (cv'?). , (4)" Y LAOB (1AOB :90") BO : 8 cvt, AO : 6 ctrt, I / roAiAB: 10 cnr. V Pa"r: 4AB, P*,,o: 4A cw. D Bi6noai0a: 40 cu; 96 civr'z' 866 l:id: TEOMETPIS Eees F. Il. ra in. flpoQinuxhfi piBeHb ii#;i:.::iiiiii:;l:'+i 'I A,B - CD =' .ZA = 13 (cm). 2 Bncorr pisuo6e4peHoT rpaneuii siaruuarors eir Hux Hsoi ocuoeu pieHi ei.qpiexfi. roti Atr = M -^BC ,''22 3 LABH (lAtrB: 9o') Br{ - ,,En' - dn', BH = To,qi paAiyc BtrtrcaHoro Hora 4opinHroe l-at - o "Bidnoeida: 6 cyt. z 30. Hexaft y AABC (lC : 9O") CK - 6icerrpuca, AK 3a eracrrreicrrc 6icexrpucu Kyra rpunyr'."" AC204 BC153 ,flrcrqo n - xoe@iqienr nponopqirluocri rarerin AC 4x, BC - 3x. Y AABC AB: AK + KB, AB: 35 cm, AB2: AC2 + 35'! - llx)t + (3.r). 25x' : 352, (5x2 : 35', 5x : 35, x : 7; AC:7 .4:2a (cu), BC :7 .3:21 (cm). s.^"- 1AC.BC. s^", =:-2a.21 -294 (cM,). 2^''2 Bidnoeida: 294 eM'?. i;:li:ilriili*t:ilii;::i TEOMETPI9 6ees F. Il. ra ix. flp to* sarefi, Ba Hac,ti4xom I TeopeMr,I lcocuuycrB, Bo2+Ao'-AB2 t4 62 +76'-142 | t
  • 7. 31. Hexaft y rpaneqii -43CD AB : CD : 13 cru, BC : flpoue4euo BH LAD, AII -27 -^11 = 5 (cu). 2 B LABH (ZAHB : so") cos/A= + = +AB 13 3 LABD Ba reopeMoro xocnuycin BDz : AB2 + AD2 - 2AB .AD . cos /A, BD2 =132 +212 -2.L3.2L.1 = t6t + 441 - 210:4oo, BD:2o cw. 13 BiOnosida: 2O cv. 32. Cyva rcy"rin onyx.noro n'R'rrxyruuxa SE :180' (5 - 2). S; - 540', flr.cuo rcoe@iIlienr nponopqifiaocri xyrin - x, ro Benvq'fi:F.u nyrin 3r, 4x, 5x, 7 x, 8.r i 3.r + 4x 'l 5x * + 7x + 8x: 540; 27x : 54O: x: 2O, roai 3r : 60, 4r : 80, 5r : 100, 7x: 140, 8r : 160. Orrxe, ryru 60", 80", 100", 140', 160". BiOnoei1u:60', 80', 100", 140", 160", Bnyrpiurnift rcyr B npa- 180"(n - 2) Cropoua rrecrrrnyrr{vfia t" =:, ", = f = 3 (cm). Cropoua onr,rcanoro npaBfirbHoro rptrnyruflKa a" = p" - Ba", p3 = lSJa cM. s3 = $, U - {e "61' Bilnoaila: 18J3 cm; 27^/3 ctn'. 37. Hexafi pagiyc xola, oracaHoro HaBHoJro rrpaBaJrbHo caHoroBHboro -- r. R: au-4c,r, r=+, r= fl.norqa xinrqa S,. : So - S", S,,. : nBz - na : n(R2 S" : n(16 - 12):4n (cu'). Bi4noeida: 4n cv2. 38. ,flrrqo A(3; a), B(-}i 4), c(-3; -4), ro AC= G+3)' +(4+4)' =LO; BC= J1_3+O:L(4 poHaMrr 6, 8, 10 - np,fiMoltyrrruft e rinorenyaoro 10 gopinHioe noronltni go6yrxy xarerin, roAi S = f o Bilnoailu: cropoHrr 6, 8 i l; cM; nrroura 24 ." &,. 39. Hexaft y rpzxyrnr.rxa *43C A(7; 5), B(4; 1), C(-4; "(z r+.r+taM - cepeAtrHaAB; M 2-; 2 ), M(5,5; 3), rosi uesiana CM = Je+ - S,Sl" * tl - s (4-4 t+7 6)N - cepeAr4na nC; w[ ,j t;), N(0; 4), ro4i negiaua Ar,r = !/47 - 0'/' + (5 - 4)' = "6 /" -4 5+7 s)K - cepe4r,tra Aq Kl;r;), K(1,5; 6), ro4i ruesiana ,B1{ = G- 1,5)'+ (1 - 6)' Bi0noeiOt: ; rJi; 11 cn. ,4-D : 21 ctu. 811 a 2L 360" 33. I]enrpa.nrnrft ryr c' irpaBr.rJrbuoro rl-nyrunxa o = -.Brzrrb'oro z-r(yr*r,.*a 1] - 180'(z - 2) n Ocxi;rlru o n 4 paara menlrprft ni.q p, to 4cx : ll, 4. 360' - 4 . 360' : 180'(z - 2), n - 2:8, n: 1rO. n BiOnoei1t: lO. 34, 3a yuoeoro D : 6 cnr, roAi fi : 3 crvr, Hexafi cropoHa onficaHoro Muoro(yrurrxa o, enucaHoro - b. a) n:3; a=2RJ3, a=6Ja crv.; b=R&, b=3Jd cu;, L = Ba = 18nE cu: pa = gA = 9JE ciu: 6) n: 4; a: D, a: 6 crrr; b = RJr, b =3J, cM; P, : 4a: 24 cNr; P, = 4b = 72yli ev: olo a) n:6; o=o!o R, a=2Js cwib:R,b-gcM; P"= 6a=l2JB ev., ,' Pb: 6b: 18 cu; r n: t2: a =zntsff, a:6tg 15'; b = znsin{fr, b - 6sin 15"; P. : L2a : 72tg 15', Pa: l2b : 72sin 15'. 35. Hexafi pa4iyc onrzcanoro HaBKoJ'ro rcBarqpara i uecrrrcytnrzra KoJra * R. Cropona P 2J = 6 (cru). Toai R = !+, n= * = s.E tcMl.KBaAPara an = Z, an = 4 ,12 lz Cropona [recrr.rxyrHuxa au: R, au: 3rE crr. Pu : 6 ' da, Pa = 18rD (cm). ",=ry, " -g (a€)'J5=2z16 (cm,). Bidnoeido:18uD cm; 27J3 cvt". 36, ,{rs [paBr{JrbHoro rprrr(yrH{r(a gaAaHe rco,'ro e pa4iycom -R e nuucauzu, a Arfi npa- Br4IAEHOtO mecTrrltyTHr4ft a - O[r4CaHrrM. 868 iil: TEOMETPIf, Eeee F. l'1. ra ix. flpoQinrxufi piBeHb iii{i:.:+liii::iiii:*ili 40. Hexait KoopAtrHarrr ro.rxu N(l; y). Tovxa K K(2 =6;l+rl, K( 2: _t). 2 2 ) P * cepe,4una niapigna KN, P( -2 *' :-1*vl, a P(-6: 2t,'oai -21* = -6,rtrl 2 '; o =2' -I + a:4, u:5. or:+ce. N(- 10; 5)' 2" Bidnoai1u: N(-10; 5)" -Ebvb 2 -t;Oi I 2 - iiiai{lt{:iititli;:* rEoMETPle 6ees r. fl. ra ix. Ilp
  • 8. 41. Hexait roq:ra M(x; O), rcai MO' : (r - 0)' + (0 - O)'z : x2, M P2 : (x - 2)t + (0 - 3F : : x2 - 4x + 4 + I : x2 - 4x * 13. Ocrinrxr.r roqxa M pinuoei4ga,reHa si.q O i P, to MOz: MP2, x2: rc2 - 4x + 13,4x:3, t= +=3,25. Orxe, M(3,25;O. Bidnoaida: (3.25;0). 4 42. Pir;r-ftsss npauoi, flHa rrpoxoAurb vepea 4ai roqrcu A i B: *-l =a-4- x-L=a-4. x l:u_4.x u:_8. -2-t I-4' -3 -3 -- IIs np,nua ni4rzHae ri4 ocefi HoopAuuar npauoxyrunfi LCOD. TTpn r : A -A : -3, y : 3, or:rce, C(0; 3); rpv ll : O r : -3, otrxe, D(-3; 0), ro,qi CO : 3, DO : 3. 11 S,oo= rDO.CO. S^,"" . ;'3 3 4.5 (^".oa.) Bi}noei1t: x ll : -3; 4,5 ",. og. 43. Croponn LABC: AB = JC 4r' - (4 - rf = v6+ 36 = "4f = sr6; AC = ^tE + q" + @ -rf = !66 + e = J+s - z'ls; BC=Jt6-3i +( ,-1)'=Jsl*s=Jeo 3J10. AB : AC, roti LABC - pinxo6e4peHufi. Me4iana, trpoBeAena Ao ocgoBr BC' 6y'4e (a .z -ze t i srcoroio. Hexail cepe4nHa BC - toqxa ,, Ol.t, , Z l. -r{(1,5; b,5), rogi nzcora aa = 1fu - r5y * 1+ - O*ly = 'lt+s. ^ 1-^ -,"=;.sJlo.Jrar=4f ,-""o.,Swc - 2BC All , S*, - t 3Vt0 Vl4.5 - 2- Bibnoeiu: "=t.y *.o. 44. }toop4uHaru cepeahHn.qiarosarir", l+n,+) ' (6:4)' KoopquHarh cepe.{Iu- uu aiarona,ci "o. (ry ru ;'), (6: 4). orxe, aiu"o*uri qorr4phKy.rHItK a ABCD treperr.rnarorbc;r i roqxoro treperuuy 4i:rarsca uanni.n, a orxce' ABCD - rapareJro- rpam.AB={tt-4)'+(1 -6y =,tia, ao=v!t-8,t;G2t =,E6; p=4'126 ot' AB : AD, roti ABCD - Pou6. 3xafi4euo 4iaronari: AC = ^1@ - s!" 41 *y = Js6 * 36 = o"D, BD = Je - 8)" + (6 - 2f = 4Ji. s*"o =! :ic' no, s^"." = 1 alz.ali 24 t^, ot. 2 Bi/noeiu: 4J26 oa.; 24 *". or,. 45. a) Touxu, cnuerpuvni siaHocHo noqarKy noopAhHar' Malorb trporuJrexui xoopau- narz, rouyA'( 3; 5); 6) roqrcz, cumerplruHi ni4nocno oci a6crlui, Marorb nporzJ'rexui op4unatu, ornte, t,(B; S); e) rouu.r, cuuerpnuni uigHocHo oci op4uuar, Marorb npor[Jrerxni a6crlNcu, orxce, A'(-3; -5' Bi1noai1a: a) ( 3; 5); 6) (3; 5); e) ( 3; -5). a7o lilr reouerplg Eess F. rl. ra in. rlpo$inaxnfi piBeHb tlitii{lll::fil*:dl 47. AB(3 -I; 2 - 6), AB(2; - 4); CD(2- 0; - 5 + l), C ru eexropia pisHi. ro AS Ci. 4a. 2d.(2;6), 3i(,6; o), FQ+ 6; 6 - 0), F(s; o); lFl = Bidnoeidu: 10. 49. Bercropn mo,rineapHi, nxuo { = ! , ,*':8. x'- Bidnoaida:x-=!2. 4 2x 5O. Bexropn nepreuArrr(yrrfipui, axrqo 2. x * x (r + x2. + 5x :.0; x{x + 5) : 0, eni4Hrz r : 0 a6o x : -5 Bi1noai}u: r : 0 a6o x : -5. 51. PoausHeNro LMAN i LMBN: MA: MB, AN : B MN - criit:,;.a, orx<e, LMAN : LMBN (sa rprona Hauu), ro4i /AMN: IBMN.. . Posrrr,rueuo LMAB (AM : MB): MO - 6tcexrpuc O -. ' rouxa nepermny AB i MN. y piuno6e4peHoro r HnHa 6icercrpuca, trpoBeAeHa Ao ocHoBn, e Bvcororo MO L AB, MN L AB.IIdo it rpe6a 6yno 4onectz. 52" Hexafty LABC AB: BC, AI< - Bucora, Al{ L BC AC:30 cv, BK : KC:7 : 18, roAi BK - 7x, KC - 1,8x,.4e .r - rcoe(birlienr nponopqifinocti. 3 LABK (ZAKB : 9O") AK2 : AB2 - BK2, AB : 25 AK2 : 625x2 - 49x21 AK2 : 576x2: AK : 24x. i:if:*ii:r..: ]i:.!:.i:.j. 3 LACK (IAKC :9A")ACz: AK2 + KC2,9OO : 57 900r'z :. 9OO, x' : 1,, r - l. Togi BK : 7 clr, KC : 1"8 c*r, AK : 24 cv. s*" =|xc . A.r<, swc =* t" 24 - 276 (cur2); s*, =f,ax.Ar{, s*;-=} z z+= 84 (cu'z). Bidnoei)o: 84 cut2i 216 cnr2, 53. Hexaii y LABC BH L AC, AM : CM, AC : 34 c*r, BH : 24 cm, BM : 25 clr. Posr.r,sneuo LHBM (IBHM :9O"): HM =.t85' 24' = JAs - r4r(rb - 14) = ? (cm); AM = MC - 1.tn = 17 (cm). 2 AII:AM HM,AII:17 - 7:10 (cu); HC: MC 3 LABH (ZAHB:90"): .4-B - Jr4" -iro'= 26 (cm s LBHC (IBHC :90"): BC = .,t876 + 516 = 24nE @ P*" : AB + BC + AC, Pffic - 26 + 24Ji + 34 = 60 54. Hexafi y rpaneqii,4BCD BC : 6 cw i AD : 18 cl O - roqxa nepear,ruy 4iaronaaeft. Pogr.ngseuo LBOC i LDOA: IBOC : ZAOD, ZCBO : IODA, BOOCBC6IToTiABOCNLDOA.::=- ODAOAD183 iitll:iillilllii:i:rt rEoMErPIe 6ees F. fl. ra in. Ilp
  • 9. Or.nre, 4iaroua;ri tpaneqii nepernrrarorbcg i roqrcoro [eperr,rHy gi.nRrr,cR y ei4no- urenni 1 : 3. Bi0noai}u: L : 3. 55. .{onxcrzna xoJraC : znR. ,{on:rcana a6i.nr,urenoro xona C' : L,2 . 2nR : 2 (L,2R)n. Cropona enracaHoro B rcoro npaBuJrbHoro rplnyrHurca a = RJ}, nJrorqa rpranyrnlrxa ^ o'Ji ,R'8'6S=-':'- =-' :'-. CroponarpaBr,rJrr,Hororpr{Kyrnrrna,Brrucanoroea6iruuene 44 xo.no, a' = r,zR,h, rrrorrla rprr*yrnraxa s'- (r'znv5)" JE =t-148-bF44 L,A4R'z.3J3 , n = = 1,44, sa 44 o/o. R' .3J3 v LABH (1AILB: go") BH =,[+s." -16.'= "J43 Tori sinz =BH "inza-'n6s =@-. ou"--AB'7xL loo sin zABC = sin(180" - ZAI = sin z2! = !2! . ' t;; Orrxe, sin 1A = sin lB = sin ZC = sin ZD = vl" . 7 ,C; 11;7rn6i6n' 1!9. ( 59.' SacroconyeMo reopeMy xocnuycir go LABC: a) 3a nigorrurrz croponarvru. 3 : 5 cvr, BC: 8 crvr i g.er,ao AC. AC2 : AB2 + BC'z - 2AB . BC ' cos lB, A AC' :89 - 40: 49, AC : 7 cltt. 6) 3a ei4olrarzur,r cropoHarrv,AS:6 cu, AC:4cttr _s's'3nafi,qervro nianouennR - : - = ssBi0noei0a: sa 44 oh. 56. Hexafi y MBC (tC : 9O" ZB : 30', AB : L2 cv, B - qenrp xora s pa{iycou fi i n.noula cerropa S"""*. s qerrrpalrbrr,rM nytorvr 30" i pagiycorvr R topinriroe ]S*".3 LABC: BC : AB cos ZB, t: BC = 12cos30" = 12 V: = 6Jd (cu). 2 1 S.^"" =:AB.BCsinlB,2 1- s.,"" = + 12 6Ja . sin Bo" = tgl6 (cM,). 2 =Ir*"= .1. rsJS; @ = sJzt R, -t2'9Js . A AC2 : AB2 + BC2 - 2AB . BC . cos lBi 16 = 36 + B ""' -'*:" + 2o = O; 3Bc2 - 28BC + 60 : 0. Hexair BC- r, roAi 3x2 - 28x+ 60 : 0, *r,, =4 ,, =+= a|, x, = o. Orrxe" BC = 3* c* a6o BC : 6 cu. F; I; Bidnosida: 6^i ovo. i n 57. Hexaft AB - epicr 6arrra, ArB, - apicr 4ou*u, AC - Aonxczna r:iui siA 6arr- xa, ArC, - AoBrrirrHa riHi ei,q AoHbKrr, AC : 3 u, B N ArC: 2,5 rvt, AB : 1,83 ru. ZABC : lAtBlCl, ^ B.r ocxi.nrxn aonr,r 4opinnrororb Ryry HaxrrJry uportenia I |t consiZACB:ZAtCrBt:9O", I I '.oaiMBCNaAlB,cl ,#=#, L .[ACAtCI aei.qxr ,4"8, = *',!"' , 'AC 4p. - L'83| z's = o,6r.z,b = t,b25 (u) : 152,b cm.t Bi1noaiSa: 152,5 cu. 58. Ilexai yrpaneqii.4-BCDAB: CD; BC:AD:3:11. flpone,qeruo BH LAD. e) 3a yuosoro-4C - AB:6 cu, rogi AC: AB * 6 3nafiAeuo,4C ! AB. Ba reopeMoro xocunycin AC2 : (AB + 6)2 : AB2 + 64 - 2AB. 8' cos 120"; , AB2 + I2AB + 36: AB2 + 64 + 8AB; 4AB :28; A coaiAC:7+6:13(cu). r) l{yr /A: 60o rer*urr rportr croponrr BC: 14 Toti BCz : AB2 + ACz - 2AB .AC cos 1A; 196 ro1i,432 - 648 - 160 - 0; AB : 16 c*r a6o AB: - BiDnoaila: a) 7 c*r; 6) 3l con, 6 cnr; a) 13 cu; r) 1 J 6O. Cropucraennocr Sopuy;rarvrn me4ianu, 6icenrpuc fioro croponn a:30 cu, b: 11 cm, c:23 cttr. Megiaua, npoBeAeHa Ao croponr4 ai m' =lQ@' * 4' ^1 =7 1z1tzt+b2g)-900) = 1(ts00 -900) = too; " 4' 4' 6icextprzca, rpoBeAerla Ao cropoHrr a: l! = "*L- @+cr ,, 11.23((11-23)'?-900) 253 256. , _16 F'" (tL + 23)' 34' "I 4 .. Blrcora; npoBeAeua Ao cropour.r a: hi -- ,- p1p - a)( -a' nRz.30" R = o,/gre. l n S""". = 360' :.iil'i:-l:':. ,l+il.iili.i ,flrcrqo r - rcoeipir{enr nponopqifinocri, ro BC - 3x, AD - tLx. Ocrcl.nsr<n s rpanerliro Molr{Ha Bnr,rcaru noJro, to -43 * CD : BC * -4n, orrxe, AB + CD : 14x, AB : 7x. du=M-BC, Arr=tt'."*=n*. A 22 D 872 lii: TEOMETPIS Eees r. n. ra ix. IlpoQinuxur piBeHb *liil{il:llfiilii{l:l i:i:i:ii:i:1:i:i:i;i:i:ii:ii::: FEOMETPI9 Eees f. Il. ra in. Ilp
  • 10. p - 7 + 23 + 30 = z2; hl = ft ' eua+- 80x64 - tL)ga - %); nl 2, .8, .82^.b5. 4I . h = 9 JOS5.J6 "*.go' 15 '"":"" ""' 61. Hexafi y LABC AB : BC : 6 cm, sin lA: O,4; BH -rrany me4ian, M € BH (BH - nucora i ue4iaHa pinHo- 6e4peuoro LABC). 3a nac,ri4xorr 3 reopeMr4 cnHycia ++ = 2R' ge R paaiyc onncaHoro Rora: sin 14 ,R- 6 =7.5 tcnrl. 0,8 R > BC, rogi qeHrp O onzcaHoro nora .nerrutr sonsi rprrnyrHrrr{a, BO : R : 7,5 cm. Is M,BH (IAHB : 9O") BH ="tZn, - da,, BH -- fibL y = 12 (cu). orxe, pa.qiyc rBlr{caHoro Hora: r -lur, r- 6 2 Ocuil.sxu AII : 9 cIra, ro DH : AD - AE : 24 - I Is LBHD (IBHD: 90'): BD = Jt5' + 72'z = 3'14L noro rpatreqii ABCD, e.pa4iycov ltoJra, otrucaHo o _ AB.BD. AD . 45 no,o lt S*,,= Z AD BH. S*o--:.24 1,2 144 (cu'z). Lb.sJn 24 roai R, Lo'oa!-2 =7,bJ4l (cM). Bidnoaidu: 6 cu; 7,5J41 cv. 65. Ocrcimxz AB : L0 crn, AI( : 2 cw, ro BK : 8 cn ocxi,rrrcu BC : 14 cut, TC :9 cu, ro BT : 5 c*r. s^"".=1BK.BT.sinu,B, 2 1 S.,". . ^ AB.BC.sin/8, 2 za-l ro .L4sinlB, sir*e. l.25 1' roai S,"". : 8 5 :=8 (cu'?). t4 Or:xe, S*r" : Sooc - Soorp S*"" : 20 cm2. Bi}noai]u:2O cv2. 66. V rpaneqil ABCD AB : CD, BC : ll cm, AD : ' BHLAD. att=M.8,9, Alt=21 =11 =s ("* 2' 2 Is IMBH (IAIIB:90') uaemo: AB: 13 cv, AE: BH : 12 ctut.Is LBHD (ZBHD :9O': BO = nEn' * ocnirrxra HD : AD AjI, HD: 16 cu, ro BD = a) Pagiyc -B Hora, ouucaxoro HaBRoro rpaneqii' A onucauoro HaBRoro LAfD (ABCD - piano6e4peua ro.ai R. M- 82-!2. s.^.^ - I AD BH, s^^- ASno"o 2 *" roAi R = l3 20^27 = lo! 1cu.1.'' 4.126 6 Pagiyc r RoJra, Bnr4caHoro B TplnylHlrH ai cropouamu - S a-b+caoo r=i, o" p 2-. 6) V MBC: AB : 13 cn, BC : 11 cu, AC : BD : s*"=.[p@ o)(p' b)(p-, Ae p=t]fff'=22 ro4i S-r" = Jz'L 9. 11. 2 - 7I. 2.3 = 66 (cu'?) i Btrcora,M-rovxanepe- B BH : AB - sinlA, BH : 6 - O,4 : 2,4 (cm). Megiaun rprnyrnr4na nepernHarorb- cs i roqmoo treperlTuy 4i.nnrr,ca y nignorueuHi 2 : 1, paxyroun sia sepuunrz, ro.ai BM ?"n, nl'f =?.r.n= 1.ti (cn). BiacraHs ai4 rovrr neperuHy 4iarona,reir td Ao rleHrpa onrlcaHoro xona MO : OB - BM, MO : 7,5 - 1,6 : 5,9 (clt). Bi}noeiOa: 5,9 cw. 62. V LABC AB : BC, AL - 6icexrpxca, ZA: 2a, ror.'ty ZBAL: ICAL: u, lC :2a. y MBC tB : 780' - 2ZA: t80" 4a. ra LABL Ba reopeMoro czuycie - P+ - = ,N:=," sln ,/RAL sin ,/R' a sin(180" - 4a) _ a sin 4q S]N O( SIN C( Ocniarxil rpr4rcyrnr.rn pisHo6eApeHuii; ro 6icenrpuca nyra C ranox =*# d sin 40 stn c( TeLALC AL - LC . rc= ;no slno-asin4cx =2acos2a. sinlC slnlCAL - sin2o sin2cr To4i BC : a i 2a cos 2s : a(I + 2 cos 2a) i A-B : o.(1 + 2 cos 2c). t,t LABC AC = "c o" sin -B sin -A iel.l.l;!i.l a(1 + 2 cos 2o) Sin 4c = 2a cos2<t(l + 2cos2a). sin 2s Bicerrpucoro nyra B e Brzcora, fipoBe.qena Ao ocuoBu, BH LAC. ls LA-BH BH : AB sin1A, BH : a(! + 2 cos 2oc) sin 2o. Bi)noai4u: cropoHr: 2a cos 2a(7 * 2 cos 2a), a(L t 2 cos 2a), a(7 + 2 cos 2a); 6i- a sin 4C s si;, let ceHTpEcz: s1n cx srn (I 64. Hexafi y aa4anoi rpaneqii AtsCD BC : 6 cu, AD : 24 cm. Mo)fiHa Brncarr,r r{oJlo, ro AB 'l CD : BC },4n : 30 cm. Ocrliarrcu uaeroro rpaneqii uoxrrra onlrcarr.r noJro, ro Aarra tpaneqis pinxo6e4peua, AB : CD: 15 cm. Bucora rpa- neuii AopisHroe Aialrerpy Bnncanoro e sei xora. A flpone4<fuo eucory BH, BH IAD. AH . AD' BC . AH '24-6 =g t".*t. 22 OcHilr,rcrz B Tpaneqrlo B roai ia LABH (1AIIB : 90"): 974 ii:i: reouerprs EeBs r. n. ra iu. npoSinsnufi piBeHb liiilijiililllilt'.:iii: i:ilitiil{,:*:l:l:* TEOMETPI9 Eees f. Il. ra in. l-l
  • 11. B) yAAcr: cD:LBcl*,AC:20cm,AD:2:_ew,togi p= 1'3+29+2L =27 (cv, S*,,=,lir.A 7-=3.7.6-126 {cn2); r =!#-+f,rca. Bidnoaidu: ar I0l cu;6) 3 crvr: nr +f crr,. 68. Hexafi cni.nrna xop,qa.4B HiJr ia eiAnoni.qEo qenrpoM Or i pagiycou r ra qenrpoM To.4i Aiarrnerp naft6i.nrtroro Hpyra D : d', I d'" t t, c -n(d!+ d)2 -( y: -4J = I td? +2d.d^+a! -- 4 4 4) 4-L --'-z LADC - npauoxyrnufi ia npamum xyrou ,4, *cr AH - rln.cora npfiMonyrHoro rprrnyrHrrrca, onytq :HD cH, AH=#, ;ur=t, li)'=a,a". Orxe, nnoula S BanrplrxoaaHoi (pirypr,r: " =; l;' 3r6ro"r6o,t 4. 8 73. ItoopAlrqari(x;ilrorrrcu,r{oecepeArrHobni4pie o6.rrzc.nroemo Ba SopMy,'iaMfl : ^- xrtt, _._attUz 2"2 Ocxi,'rrxu I((1; -1) - cepeAr,rua ni4piaxa BC, p,e B(-2;1), ro 1 =-3:t' aai4xr'r x': 4; -1=+, sni4xn y : lr, uc(4; s). .L - cepe4rzna AC,ro4i t=-3ia =0,s, ,/ = -!-3 = -2. o.n.e, L(o,6; -z). "2 O, i pa4iycolrr R; ZBOA:124", ZBOA:60"' Togi AABOT, ge BOr: AOr: p, pienoctoponuiii, ocxi.nrxu IBOA: ZBAO2 : /ABO, : 60'. Y LO'BA: BO1: AO1: r, BA: R, IBOA:72O", rogi IO,BA: IO.AB : 30". Sareopervrorocu'ycie . * = o'B sin ZBO.A sin IO.AB' RTRTRG "*120"=sin3o" =T' ;=uo' 22 2 Bilnaeila: J3 1. 69, 3a uar. 33 niApy.rnura rcBaApar yntrcaHo B IcoJIo' rra cropoHax r{BaApara s}c Ha 4iauerpax no6ygorani nirxora. Hexafi cropona HBa,qpara.r, ro4i fioro 4iaroua.nr', sxa e AiarrerpoM orrilcauoro KoJra, - "Ji, Tlnot4u KraApara S*". : t2, trJlorqa orrrr- nr12 nx' 2 fix' canoroKpyra- S*0.= ti , ".".=T =f. Cvuan,rol1cerueurie,,sxieir- rrrHarorr,c,s niA rcpyra cropoHaMrr KBa,4para, S*",. : S*. - s.", S""." = :L - x2. Cyua n,roq nicnpyriB, no6y,4oaauux Ha cropouax xea,4para, Aopinnroe 25 = 2 4 = ilx' ^ nx' lllx2 ,l rtx' Tx' =:. Cyuan,'rorqcepnuxia S.*"' 2 -l Z -*')= 2 - 2 n*'. Orxe, S""n". : S*,. : 12, nrorqa cepnuxir 4opienrce n.nouli xra4pata, llo ii rpe6a 5y,ro 4onecru. 70. @irypa, Bo6paxceHa Ha MaJr. 34, cx,ra4aerr,cs B np,ffMoxyrxur{a 3 poauipauu a i 2a ltrJrorqerc 2a2 ra ninxpyra s pa4iycom a i n.norqero S. nnoulu 6iryp _-.2 Sa = 2a2 + "* . IIo6 orpuMarl4 Barflrpr.rxoBaHy Sirypy, rpe6a nupiaarr ia aaaairoi I _-2 Ta' girypr 2 unepri Hpyna a pagiyconz i nrorqero 2.i-xa'= -11. Ilton{a Barrrrpr'l- " fia' na' xoeaHoitPirypu S=2a"+ : - = =2a'. Bi}noei1o:2a2. 22 71, 3a ivra.r, 35 ni4pyuurxa MaeMo: n,'roqa S aanrtpu- xoeaRor ipirypz gopinroe pianuqi n;rorqi Sr. xpyra s safi6immiaM giamerpou ra cyMt4 Sz + SB 4rox in- urux xpyrie: S: Sr - (S1 + Sr). Xopga.4ll, ilIo rpo- xoAr4Tr, qepes roury 11 AoTr{}ty ABox meuuux rpyrir, gopinnroe a i nepnengnryJrspna 4ialvrerpy CD safi- 6i.nrnoro xpyra. Hexait AiaMerp oAnoro B Merlurux -)2 -/2 xovrin d, i S. - 11. a rDvroro - d" i s" = 1:? . 4''4 Bi}noei.Oa: (O,5; -2). 77. [,tts xea4para ABCD ni4pieon AC - 4iaronaar'. H ronareft. ocri.nr xuA(2;*5),C(2:-tl,ro xl?!3t =#K(2 ; -B).,{i a rona,ri KBa,iIpara BaacMHo nepneu4nry i 4i;rerrca ror{Koro K napni.n, AoBrnr,rua no.nonzura na.ni 4opienroe 2. ToAi xoop,4uuarr{ Beprflugr. B(0 Beprrrlrrt4 D(4; -3. V rra4pari Torr(a treperllny 4ia .reii e qeurpou Btrucauoro ft onucanoro ri.n. Ilenrp caHoro norra K(2; -3, pa,4iyc fi : 2, ro/-i pinnarna caHoro Korra (x - 2)'+ (u + 3)':4. Pagiyc nnrcanoro r{oJra gopinnroe no.nonragi cropoH nBaApara /3 = V'(2 0f + (-5 + sf = 2ii, rofi p otr*re, pieHatn.fi Blrrcanoro xo.na (* 2)" + (a + 3 Bilnoailu: or{cane xo.no: (.r - 2)'. + (a + 8)'z : 4; Brrr4caue Ho;ro: (.r - 2)" + (U + 3)2 : 2. 78, Pinn,nnnsogHororcoJra: tc2 +A2 - 4x+29:O;x2 - (x - 2' + (y + l)': 5, fioro qeHrp (2; -1). Pinnnnua4pyroroKorra: 12 +y2 + 4x*6y:3; x2 *4 (r + 2)2 + (A - 3": 16, ftoro qenrp (-2; 3). Ilp.fiMa,.ffKarpoxoAr4rr,arepe3rourv (2;-1)ta(-2;3),u x-2 u+I x-2 u-l ^orfir€,4-t ., - --! 'i2 x-q+l:x-"'--'-2-2 3+1' -4 4 BiOnoai0u:)c+V:L. [i{:il:i{r,{ii{iii;$i: TEOMETPI9 Eees l. fl. ra ix. Ilp .,;rt.!.1:._' az6 li:;: rEoMETPlg Eess F. t-1. ra ix. IlpoSinrnut piBeHb lJJilt::ltlit:t:l:ift
  • 12. 87. J{nrqo m = d + 26, ro lfil= Jrn' = J@. r6f ='td 4d6 + 46' = = 'rE + 4di | + r" - .tTa-b - 8. Crarapnzfi 4o6yror a5 -lal.l6lcos4a,6l, au - 2 1 cos60" = 1. roni lrzrl =J+*e-zG. Bidnoaia: 2J3. 88. Poeussevo s LABC reHropu -4-B i Au. Ac =lAul lAcl.cos.1A, AB. AC zsitltu cos/A I'qnl lac AB(4; 3), AC(3; -3), to4i AB. AC = 4'3 +3'(-3) = 3; @E1= "@ +s' = s, Acl = vEL (-sf = SuE. orxe. "o" zt = --j-^ :?5 3J2 10 -t; Bidnoeida: Y. 10 89. flxqo xyr ryuufi, ro siu Harexzrr II roopgrnarniit usepri, a Kocuuycu'ltyriB Ilvsepri ua6ysarorr,ei4'emHux3HaqeHb. "o"o-,,2 b,'. ,n t0, lb >0: cos o o, nxulo m .b < o: lml bl nL.b =6.(-5)+a(a 1)- -30 +a'-a, a' a - 30 < 0, nopeni rpuureHa ar: 6, u^ - -5-'l Ilpouinrnn straxocra.nocri: ---1;--'t--i a e ( 5;6). ,flxrqo a e (-5;6), ro cr tynufi Hyr. . Bidnoaidu: ae( 5;6). 9O. 3a ymonoro BeKroph 5(i - 4b i d - 2b nepneHgnxylapni. a orxie, ix ctcaaapuuil go6yrox4opinHroeHyrro, a caMe: (54= -4txo 2tt1-O, 5a'-nab;4oi +862 =O; ypaxoByroqu, ,qo ]al = 1, d2 =!, Ul=t, U" = 1, uaemo: lkdb =13, aa :11. g*rqo 13 14 xyr mirr a i b -o, ro coscx o, f-, "o"o = .'o- -!t.lal lal 1 .1 L4 Bidnoeidu: E, 14 93. Hexafi HoopAuHartr BepurHu O Aiarparrarz BopoHoro O(x; U, rovrca O, - roqna nepernuy pe6pa OM a AB, Ot(2; 4); M rouHa O, roqKa neperrrny pe6pa OK a AC, Or(0; 0); rovHa O, roqxa neperrHy pe6pa ON s BC, Or(3; 1). 3Hafiaemo KoopAilHarh nepneHAnHyJrffpHlrx sercropie: I ) OQtx - 2: s - 41. enrc: zl: Oq an - O. 6(1 2)+2(y-4):O;3x 6+y 4:0;3x+ g :10; 2) oo,(x - o; a - o), oS@: y1: cd(-z: q: oq cn - o, -2x + 69: O; x + 3y: O; S) OO"(x-3;a-1); C,e(+; S): OA" CB=0,4(x-3)+8(y 1):0; ')c-3+2a-2:O;x+2y:5. 878 :::;i: FEOMETPIS Eees F. Il. ra ix. l-lpoQinuxufi piBeHb :iiiiiiiiiiiiiii::ii:i;i:i Poas'srxeivro,czcreuy pinnaHr,: lxt2s=5, )x+2y=5. lx+2u--5, lx=3. I xtss=0: l5y=5: Ia L [y=1. Itoopgunarz rouxu O(3; 1) - rouxa Or. Piessxss oM a6ooo,, * -3 =g -I. * 3 .a - ' 2-3 4 l 3 3r-9- A-1:3x g-8. --3_g-l. x-3_APisHsHHsOKa6oOO,:l x_}u_O. '0 3 o-I' -3 -l SHaiisenrro axy-ue6y4s rouny X na ON: X(m; a), ro +(z +3)2, XBI:(m 5)2 +(n-5)2,ornce,(m 7) nlz 2m i I + n2 + 6n * 9: m2 -lom + 25* Hanpurc;ra4, m : I, n : 2 i X(1; 2 uatetxurt ON ro4ipinnrnna oN a6oox: # =T: *-,-L- t t,,:a -v Bidnoei}a: (3; 1); 3r - a : 8t x - 3a : O; x + 2y : Po3.qln il. BCTvn AO CTEPE AC: ix cxa,rnpHrfi ao6ytox A(-L;2 ,:{i.Li:.i B(5;5 1 14. a) 1) AD; 2) AA,; 3) CC,; 6) 1)A e (A4B), A e (A.BC), A e (AA'D,); 2) C, e (CrCD), C, e (BtCtDt), C, e (C'B'B); 3) D e (ADC), D e (DD,C,), D e (ADD). n) 1) B, e (ABC); 2 B, e (BB'C'); 3) B' e (A$tC) 115. a)AB; 6) B e (ADC); e) AC c (ADC, AC c (AB 117. 1 18. .t'/ /t"/ ryp/ l2O.,flHuro4sinrorqununapalersHimimco6oro,ToBo qacrrrru. .fnrqo 4ei uoilIfiHtr treper[Halorscs, To Tlrpn llacTuHv. l{i:ifii;tidiiii.;i TEOMETPI9 Eees F. Il. ra ix. rlp B{3;5) S 3. OcHoeHi noHcrre crepeoMerptt- OcHossi $irypra crepeomerpii: rovra, fipsMa, rlJaonl fIoanaqeHH,f : Toqxui A, B, C, ... flprmi: -4-B, AC, BC, ... a6o a, b, c,... flnorquHu; (ABC), ... a6o a, B, y, ... flpnua.n6rxnicrb roqru 4anifi upamifi: A e a. Heupuuarer+cnicrr, roqxu ganifi npauifi: B e b. flpuna,rerxnicrs npruoi 4axiii rronluni: a c a. Heuprzna.ne*cnicrr npauoi ganiit n.norqzui: b cl 9. fleperzH npsMux a i b s ro-rqi C: a n b : C. Ileperuu npanroi a i n,ronlaHu o e rovqiA: a ^ cx Ileperuu trrorqr.ru n i B no npsuifi c: u n B : c. o(x;y)
  • 13. 125. flxuope6pory6a a,V:43:8 cm3, rogia:2 clrn' S""".:642, rogi S : 6 . 4: 24 (cv2), Bi)noai)b: 24 cM2. $ 4. Axciorvrn crepeoMerpiiixacni4Ktt s xtax Cracrelira arciom crepeouerpii cx.na4aerbc,s a axciolt n,raniuerpii ft aHciouu npz- naaerxHocri roqor i rrJlou{vH, axcioun neperrHy nJrouII4H ra axciovn nr'loruuHu, .fiHa rrpoxo4r4Tb vepea 4ni np,sui, uo neper;4rarorbcfl. Ancioua [pfirtaJreilcuocri ro'rorc i nnouluu. ,flxa 6 ne 6y.na n.norqana, icuyrom Toqnu, rqo HaJrer(arb qifi n.norquui, i To'rnrl, rqo ifi ne HaJrerfiarb. Ancioua neper{Hy nnorqun.,flxr4o 4xi pisni nJronlunri :uarorr, cni.rrruy roqny' ro Borrn neperr.rHarorr'cs IIo npnuifi, nlo [poxoAr,rrb qepeS ilro rotlKy, Arcciorra lJrorquur{, sfta npoxo4urL uepea,qni npgrui, rqo neperrrfiarorbca. ,frqo 4ni piani uparvri marorr' cni.nrry roqrty, To qepeg Er,rx Mo?r(Ha npoBecr[ nJIorqflsy' i 4o roro x ri.ntxa o4ny. 3a nac.ni4xallru a a(cio1u crepeonrerpii, e4vuu:ol rIrrHoM npoBecrr{ nJlorqr'ruy MolfiHa: l)uepea npauy i rorrxy, rqo rre Jrexcllrb na qifi npgmift; 2).repes rpr.r roqxr,r, qo He Jrexiarr' Ha o4nifi npnmifi. ff&qo,qei pieHi rouxu nparvroi naaexarr n.nouluai, To np,rMa' rqo flpoxoAr{Tb qe- pea qi rourra, Jrelfirrrb y 4ifi n.norqzni, Ilaorqltra poa6uaae npocrip na 4aa ninnpocroplr. ,flxrqo roqrcrz A i B na.lexarr oAHoMy ninnpocropy, ro siApisox AB rre fleperrrHae nJrolqrlHy. flxqo rouxu A i B Harer{darb pieuuu ninnpocropaM, ro nigpieon -4,8 treperrrHae nrolqr'ruy. ,{.nr 6ygr-rxux rouox A, B i C npoctopy Br'rnoHyerbcr nepinuicrr rprrrcyrnrrna: AC<AB+BC. 1 42. rlepea Asi Aani piani ro.rrtr Mo?rcHa [poBecrrz np€My i rinrxra o4iry. rlepee npauy Mo)rcua npoBecrx 6eaJri.r nJIotrIuH. 144, flpwrrycudMo, rqo rosxa M - cepeAr,rna siApisra.A.B -. ila.nerxlrrr c. Ocxi.nrxr'r A e a, M e a, ro npav'a AM c o aa axciorvroro Cr, a orrxe, aigpieorc -43' qo Jretfirrrl na npalrifi AM, new-:rrrr y flrorqaui cr,, roqrca B e cL Iqo ue ei4noni4ae yrtori ea4a- .ri.Orxe,Mes, Bi 0n oei0a: He HaJren(ilrb. 145. Ocrinrxu B e a, Toqepea 4 i B MorrcHa npoBecrr4 rlJlorqrrHy o, i rilSxrl ogxy (ea nacnigxou a anciorvr crepeonrerpii). Orxce, B e a., a r_o. ToAi Bci npaui, r4o npoxo- AfiTb rrepea rouxy B i neperunarorr trpflMy 4, Marorr' 3 nJrolquHoro oc no 2 cni.nrnr'rx rouxra (r. B i rovHy npnuoiA), rogi aa arcciouoro C, rcorxna s Irr{x fip,ffMllx Jrexcr4rb y urorquni cr. IIdo fi rpe6a 6y.no AoBecrrr. 146- Hexaii y rpaneqiiABCD rouxnA, B iO - rotrfta rtepertrHy 4iaroHa-lefi Harexcarb nlonlr.rni cx. ToAi aa axciorrolo C, npnui AO i BO Jre]$arb y n.riorqnni n. Ocxi.nr'xra ro'rxa C e AO, ro C e cr; ocxirsxu rorrxa D e BO, ro D e o. 148, Ae a, BM c cx, To.rxr.rA iM sanetxarru,roniAM ca i ro.rxa C e AM, C e c, To.rxu Ai B na.rexatb o, roAi AB ca. ToqrcaNe ,4-8, roaiNe cr. Toqr<rz CiN sare- rrcarb 0(, orraie, Ba axciouoro C, aucora CN na.nerxurr cr. 149. Hexaft tpu rovxra A, B i C ,rexarr y r;rotquni o. I]i rou- Krl e BeprnrruaMu LA-BC. Ba axciouoro C"AB c a, BC c cx, AC c c. Ornte, no:nua ro'Jrca A.ABC Jre)fitrrb y n.noiqnui a. 150. Hi, He Br{rrJrrrBae. HaflpuxraA, rprr roqxr cropoHr{ AB rprzxyruura ABC uate- ?*carb,nrorquHi c, a rourca c e o, 880 {::i: rEoMETPlg Eees F. Il. ra ix. IlpoQinuxufi piBeHb;i{iiia:::ii::*!li;iii 1 5 1 . 3a uac.niqxopr a axciou crepeome'lpii, vepec 4ni np Ha rrpoBecrr?r ilJrorqrrHy i rr.rme oAHy. Hexaft MA MB c a, By4r,-srci npcmi, rqo neperr,ruarorr MA i MB Epoxo,4fiTb qepea Toqry M, lratorr B nJrotqt4lrorc c 2 Torrnrr, a orrrce, Ba axcioMoro Cr, aerxarr y n.norqr.tri o ft rpeda 6yro 4onecra. r{epee rouxy M morxna flpoBecrr,r np,fiMy, sra r{ nit4Tb y n;rorluni o, a neperaxae ii lurrre n o4nifi M (nanpr.rrrag, upaua c). 152. flparri-4Il iAK neperrnarorbcff, orsr{e, qepeo Hr{x fiKa neperrnaerbcfi B nJrorqlrHoro cr rro npsuift BC. 3a orxe, P e B. Tourca K e AC,K e B, To,qi npnua KP .I(, ro6ro Mae B nJrorr{rruoro B .nrrre o4uy cni.nrny ror ,43. Orxe, npaua KP He rreperr{Haerrca a nparvroo /- 154. Ilpana BK neperrnae rrrrorqfifiy (AAp) n rovqi K urorqrzHy (ABC) n ro.rqi B. Ilpaua BK fiepervnae nJr (ArBP). Ilpavra CK reperurrae nrrorquny (A4B) n rovqi,K, nrrorqr,rHy (,48C) n touqi C, n,roqrany (BCC n rovqi C, Ilpama CK uepeurnae rrrotqrrHy (ArBrCr). 155. Ocri.urxu npnmi a, b i c.nencam y n.norqzni oc, a r rpsMr4x,rode o, Beo',C e cr.SayMoBoro/.e B Torrnlr ABox rrJrorr{aH Jrerr(arb ua npatrift neperr4Hy q B i C.nqnrarr na ognifi npanift. IIfo fi rpe6a 6yxo 4o 156. Ocxi,rsxu nJrorqtur4 cr i B neperrzrarorbcff rro npau yci cni;rrui Torrxrr rrJrotqlzu. Ilpaua D, arca .nexrarr y o cni,rrny roury K. To,qi K e a. ll{o ri rpe6a 6y.no ,4o 157. a) Ilparvra DrK.nexnrr y nnorquni (lArDr), nxa nepernrraerr,cfi B nrolp,rnoro (:4BC) no npanaift DA. To4i cniarna ro.nca npauoi.QrK i n.norqunn (,4-BC) Cl Jrex{rzrb na np,anniii DA. N - rovrca neperrruy np,fi- mux DrK i DA, a orlr{e, fipaMa DrK neperrar:ae nrrou{rny (ABC s ro.rqi N. 6) flparraa BK ,rerxam y nnorqlrHi (AArBr), at+a rreperzrraerbc;r B [ro]qr,(Horo (ALBP) no npavrifi BrAr. To4i cni;rrna rouxa nparroi BK i n.norqrzura (APP) rrelr(arb na nparrift BAr. M - roqr{a re- peruny npsuoi BK a n;rorqranob (A'B1Cr). 158. a) Ilpnni MN i ArBr re)r(arb n o4nift n.norqr,rni, ro,qi -O - ro.rxa ix neperrrny. 6) Il;rorquna {B$C) i rrlroqlrHa (A,B,C'), axa micrr,rrr npamy MN, neperffHaorbcs no npamift CrBr, Togi rovHa neperr4rry npfirroi MN i trJrorqraurr (B'BC) nexurr na npaivrift CrBr, rour.ca F - rorrna ix neperrany, e) CiuHa naorqr.rua neperlrxa€ ocnoayArBrCrD, no ni4pislcy M.|y', n.norqrany (BBrAr) no nparvrift ,EB, a rpaub BBTATA * no ni4pisny PB; n,rorqzny (BBP) * no nprnrifi BF, a rpanr BBTCTC - no N A :iiii::iiiiiiii:iiii rEoMETPlc 6ees r. It. ra ix. Ilpoe
  • 14. BiApigrcy -RB; rogi ryaus DDTATA - no ni4pisny MP; rpaxs CC.DtD - no ni4piexy N-R. fl'aruxytsux RNMPB - nepepia rcy6a nrorqzuofi (NMB)' lllrouqnua (BBp) treperrruae nparvry MN y roqrli K" Dl llepepia RNMPB i nlouqzua (BBp) rvrarorr 4ni cnimni N 174, Cissa nJrorqrrna Mae s trJrorJnHon (ADC) 2 cnilrs KC; a n.norqranoro (ADB) - roqxlz K i B, a orxe, npa B nroqzHaMr {ABC) i (BCD) - roqt<n C i B, a ornre BC. Ilepepiaou e LKBC. 175. flrcrqo civua naorqrzna [poxoAr4Tb qepea nprivry,43 t<u A i B Halexars ciuuifi nrorquHi. Ocxi,rrxu ciuHa Ha lpoxo.4r.rTb vepes rovru i4 i M, ro AM - cni,rrua civnor n.norqrzHz i nrorqunr.r (ADC'1, aua'lorivno, ciun rquna i nnorquna (BDC) lrarorr, cnimuy nparty BM [Jrorr]r4Ha neperuHae rrJrorr{r4 Hrr (A-BC) r (BAD no npam Ortxe, A,BMA - uryHanzfi nepepia. 176. 1 7 7 . a) Ciuna n roquHa neperrrnae rp ans D D rC rC no ni4p PT; rpanr BBTCTC no nigpiary -KP; naorqlrny (-A-BC) n r S, S e I{P, S e BC ra no upamifi S?. Sf treperrxae e rourli M. V ci.rnoi trrorluHu i rpani AATBTB 2 cni ro.rnrz Ki Mi ni4piaox KMi y ciuaoi n.norqraHt irp ABCD - ni4pisox MT. Orrlrc, myxauuii nepepia - q prxyrHrrft MKPT. .. 6) Hexafi aa4aua nipauigaABCD. CropoHz nepepiey -i ?I{. Ciqsa nrorrlnHa nepernHae pe6po CD n rovqi S r4nny (BCD) no nprmifi SP, arca nepertluae BC n N; rpanr ABC - ro nigpiany KN. To4i qorr.rpr.rxy KTPN - uyranufi nepepia. e)Dr) 180, Terpae4p ABCS -npanunr,nlrfi, ycipe6pasnoro4 peguHu pe6ep S.4, SC, SB, ro4i uyxanzvr nepepiaou e L Ocxi,rrnu rouHu K, M, N - cepeAr.rnu pe6ep, ro KN KM - cepepai .niHii ni4noni4no rpuxyrnrzxis AASC, AABS. ^ KM,Js li)' It o,JB 5rrr=-4 4 = 16. Binnoaint: +t + iil:ji;:li:ii{::r{::* TEOMETPIC Eees f. Il. ra in. IlpoQ M roqnv B i K, roai KB - nisis fleperuHy n,rorlux (MNB) C, i (BB1D1). r) flrorqrzna MND neperuuae rpaHtAlBrCrDr no nigpiany MN, rpaas AATDTD - no ni4piaHy MD, orxe, linia nepe- rtrHy nrrorrluH (MND) i (A D) - MD. 159, a)MN iAC .nerxarr y n.norquHi ADC i neperunarorlcR n rouqi K. 6) Ilpaua MP tetxvrr' y n;rorqrtiri (ADB)' axa neperr- Haerbcfi s trro[{r{Holo (ABC} to npsuift,4B, or]{ce' roqxa S neperuuy MP i nlot4unu (ABC) newurr na AB. n) flpaua NP ,rexurs y uorilnui (BDC' axa treperr- Haerbcs B rrJrotr{r.Inoro (ABC) no nprnniii BC, o'nxe, rouxa T neperrany NP i nlorqxHu (ABC) textwrr na BC. r) Il.norqzxn (MNP) i (ADB) narorr 2 cninrsi roqrtn M i P. rofi MP linia neperuHy rlrx n,'rotltrH. 16O, flxrqo MN ll AC, ro flpsMa MN i n,'roulzua (A-BC) He neperxHarorbcfl. '162. Hexafi aa4ani npami ai b, npava a Jlextzrb y uorquni 0(' trpfiMa b nepetnuae qro trJrorqrHy e rouqi A, A e a. Hepes trpfiMy 4 trpo- BeAeHo rrr'rou1hHy o, flKa fleperilHac npRvy e toutli A. gepee npsrry b trpoBeAeMo rrJroflIuHy 0, sna uepe- rnHae [psMy a e rouqi B. To4i nJrorrlrHrt cr i B ue- perzuarorbcfl rio npauifi AB. Toli n.lrotql,rntr c[ i p treperrrnarorbcs no npamift,4-8. To.{xaA € b, B e a, togi upama l-B treperuHae i a, i b. 166, Hexait y ny6a ABCDA.B.CrDl giaroHarr CrD rpani DDPTC gopinuoe: C1D : d, rogi pe6po xy6a cropona RBanDara DC = L. ,lz .l O6'ev xy6a V DCr, V -' d ;"0;,,"",;:' ft -l'Dr $ 5. flepepisn d" 2J2 clB Toni KM NM ' KM =f,,, ,"u, ".;=+, IIpu no6y4oni nepepisie Br{Rop14croByeMo naacrusicrr: flrrilIo AB1 rrJrorr{uHr4 Ma- rcrs 2 piaHi cniasni roqxu, ro ui nnorqnHu nepertrHarorbcfl no npnuiii. ur,o npo- xoAr,{Tb qepea qi rounu. 1 73, a) Ciuua nJrorqtrna Mae 3 fiJrorrlnHow (AtB rD t) 2 cnrnr'nr rouru B, i Dr, to4i ix cli,rr,sa npRma B,Dr; y civHoi n;ro- Ar quuu i tr,rolr{uHu (AAPJ - npsua A-Br; y ci.unoi trJror{u- Hu i n.nonlnnu (AAp) npsrra.4,Dr. To4i myxanufi ne- pepia - LABID 6) Hexafi cepeAurra pe6pa DDr - ror{fta M. Ci'tHa trroqrr- Ha, nro npoxoAurb qepea rout<u A, C i M, neperunae rpaHb A AAtDlD no ni4piexy.4M; rpant' DDTCTC - no siApiexy CM; rpausABCD - no ni4piaxyAC. Orme, uryxauur? nepepia - LAMC. AA2 ::::::: fEOMETPIg Eees f. Il. ra ix' l'lpo$inuaui piBeHb:i:i:i:i:1:i:i:i:i:i:;:i:!:i:i:;:i
  • 15. 181 . Hexai y rpsMonyrHoMy napa;re;reninegi ABCDATBTCIDI AD :6 cwr, DC : 6 cu, DDt:8cluiM - cepe4unaAD, T -cepegnnaDC,K * 81 cr cepe4rzna DD,. Illyxaxra* nepepis - LMKT. MT - cepeA- ns ninis LACD - npfiMonyrrroro rpr'rryrHlaxa 3 lcareraMu 6crvri6cu; AC=6J2 cu, ropi MT=1'u"1,=aJZ (cru). '' 2 MK - cepe1:r..a ;rini,n LAD.D - np,fiMoxyrrloro 3 Karera- rvrz 6 cm i 8 cu;.ADr : 10 crvr, roti MK=1 16= 5 1"*;' KT: MK. v ^MKr o=Mx+t{+mr =4#=L5vE+5. S*"r=@, = J1s * r,sJz). r,s,D.. t,rJi. (s - r,sJz) = = L,lJt. J% - r,r*, = L,sE'E = r,sJm rc*t' Bilnoeila:1,5Jfi cu'. 182, Illyxauufi nepepia xy6a ABCDATBTCTD, - MB:C - pinnocroponnift, ocKi'nr'ltr'r #;.Ur" = AC = aJ2 - giarona;ri nna4parie ei cropo- orl,ce, P14s =zali. ,*"=# - (o'li)'Js o'JB b*s=-T= 2 . Binoeidu: ,"A, #. 183. IIIyHanufi nepePis - LMNC1. MN - cepeAn s ni:rrir LBCD, ge BD = alt, ,osi MI'{ = $. Ie ANC'C (lCrCN :90"): 41 K A _"Ji .a 448 I 88.Cnig nroqranz nepepiay:. - na rpani - ArDtD - KDi - Ha rpaui DDCp * DrP; - na n"noutrrani nrzrrtnroi rpani .- r. X i r. Y i npaua XY, sua [eperr{Hae ,48 n r:ouqi S, BC - e rovqi I{S, na rpaai BBpp * TP. ToAi uryxanufi nepepis - n'arrxyruax KDTPTS" Clqnoro nrorqnHorc BaAaHoro reapaeApa (napa.ne.nenine no o6u4na 6oxra ni4.aroi e roqrr BaAatoro rerpaeApa rrJror{r4Ha neperuHae rparii rerpae4pa (napa,re.nenineA rcyrrrrrx, cropoHaMn fiKoro e qi nigpisnu, HaauBarorr n pa.neaeninega). ,{:ra no6y4onr nepepiain,qocrarnr,o no6 civHoi n.rorqrauu a pe6pannu 'rerpaeApa (napa:re.renine xorxui 4ni no6y4oaani To1rrcr4, rqo reJfiarb n o4Hifi rpau Ocrcinrnlr rerpaeAp wrae 4 rpani, ro fioro nepepiaauu M a6o vorapuryrHurcu. Ocxi.nstu napa.ne.nenine4 mae 6 rpanefi, ro fioro nepep r(yrHrrxrr, rlorr,IpurtyrnlllcH, n',*lturtyltxz(8, mecrlrxyrn 194. Illyrcanuii nepepio - MDM. flxuqo DA: DB : DC : 5 crv- AB : BC : AC : ro MaeMo: AM - nucorya piunocroponnroro AABC ACh 6.,5 -AM= 2 , or*te, AM= 2 = 3V3 (eM). crB1 Ar V piano6egpenoro ACDB DM - eueora i ue4iana ^^CDM - npauonytnlrft, Clt =nC = 3 cm. 2 la LCDM (ZCMD : 9O"): DM : 4 cyt. P*o, = 5+4+ lliiL..--...-. ';:t{.:i.:.:.:{'i!. C1N : MCy llinnepuuerP LMNCT: t( ali "'6) o'[i "JiP=rz*''r,=4+2' _ ro4is-u.' =iit ,.2,1 [ 4*z-2)l 4n2-2]= G$ '$,("i a'[zt(alr'z =ll a * z t z - 4 ) 4 )= 884 l{i;i rEoMETPlg Eees r. rl. ra in. l'lpoQinunrr piBeHb tli:iirdil:il:**;'-:i': p = 4,5 + 5uE cu. S-.= =v o =@=t,b.,69 (cm'?). BiOnoeiu: g+3rE cnr; 1,5169 cu'z. 195. IIIyxaNzft nepepia - MMC, y slcoro AM : CM, A Posussemo LCMB: lB: 60o, ocrci.nsnu LCDB - pin poniriii, CB : a, *U = t. 3a reopeuon xocunycie CM' : MB2 1- BC2 - 2MB . B CM2 =a2,o' -2.a.g=a, *t--4=a, -U' =79693993 itiiji{iri;:iit::{iti TEOMETPI9 Eeee F. fl. ra ix. IlpoQ *" =t, CrC : a, rogi CrN =
  • 16. 197. C:ria trJrorr{rHr4 fiepepisy na nparvriii -43 - roqna X, na npruifi BC - roqxa Y, 199. Ciqsa nJrorqlrna neperrrHae nJron{rHy (AAB) no n nJrorqr.rHy (CC$) - no npanifi BrC. Idi nprui neperunarom ArBr - y r:o.rqi M, BrC, y rouqi N. IJJyxawir repepia - rpareqiaAMNC. 2OO. Ciqsa r.noqrirHa fieperrHae ntorqrany (-4ArBr1 no np AB", axa [eperrrHae ArB, e'rouqi M; trJrorqrruy (BBp) - no npavifi B"Cr, axa treperr B,C, y rovrli Cr. C,riA ci.lnoi nJrorqrrur{ Ha n.lor (ABC) - roqnu A i Cr, orxe, npav'a AC2, srca fiepe DC e rovqi N. To4i uoraprnyrnunAMC.N - nynaxzii nepepis. Oc o,li "h t zJt a oJ7 To4i P-.n,=a+2'l = g (cM).P=2* B ffi,J, oJit' f'tT 'le, "A's*-'=/lt.- Jti' t -o)z*n - s, ) !l; *zlt s -tl r) - a 2 altg _o'Jts 612 o" JLg cM: - L2 (c.'). cu2.Bionoeiov: "6 * z"li) 3 ua n.norqzni (13C) nparra XY. 3a yuonoro 1 1119 -'- BB,, NC ll BB,, orxe, NC - cepeAHs 2- rinia ABBTY, ro4i C - cepeAr4r{a BY. Asalo- riqno A - cepe,4uaa BX, or:ne, npaua XY npo- xoA[Tb qepeo rovxy D. Yotzpurytnux MBrND - uryrcauuft nepepis, HolKBa cropoHa fiKoro c rinoreuyaoro npamo- Kyr'oro rpurdyrnrrxa B RareraM[ a i 9, orxe, MB.ND - pou6. B cr guaa BB", ro MB, - cepeAu,ff ninin MBB, i Arra.noriqnoN - cepe4unaDC. V voruprarcyn- nuxaAMC,N yci cropoHu c rinorewyaa:vru pis- Hhx npsMonyrHr.{x rpriKyrHr,rHis o Hareral{u a i l, roAi AMCTN - pom6 ai cropouoro 2 E 4!D 2 mn,- f t,r' !""*lzJ ls LDNC (,/DCN 90') ,N = Fe t- t- . dvD avc , roAi Pr",n, = 4' Z = Z 3Hafi4emo 4iaroHa;ri pom6a MBTND: MN = AC = axii; giarona.nr BtD ny6a: BrD2 : 3a2. BrD = aJT. s"qno = )u,o **, s,^nn L, "A "Js="'!9. E A./b 2 P*","=+ f =2"'tE. Bi|noeia: 2aJ5. 2O1. Hexafi y napa,re.neninegiABcDArBrcrD, r.M - ce , fieperrHae rrJrorqfiHy (ABC) no npamift BD. Il.norqr, (AMC) rluicrurl,, tpavy AM i npnivry CC, neperunae (-4- ro AC. Toqna O neperzny npfiMux BD AC HaJresfi i (BDA), i (AMC). To.rxaA,,ranexrurci (AMC), i (BDA or:xe, ArO cnilrna rp.sMa nrrorqzH (AMC) i (BDA'). Bo neperuuae nigpiaor AM y ro-l.4i N. Orxe, N roq nepetu:ry AM i (BDA). 3a4aqi sa roroBr4Mu MarpHKaMu BapiaHr A 1 . Ilpaui BA, BC, MN .ne:*rars n oAniii nroquui, fiKa nep nci cni;rrni Toqnrr rrJrorrluH Jrelrcarb sa AC i roqxa O np D flpami DN i DM teperlu':r.aytruc Mo)fiEa lpoBecrrz nJlorquHy, s vift KB, ae K : DN a AC.lif B n.rorqusi HaJrerfiarb roqrq N M, a otx<e, e uifi renturr, i npn ua MN, arca reperunae (,48C no cni.nrsiit npsmifi KB trroulzlr y ro.rqi X. Toqna X uyHaHa. 3. Fy,4yeMo nepepisr rerpaeApa rrorluHaMri (MNP i (EFK) - qoruprrnyrrrux MNPY i rpzrcyrnnx EFK BilnoeiOo: E ,E aJb a-!b 198. Cri,q ciunoi n.nonlrznu na DC - X, Ha CC - Y, ua BC Z, rogi na nnoulzni (ABC1 - npstwa ZX, ar'a leperrxae -4-B e touqi P, AD - r touqi F; npstma ZY rreperrHa€ BBr y rovqi -8. IllecrNHytuux PEMNKF uryxaunft nepepis i eiu c npasnlstllv, ocrci,rsxn eci fioro eepurrarra - cepeA[Hu pe6ep xy6a i HoxtHa cropoua rooieHloc: KN o Jz. rofi P"= 6 lJr' sarlr.j"2 fhouqa rpaera.rr,Horo rrrecrlnytuzxa o6quc;rnerr'cfi 3a ^ Bxrn6 OopMyroro So = o , .qet -cropoHarrrecrflKyrHrrKa. BI'"El Js so" z,[s ^ztit I n dqlo Toai Su =-'- 4 ' o^z I Bidnoeioa: go1Ql 3{11 . 886 il{: reouerplg Eeee F. fl. ra in. l-lpo$inourafi 2. piBeHb ii:i:i:i:i:i:ii:::iJ:ii:ii iiiii:fii:l{i;ii:*::i: FEOMETPIS Eees f. rl. ra ix. rlpoQ
  • 17. EF a MY : S, PN a FK : ?. To.rrcrl S i ? ralexaru o6ovr n.norqranaM' orrrce' rrJro- ururrr4 rreperr{Harorbcff no npswrift S?. 5. MN c (BCD), CB c (BCD), MNaCB-X; KX C (ABC, AB C (ABC, KX a AB : Z; AC c (ABC), KX ^AC : Y. r{orzpuxyrnrx YMNZ - uryxauufi nepepia' 7. Terpaeap A,B CD - npaznar,nufi, xorxne pe6po ano- ro a. lllyraunfi nepepia - Aa4NP. $n LADB MN - cepeAurr aiiria, toivry MN =2.2 -- ,l Bidnoeida: oJz: t.lss -2]j1.'8V 2 Eapiaxr E 1 . Ocni.nrxu rA-BCD - ponr6 i rrpsrui CM i CN, .nxi uicrs nn M, B, C, D, A, N.nexaru e oAHift n;rorrlrarri, flrra i To.rfia A Mae ,{erxarrn na MN, 2. Toqrcu N i M Harex{arr, ,4BoM rrapaJreJrbnraM rrpgMu AAr i CCD rogi norra HaJrelrcarb n.noulraui (,4ArCr), i np ma MN c (AArCr). I-{a nrou4raua treperuHae (ABC) npsnrift AC, r:o4i rovxa X neperray npalror MN .nentr . na npalrift -4C. 4. fl.noiqlrna (P.Rl() nepernHae xy6 i yrBoproe nepepia WTPRK.Ilpaua MN J-rexrrrb y nnorqrani (AMC). Cni;rrna roqnax qfix nrnrqun - S. Ilpnua S-R Jrefrr.rrb n o6ox naorquHax. flpone4euo e n.norquui (,4MC,) npslny MN, sxa [eperu npnrry S.R y rouqi O, a orxe, i n;roquny (Pl?K). 5. MN c (DD$), DrC, c (DDrCr), MN .i DrCr: X; KX c (ArBrCr, KX et BrCr: L, KX a AtBl : K, KX a ArDr: Yi Y e (AApl, YN c (AArDr); . YN n-AAt:,$. fl'arzrcyrazx MNLKS - lryrauNfi rrepepia. 7. flo6y4yerro nepepis rrrorr{riuoro (MNP) - xy6a ABCPA PN ^ PtCt: Y, MP a ArPr: X, XY c (ArBrCr), B, e XY, ocri.nrrcu MA, - cepe,4rrfi ninln LXPTP, CrN - cepe4na ninia AYPTP, AtBt i BtC! - cepe4ri .ninii'AXYP,. Otr*te, ruyxanzfi nepepia - qorrrpnnyrHr{r{ MPNBD nnr.rfi e pou6ou, ocxi.nrxn sci ftoro cropontr MP : PN : NEl : MB, e rinorer^y- aaMu npfiMotyrHrlx rpnxyrnrrxie o xareravu ai!. tvtp=^F*--tJt.242 t= o 15 P..-.,^ =4.:=2a45. V LMDP: *O = t, DP = ?o (sa yMonorc DP : PC - 2 ; L. A ZD:60" (LADC - nparrznrnuit), ro4i aa reopeuoro ftocr- nycit: MPz : MD'z * DP2 - 2MD ' DP 'cos lDi *r'=I.{-, }?" cos6oo. MP2=o li-;-l)= *, = +. Ocrtilrrnru LADC : LBDC, ro NP = MP - "Jt36 -ar"[SooJlzIMNP- 2' 6 , Y- 4' 12 , t-T l( alts a li cJt g af a .aJ13 l - tz '4Jrr t2 4)4 L2 l (a ofiB) @'rc ,' o'(a - fi3) =lA-n)l 144-t6- n ' Bidnoaioa: ;. +, ": (s --Jn) . 8. Illyrcanuft nepepis - LMNP. NP l1 BC, SN: NC: SP: Pts:3 : 1; ASNP - ASCB, roai NP: CB: SN : SC :3:4; Np=!a. IcAMSN, rc MS=9, slv=3o, 4 2' 4 IMSN :60'(AASC - npar.unuufi), Ba reopeMoro rcocunycin MN2 : MS2 + NS'? - zMS ' NS MN'=a' *g::-2 416242164 -, t; MP: MN. P,n,=4 +="Ji. p,,,=;J7. , - - Er ",,"-,/; Ji(ili i,)l;e -uu^'I = ,{iaroHa;ramr.r pou6a e ni4piaxr.r MN = AC = onE G BrP = axlS (AopisHroe 4iaroHa;ri xy6a). ToAi sr"ru =tr** . Bjp,sMpN4 = I ",t, ,Js = "'f LBa2 36 cos 60o, Bionoainu: zoJE; ""f. 8. NP n BC : Y, MN ^ AB : X; XY c (ABC, XY ^AD: S, XY a DC: T. Illyxanuft nepepis - n'arraxyrnux MNPTS. (g BBB li{: rEoMETPle sees r. rl. ra in. Ilpo$inunufi piBeHb l{i{tiill;ti*:iii;: ili:iii;i::i:ii*iii::::i TEOMETPIS sees F. Il. ra ix. IlpoQ
  • 18. Ocri.nrrcra pe6po nyda aiArM : MA: CtP: PC:3: l, ro A-M=?o. ua=!.'44 f ,"^' sa MN=NP=.la'+l ""1=::: 4) 4 _sa,[, fftuo" BoJ2 a 1- E; Ba,JsT.-!Z VJ 8V648832 s,,"." , t-f ,po'-J|i -fi tra*3v6?). Bidnoaida: TW ,z), I'zbJit+e,67). Tecroei gae.qaxHc Tnnosi sagavi ,q,nq KoHTponbuoT po6orra 4. Hexafi y rrpaBrrrbHoro rerpae,qpa SABC SL - 6icenr Ocni"nrnu eci rpaai rlboro rerpaeApa npaeumui Tpux ro SZ - uegiana i nzcora ACSB. IllyrcaHufi nepepia Ae AL sucora AABC. SL= AL -& . 2 ,qe a - cropoH a LACB, roai SZ = AL = *- erf ( 2 v AAS,L: As : 6, sL = AL= 8.r5, p^, = *# = 'ts LBCT (tBCT :90"): BT = = Ja'taG - e)(s,6 - si, s s,[z = s,[, rcNo. Bi|noeilu: 9Ji cm". 6.MLaDDr:X;KXaADTN; ML o D,C, Y: KX n Arb, - Z; ZY r, ArB, : S; ZY o BrC, : T. Illecrznyrunx KNLMTS * r:lyxanuir nepepia. ;rili,i!:.:.:;i:.:. '" r 1..1J:i'l.ll.l.l MS=PT oJ2, sr-3orlz. 44 Pvtprs: ST + 2MS + 2MN, P**n" = |",0 * 2. i"E +, T = rf .+ = lW . r). Pogi6'eMo n'aruryruzn MNPTS na 3 rpuxyrnrznu LNPT : ANMS i pinno6e4pe nplri ASNT. _ "Jn4 Is ANB? (INBT :90'): N? = i2l-a ^la" + -v16 3uafi4euo ttorqy ANP?: o = *' * Y * n', ba oJrs oJz o-4t 4 n 4 -sg-@rofi.'2888 il so ,Jsg alz ( so ,.6s "Ji sa) s^u' 1fl s-- * a it 8o s *-s -T)" l( so oJz ) '!la*-r a JIs*-s ' a- q ) _,li i +-ll- +- i-+- --i, -;-li 8 8 8ri 8 8 8r8 8 8 r8 -8 8 l rT- -z /l I aJeg ,alz) _2sa' ll ( lnY I a,l3s aJ2 | {tt t'sJ-oaJilsl t 8 -8ll= l -z t - .2 =ilfn((Jvi *Jz)" zsl-ft(s-{Jse -.lEf )= - o',C.,tallo)G'ta-ld =# o^=ry oJss B r- z. L + ^alz orl11 SaJz PsNr- 2 4 g ffitsl-loJbe saa Bov2 l s""=1|IJ * s Jt-a - 8 4-)1 + 8' 4 )= 8. Il.norquna (ArCp) nepervHae rpaa; AATDTD no ei4 A,D, a rpaxs DDP$ no rigpiaxy CrD. fltoulvaa (A neperrHae rparl.s AATDTD no nl4piaxy AD', & rp&Hb DD no eigpiaxy DrC. ADt i, AtD treperrruarorbc,s n rou4i O JTflTbcs Hero uaenin (ex giarona;ri xeagpara). Asalor D1C i DCI treperuuarorbcs n ro'rqi O, i gi.narsca Hero ri;r, To4i OrO, .niHia neperuHy trrrorrlriH (ALCp i (A 1 Bi4pieoH O,O, cepeAHfl ninia LATDC, i O1O, = ;4 4C, arli, roai OrOr=ol2 . , ",2rou,Uo, ?12. 2 9, M e (APC).Ilpoee4ervro vepes npami PB i PM, axi rre Harorbcs, nJrorquHy, flKa rreperuHae,4C a rouqi S, r.n ny (ABC) no SB. To4i MB c (SPB). I 2 3 4 5 6 o 6 I B a a , *17a2 - o"65 890 ii::: TEOMETPIS Eees F. fl. ra ix. rlpoQinunur piBeHb i:i:::iii:i:i:i:i:iri:iii:i Iiiiii{ti{iiiliiij rEoMETPtc Eees F. It. ra ix. Irpog
  • 19. Ilo6yAyeuo nepepio rerpaeApa nrrou{rruoro (APK); (APK) a (ABC): -A-I{. Ilaorqnnu (SPB i (APR marotr cninrny touxy ? u n.norqrani (.48C)' toai Bonrr neperra Harorbcs no uprrrniii ?P. To4i rrpav'a MB rrrrorqrruy (SPB) nepetuHae rrrotr{rluy (.A?K) r rouqi O, gra nanerxrlrr ix cni;rHiit upauifi TP. POSAln lll. nAPAIEnbHICTb nPcMl4x I nrlOl4l'lH v flPOCTOPI S 6, Mmrvro6ixHi inapatenbHi npeMi V npocropi e rprr Br,Irranrr BsaeMrroro poeuirqenna 4nox pisnlrx lpfiMrrx: a) nparvri neperrrrrarortcn (nonu Marorb oAHy cui.nrny touxy i Jrer{carb e oAHifi n.nou{vni); 6) rparvri napa.ne.nsni (nouu He uatots cnilruzx rotror i nentarr n o4nift n;ro- rqr.rni): e) npcui uuuo6irxni (ne nepetnnarorrcg i ne napanentxi, qi npnui Ee Jrelr(arb s oluiii n.norqlrni) IIapare,rr,u i npsui lra nrr' sracru sicrb rpaHcurr'rnnocri. V uaremarurli aiacmu6icmb mpaHJumuaHoclai o3Ha'rae: flKIqo.4effHa e,'ractugictr Br4xouy€Tr,cs A,rIs [apri m I n t qna.napu n i k, To AaHa s,racrueicrr, Bunouyerb- c,xi4.narapwmik, 21 3. IlpunycrlrMo, rqo npaui a, b i c aer*carr s oAHii n;rorquni B, arcift uaaerxar:r rou- xu A i c. To4i nnorqranu u i p neperr.rnarorbcs rro npauifi AC, sna Micrllrs yci rourra flepe'rrauy cx. i [pfiMr.x a, b, c, orxee, B e AC, rqo ne rignoei4ae yuoei. To4i nanre lpunyrqerrns EetrpaBursHe, a orlrce, nprui a, b i c ne Jrelr{arb e oAHifi naorqrani' 214. IlpunycrlrMo, lqo BhliCCy To,qi qepes Hltx Molr(Ha rrpoBecrv n.noquny B, sxifi riaJrexcarb yci ix rovxa. Otrxe, B, e B, Cr e 0, B e 0, Ce B. Ocrci;rrxz roqxn B i C npauoi ArB HaJre)+carb B, ro npaua ArB le:xnrr y B. Ilnorquuu n i B uarom cnilr- ni ro.rxlr 81 iCl, a oTlxe, uarorn cni.nr'ny upniuy BrCr, gxiit na.nexlars yCi cniltai roqrca rrJrolr{zH, rogi A, e BrCr, ulo ire ni4noni4ae yuoli. To4i uanre npurryqeHHfi HerrpaBr,rJ,rbHe i BBr)lCC1. 215. ABCD - naparrerrorpau, ro4i AB ll CD, AB : CD' ABCtDl - napanerorpau, roAi AB ll CrDv AB : CtDr. 3a sracu4sicrro rpanar4rr{Buocri cD ll qDr, CD : CtDr "to4i Ba oar{axoro flapa,aeJrorpaMa r{orr,rptrxyrHr?Ift CCTDtD - na- paleJrorpaM (gri nporia.nexHi ctopoHu qorr.rpr.rxyrluxa rra- pa.ne.nrni i pinni). IIlo fi rpe6a 6y.no 4onectlr. 216. Hexait AA-BC nencurr y n.norqusi c, M e d. 218. y napa,re.nenine4aABCDArBrCrDL AAt llCC1, ro4i n sxa [poxoAt4Tb qepeo Torrxrr A, Ar, C, Cr. Onxe, nJr (ACC,) npoxo4rrrb rrepea rouxy A,. I{o fi rpe6a AoBecrr,r. 219, 3a yMoBoro o ll b. flpr.rnycrnMo, rrro a ll c, roai Ba rrzeicrrc rpau3rrrr{BHocri b ll c, rqo He ri4nonigae y orxe, nanre npunyqe'Hn HerpaBrurbHe, to*y o/". ft rpe6a 6y.rro 4orectlr. 22O. HexaftBaAany npfiMy d [eperr,rnarorb napa.ne,rsni n rqo fieperr.ruarorr,cs n rouqi A, npoBeAeMo n'ltoqlruy c a n rouqi "8. Ilpooegemo t{epe3 rorrxy B n nnorquni cr qepeg rorrny, rr{o He Jre*cr,rrb na npauiit, MorfiHa fipo tpffMy, irapaJre.nrny aaganifi nprvrift i lr,rtre o4ny; npaui c i c'cninna4arcrs, rogi npslra c Jrelrrrrb y n.no oc, i, orrne, nci napale.nrni npaui, axi neperrrHarorb B nprrMy, Jrerr(arb n o4nift ulouqaxi. IIfo ft rpe6a AOBeCTr.r. 221, 8a yMoBoro O e ct, A e d, B e cr., 41 € a, B, e a, A OBt : BBr. rIepea irpaui OA i OB, rqo fiepernnarorbcfl, Ha npoBecrlr riJrorqrrny OAB, a:r"a neperlrnae o no np .4rBr. Posr.nnneuo AAOB: ArB, - cepeAufi ;rini*, A,B,llApi 48,=;AB. . Ocxinrlcr.r A,B, : 3,8 cu, ro AB : 2 . 3,8 : 7,6 (cnir). Bi0noaida:.7,6 cut. 222, tr1exaft rpr.rryrnr4r{ ABC nextvrs y nnorqr,rri a, O A, e a, 81 E s,, Cr e a, A - cepeAnHa OAt, B - cepe OB1, C - cepeArrua OCr. r{epea nparrni O-4, i OBr, rq perraHalorbcs, MorfiHa npoBecrrr rrJrorrlr4rry OArBr, nxa Tr,rHae o no npamift 13. Asarori.rno trronlrna OC,B, Tzuae o no npauifi BC, n;rouqrana OA'C, nepernnae npnrvrift AC. V LOALB..43 - cepegna nlnia, AB =L4z y LOBTC,BC * cepelna ,rinis, BC =I",",, y LO ec =!;*c..2" Otrxe, P4^c, = 2Pag,, repr,rMerp LArBrCt ygeivi 6i.nrn Bidnoaido: y 2 pasu. 223, B:iz-napox 1. Bigpiaox/Il rre neperr4Hae BaAaHy rrrro rca). 3a 4one4euvu y aagaui 220 npattiAArll tutlVtrllBB ro4i rourcra Ar, Mr B, .nexarr ra o4nifi npannift. AA.B AAl i BBr. 3a reoperroro @a'neca ArM, : BrMr, ocHi.irrnz AM : B rpaneqii i MMl=4:tu', MM,=6'5!8'5 =z,b Buna4or 2. Bi4pieox -u? .r"o"tr"u" ,uou"f, rrro*"n AAtll MMl l] BB, i.nexarr s oAHifi nrorurani, .fiKa trep iiii;ii:iliiiii:;';:j TEOMETPI9 Eees F. 11. ra ix. flpoQ l11.fg,+t:rj,:.:d Togi BC c c:.', A e. BC. Ilpaua MA neperutae c n ro'r4i A, iii*F.i:i:i:.::i MA e ct. Maeuo: BC c o., MAr.epern:,l,ae a s ro"iA, A € BC, :l{li,::i#;: roai MA i BC - vrr.mto6ixui. ;ll$iif$ji:ll ToqxuE eMA,Pe MA,F e BC,K e BC. Ilpanti,4?iPKne' il$i*:i:iii!:: perr4Harorbcfi r rovqi P, roAi'repes Hr'Ix Molr{Ha npoBecrt{ nlrlo- l:iiiiffi:lii:i: urn'y 0. Ififi rnoquni Harrenrrrrr' to'rxa.E (ocri.rrxz E e AP) ;:l:'.!jili;:iii:i i ue sa,terxxrr ro'rxa F. ToAi npgma -EF neperunae urorqrrHy B n rouqi E, E e PK. Onxe, npaui PK i EF - mumo6ir*si. 217. Hexafi napaJreJrorpaM ABCD ner"urt' y naouluHi cr. 'Ioai BD c o. Ocrci.nrrn b ll BD, ro qepes Hn x Mor(Ha npoBecrtr - rrrorquny. Ce b, Ce a, ro,4i bc cr,. Ornte, npsua bJlemcrrrb y n;rorqrani cr, flpflMa 4 fieperrrrae cr n rouqiA' sxa He HaJrexc]rrb b.'Ioai a i b - nurvro6ixcni. II{o fi rpe6a 6y.no AoBecrr. as2 lt{: rEoMETPlg 6eeg F. Il. ra ix. rlpo$insmnfi piBeHb il{iil*iiiii:iiii:::i
  • 20. ripfiMifiA1B1 i M, e ArBt. Branonaeuo napaJIeJTbHe rrepeuecenH,fl nigpisxaArBr y Ha- npami .4-4, na ni4pison -4Ar, o,4eprcr.ruo ni4piaon,4-Br. ArB,ll AB, ArB, : ABr, AA.B82 - naparerrorpaM, Bp2: AA. Ilpogonxzuo MM, go nepernuy a.4Br, vaemo M, e AB, AMr: B"Mz, M1M2: AAy Posr.nsHemo LABtsr: MM, - cepeaHr liniR, l,rtr, = |nnr; BB2: BBt + r,1r.2, BB2: 8'5 + 6'5 : 15 (n); , M" Bt MM2: 7,5 m; MM, : MMr- M'Mr, MMt : 7'5 - 6,5 : 1 (nn). Bi1noai)a: 7,5 r,r a6o 1 u. 224. Bzna1ox 1. Bigpiarcu AK i BM rerr{arb B oAHoMy nienpocropi ei4uocno o. . AK 1J BM, ToMy qepea Hux Mo)fiHa trpoBecru trJrolrlr4ny' .flKa Mae a oc 4ni cninrHr roqxlr A i B, orrxe, qi n.nolluun neper[HaK)Tr,cs lo npamifi AB. KM c (AKB), ro4i roHa neper.rrua€ c n rovqi C, axa sarerxzrr,4-8. PoarnsHeuo LCMB i LCKA: eoll.u no4i6ui sa 2 nyramu (/C - cninssufr, ZCMB : ZCKA aH ni4noeigni rpu BM I KA i ci.rHifi I{C), ro4i + = rytypaxoByrcqr,'' AA KA" q + =93, ru yuonoroAC, : CrB, : 2 : L i BB, :AB1 BB' o'*", l=-99', a", -8A';372 AB AA e) _ = ^^ : aa yMoBoro AB : BBr- 4 :3i AC - m BB, CC, 4mBmoT?fie. -: UL. = -3CC,'4 )aA BiOnoei0o: a) 9,6 4u; 6) 8 ,qrvr; e) 3m 4 tto AC - BC - g. *u"*o, -qC 12 =3, sei4mn: 4BC - }BC t 27, BC - 27 cv' ior,iAC-36cu. Bc+9 16 4 Buna,4ox 2. Bi4piaxu AI< i BM Jrercarb y pisnux nienpoctopax ni4rocno ctc. Il.norquHa, il{o trpoxoAr4Tb vepes AK ll BM, nepetrzn ae a no AB; KM uar'ex"urt' 4iil nnorqrzni, ToMy Mae touny C neperuny 3 cx rafiy, u{o C e AB. LAKC - LBMC Ba ABoMa HyraMu (1ACK : IBCM sH BeprnxarlbHi, /AKC : IBMC ax nHyrpilini pienocro- poHui npn,4x I MBi ciqHiitKM.Toai { 11 uo".'MBBC xoByroqrr, qo BC : AB - AC, BC : I - AC, uaeno: 16 =!- L, sag.86-7g1C,1AC-86, AC=5: L2 3 q-AC I 7 Bi1noeida:36 civr a6o 5- cnn. 225, Oct<intxuAArllCC, i] .e-Ar, ro BoHu rre)fiarl' n o4nift n;rorquHi 3 trprrMoro A'B'Ia 226, flapaterHri upaui CCt i BB i npaua-43, rqo ix neperrnae, Jrextarr' B oAHla rro- n1zui, axa neperfiHa€ a no npauifi ABp Cr e -4-Br. Tpurcyr- uunu ACC, i,4J3Br no4i6ui 3a ABoMa rcyraMrr (1A - cni,lr' uufi, IAC.C : IAB.B ax ni4noei4Hi npu CC,l] BBr i ciunifi ,4Br). To4i: AA CC. a) -" = 4; ea vMoBoro AC : CB: 3 :2 i BB, : 16 4u,'AB BB. 3 cc. 48 orrxe,1- -]; CC, - : =9,6 (AM): 5165 227. SayMoBoro g =2, rogi, axrqo r - ix roe@iqie"BC2 Bx, BC - 2x cyt. KM = *, =+ cu, ocnirsxu KM cepe4Hi risii LADC i LADB i KM llAD ll eN. Xe : M ocxirrxu KP i MN - cepe4Hi aiuii M-BC i LBCD, KP tl M N ll sc. P,,n"n ' (t; ,*) = ur. Ba yMoBoro Prrro : 5x : lO, x : 2. Toti AD : 3. 2 : Bidnoeidu:6cui4cm. 228- NlMt- cepeAns niuis ACDB, *r*, ='+ = 10 (c P,M, - eepeAHs nisis LADB, *,=+= 8 (cv). AHaroriqso KrPr : 10 cu, KrN, : 8 cu, roAi Pr,r,v,q = 2 (10 + 8) = 36 (cM). NM - cepe4Hr;rinis AN,DM,, NM =ry = 5 (cu). AnaroriqHo KP - 10 cu, 2 KN cepeAHn riHin rpane4ir K.ADNt, roqi KN - 1 rrJronltrHa rreperrzHae oc ro npauifi A.BD Ct € AlBr 3a yaa- ^ I ia.nr,seHoro reopeMoro Qa:neca AC i CB : A$r: CrBr, or:ne, A :4-t I ArCr: CrBr: 2 : 3. flrcqo ix Hoetbi4ienr nporopqiitHocri J. _ ,1--1-_L ro A,C, - 2x, CrBr- 3x, ro4i A,B, - 5x i ArB,: ArCr-- / I I l) Ex:2x b:2. > A, C, B, Bi0noei0a:5:2. '--> -f.t{z(-"/ / 24eBC jii.r.. -. . ... , Asaroriqso PM : 12 (cm). To.4i Pruv":2.(I2 + 5) Bi}noeidu:36 cu; 34 cu. 232. a) flxu1o M i i/ e6iranrr,ca, ro npsui.4-B i CD nep 6) ,flHr{o M i N se a6irarorrca, ro npama -4-B Jre)fitrrb a rrp.ffMa CD ne netxur; y nifi; npsui AB i CD M 6iatsi. 235, 3a MarrroHxoM 87 nigpyuuuna AAr ) BBrll CCr, A, rerorpaua-43CD ro.rna treper[Hy 4iaroHa,'refi O; vepe 4e 0, e o. PosusHeuo qorr,rpur{yrHrr}c AACC.. Sa yu orwe, ATACC, - rpaneqia. O - cepe4rana AC i BD. uoro @aneca O, - cepe4zHa AtCl i DBr V rpaneqii /A ss ainis. oo. A4*cc' . oo. =32 - 16 t.*t.'22 Poausneuo LDBB; OO, - cepe4rr ,linia, OO, - ! "BBr: 2 ' 16 : 32 (cn)' ------'-' --t 2" Bidnoei1u: 32 av- iiii{iilliiill:iil{i rEoMErPls Eees F. Il. ra ix. IlpoQ 894 i::::: fEOMETPI9 6eBs F. l-1. ra ix. [Ipo$inuxufr piBeHb :;:;:;:il;:;:;;;:;:;:i1:;:;:;:;:;:
  • 21. 236. 9orupuxyruux AtACCt - tpaueqis, oo, - cepegua .ninia, oo, = &:cC', ari4xu -AA' + '2 + Ccr : 2AO. V rpaneqii BTBDDI OOr - cepeAss ,'iinis, roai oo, - BB' ! DD', asiArcu BB, + DDr: zoot. 2 a) Bnafi.qeuo cyMy ,4oBlr{r4rr ri4pisxin AAr + CCt + BB1 + + DD1 : 2OO1 + 2OOt: 4OOt.II{o fi rpe6a 6yrlo AoBecrl.r. 6) Cyrraa.AA, + CCt : 2OOt, cyrr.a BBt + DD1 : 2AO, rogi' AAt + CCr: BBr f DDr, rqo fi tpe6a 6y,ro 4onecru. $ 7, IlapanenbHicrb npeMoi i nfiou-[nHl,t V npocropi BBaeMHe poarrirqenirr npamoi i fiJrorr{r4urr uolrte 6ytr'r raxum: 1) npsua rerrcurt y n,rorurHi: 2) npsua neperr4Hae nJrorqnny; 3) npnrvra i n,roulrua He Malorb cni,'rr,nrlx roqox - np.*ua i nrot:1una napaJlenb- ni. flxrqo nprua napaJreJrbHa Aearifi n.norqlrni, ro arepeo 6y.4r,-sny rovxy qiei nnoqu- r{n MorfiHa npoBecrrl npfiMy, rrapaJ'renury .4anift, i go roro rx ri;rrxy o4ny' ,flxrqo npama, sna He Jrelrir4rr, y txo4nift 3 ABox [JIou]]rE, tr{o neperrrHarorbcs' na- paJaeJrbna niHii ix neperxEy, To BoHa napareJrbna rcoxtsifi is 4a-Hux trJronlnH. .fku1o npalra napaileJ*'Ha roxnifi ia gnox u;rorqrn, rrlo neperr,rnalorr'c,F, To BoHa trapaJreJrbua ;risii ix neperfiHy, 251 ,AB ca, DCllAB,ropi Ba osnalcoro napa.nearuocri npalroi i rrJlorqrrHrz C.A ]l rri" 252. Hexatr rrJroilIrsa cr neperr{Hae trroqr,rHy rpane4i A,BCD no nparriii, ,sra uicrrzrr cepeArrro .rirriro MN. Totl MN c a, BC ll Mlr, orxce, BC ll a; l,o ll twn, omne, en ll a. II{o ii rpe6a 6yno 4onecta. 253. r{epea npsnri OA i OB rroxtrra npoBecrrr flJrorr{r{Hy, ffKa reperrrfiae rrron{rrHy c no nprnrift AB . V LAOB MN - cepeAr{s.ninis, Ae M - cepeguwa sigpiexa O-4.' N - cepe- Arana siApiara OB. Ocxiarnn MN ll AB, ro MaeMo: AB c a, M N I AB, ro,4i sa oanarcoro fiapareJlbHocri npauoi i nlo- Toai n l] z sa raacrasicrro rpansurlrBnocri, i rvraemo Ha d, rrlo npoxoAr,rrb rrepea npsMy r??, ilapar'reJrbrry rrapa!'reJrbua c, mll c, Ocxi.nrrn mll a, ro c ll a. tl{o 256. Ocxi;rr,xz BC jl cr, ro nrrorqrrua LABC nepenunae napa;re.nrnift BC, rowxa N na.nelxzrr, qiri nparrift, Togi MN ]l -AC i ea reopeMoro @a.reca N - cepegu MN - cepe4rs fi:uir, LABC. a) Ocxilrrcu MN : a, ro BC :2MN : 2a; 6) LAMN o A, -BC, Mll - cepe4u,x nini.s MBC, ro AeEUM y aagaui 254 Sr."" i S-r" : 3 : 1. Bi0noai0u:,,a) 2a; 6 3 : L. 257, Ocxinrxz M i N - cepeAuHr4 AB i AC, ro MN - ninia AABC, MN )l BC, ttN =|nC. Oclci.nrxu MN BC ll MN, ro 3a osHar{oro napa.ne.nsHocri npnvoi rqnnra BC ll o. Hexafi AB - x, rogi AC - x, BC - rnli, ** = + D _' *.rJi 22 2x + xJi AO:)2 2x + SxJiP"",r"=2 L+'Ji-+= x"li("li + 2' P",,," i P,* - *Ji('E' + s)' QryA = Ji + a : Bi]nosi0n: Ji+s lr.L' tTzuw MN ll a. IIIo fi rpe6a 6yao goaecru. (0 254. K€ o, P€ cx,, rosi I<P c a. v MBCKP- cepeAng U{),riuia, KP ll BC. To,qi Ba oauafioro napa.ne.rrnocri npsrroi i naorqrauz BC l] o. IIlo it rpe6a 6y;ro goeecrlr, LABC * MKP sa ABoMa KyraMn (ZA * crl-iaruufi, ZAKP : IABC g.t< siAnosilni npu KP ll BC i ci.rHifi -4,8). Toai: a) P*" : PAK.: BC : KP : 2 : 1 (sa n,racrznicrro cepe4nr'oi linii rpunyrnura); 6) S*" : SMp: BC2 ; KP2 : 4 : t; n) e nonepe4usoro Brtfirr.rBae, rlo Ssc : 4S*0, rogi Sr""" : Ssc * Sn"": 3Ssp. Otxce, maeuo: S*n : Sr"r" : Bidnoei1t: aJ 2 : l;6) 4: 1; 3) 1 : 3. 255. Ocri.nrxia o ll cl, ro r o icrye npama m, m ll a; osri.nrxr a ll ts, ro n 0 icHye npswa n, n11 a. 258, OcHimxu a l] fC, ro cx treperr.rna€ nrrotr{rny A"ABC - uapa-ne.nr,uifi BC. Togi NM..)l BC i aa reoperuoro (Da,'re roti NM - cepeAun rinis AA-tsC, XU =LBC.'2 3a yuoaoro BC : AC: 6 : 8. flnqo r - xoe(piqienr ix nocri, ro4i BC - 6x, AC - 8x, nN =!1rC, AN -2 l1 NM - 3x- ^S..... = - NM . AN, S",. = -:.3x.-lr= 6 2 -*' 2 --- --- rogi 6x2 : 24, x : 2, AC: 16 crr, BC : 12 cv,, AB pemn fli@aropa). CM - ue4iana trpffMonyrtoro rpn 1 reuy3fl, orr+ce, CM =: AB, CM : l0 clr. 2 Bi1noeilt: 1,O cw. 259, 3a MarrourcoM gb ni4pyvnuHa nnolqunu (ADC) i (ABC) ileperrruarorbcfi no npamifi AC. Hexaiz M, N, K, P - cepeAuHrr ei4nouigno .nauox ,'raltanoi AD, DC, CB, AB. y LADC MN - cepesne ;riuin, rofi MN )l AC; y MBC PK - cepe4irn aixia, ro4i PK ll AC. iitl:iiiiltl{iliili TEOMETPI9 Eees F. It. ra in. Itpo 29 yct rAq 10 u., c. I 896 l{ii reonerpls 6eee r. tl. ra ix. rlpo$inrnrft piBeHb {:i${t{ili:}:':l:liii
  • 22. 3a eracrueicrrc rpaHslrrtrBHocri MN ] PK. {epea napa.ne:rr'ui npami MN i PK rron*ra ilpoBecrr nrrorquuy (M N P), ro1i M e (M N P), N e,(M N P), K e (M N P)' P e (M N P)' II{o fi rpe6a 6y,ro gonecrrz. 260. Hexafi roqxa M - cePe4Iana PB. V rpaui PCB vepea rouxy M npoBegeMo MK, n rpani APB - MN, K e BC, N e ,4-8. {epes MK r MN, anine- peTuHaroTbcs, npoBeAeMo uloilItrHy' gHa neperr{Hae rpaHb ABC no N K. M K c e (N M K), M K ll PC, romv PC ll ftt twxl. n Ana,rori.*ro PA l] MN, roal PA l] (NMK). ocxi;nnu M - ce- pe1nna PB, ro 3a reopeMolo Qaneca MK - cepe4ns liHia LPCB, MN - cepeAns riuin AAPB, l/K - cepe4Ha aiHia '1 A, _BC, roai MK = NM = NK = rAP =3 cv. Bi}noei)u: 3 ctt. 261, Hexafi cepe4una BC M, roni civua n,toquHa treperuHae rpans BPC no PM' flponegeuo n n.norqugi (ABC) MK | ,43, roai K - cepeguua AC, MK - cepeAHs ttirrist LABC. CivHa naorqr'rna treperuHae (APC rc PK' ToilLKPM - ruYxaunfi nePePia. t<tw =!a: BM -9, PB : b. Ocxirsxu PB : PC, ro 22 LBPC - pinno6egpeuufi, rogt PM - Brrcora' ue4iana' Ia LBPM (tPMB :9O'): pu ^fa" ": =.E - = ur. !4'l/4 V pinxo6e4peHoruy APKM npoBeAeMo Bzcory' uegiaHy PO' ls LPMO (ZPOM :9O"): 263, Hexafi y rerpaegpaABCD M - cepegrzna BC. flponegemo n rpaHi ABC NM -4C, n rpaHi BC MMrllDB. flpone4emo naoquay (NMMr), rogi N,M, NN, l] DB. gornpnxyrHun NNIM.M - uyrauuii nep MMr , cepeAHs nnis LCDB, MMlll BD, lttttt, = NN, - cepe,+Hn ,riuia A-A-BD, NNl ]l DB, **, = V vorzpzxyrHuxa NNTMTM MMrl) NN' MM1 : ro4i aa oaHaxoro traparerorpama NNTMTM - fiapaJ 264. flxrqo civua nrorqnHa npoxoAnrb uepes M flap MK I A,D, K e CD. Ocxinrxu ciuua uorqzHa tra KL ll BC, L e DB.Ilaorqzna, u{o trpoxoAzrb uepez no MN ll BC, rpaus ABD no N-L. To4i MKLN - naparerorpaM (KL - MN. KL I MN KL MN 3 s."r, : 2(KL + MK), P.Kr.^ MK = NL =!to =!o.11 d z(?t *!").lD o I d o / - z(?t *!')J n / 1 Toai S*"" =;U* POi S,ro' t-; -tlh' 5a' Bidnoeidu: * E; J a lb- 5a' 262, Hexafi y xy6a ABCDATB.C.D. eepeguna AB - rouxa M , cepeArrHa AD - roqna N. Ocxiarxu ciuua nlorquHa napaJrerbHa CCr, npoeegemo n rpani AA'D.D NN1 | CC1, e rpaHi AA'B'B MMrJl CC', ro4i npsuoxyruntt MMTNTN - uyxanzfi nepepia. NN,_AB- r,AN=,, o*=1,t2 ro4i is nparroryruoro rpuxyrnutta AMN (IMAN : 9O"): MN = L,12. 2 orrxe, p.*"". = 2(NN1 + MN)' Pwv,r," = , (.,. L,tr) = tb t Ji). Suv,.u,N = MN NN,, Srr,,.' l,t, ,=t'J' Bionoaiua: ilz*"E); 11? 2 . . - -. - - -. - r..r;.:r:'j 898 :::i::: rEOMETPIS Eees F fl. ra in. flpoQinumufi piBeHb l;:;:;;;:;:;:;:;:1:;:;i;i1i;i;1;:; BiOnoei0v: 266. 3a MaJTToHKoM 96 ri4pyvHuxa allU, a c a, b cB flpirnycrrzmo, .qo ca. flpaui c i a.nerxars n o4Hi pa,Terr,Hi, To nepertrHarcTrca. To4i b c B, rpsua o 6irrni, rqo HeMolnJrrrBo, ocxi.nsrz Ba yMoBolo a ) b, ii rpe6a 6yro aoeecru. 268. Hexaiiunmo6ixsinpnlriaibpoanailleniynpocr - nrorquui o, a b reperuHae rrro n,ronrnHy s Aesniii npnMy c, c I a. -Iepes rfaui b i c rpone4emo nro gepea rouHy B npamoi b npoee4euo npsmy c', c'll ocHinsnz c ll a, c' l) a, ro c' ll c. 3a Aoee4eHzu paH npaui, ari neperr4narorr, oAHy B ABox npfiMnx, rqo TzHarorbcs i napa,reGui gpyriii npauiii, Jre}Itarb e n.norquni. Orrxe, c i c' .nenrarr n o4Hifi uouqnui. I{e ArfieHHfi npaBffJrbHe 4ra 6y4s-nxoi trpfiMoi, ilr,o nep b i napaae.nrna a. IIJ,o ii rpe6a 6y.no AoBecrE. 269. r{epea npalnry AC ll q npoeeAesa trroiltruna (ABC) NM l) AC. Ocnimru M - cepesirga BC, MN ll AC, otr =r KM, oM = 2 a 4 , po = '[ill" - otw", Po = 1 2 r;- -!b' - ba' 16 A1 cepeArlHa BC i MN cepeAH.fi ,rinis AABC, Nl,f = CM - me4iana, [poBeAeHa 4o rinorenyalr, roaiAB AR : 2a. BavmoeooBN NM o. BN loc o. BN' 2 2 2 BC : zBN, BC : AC * a. 3a reopeMoTo fligaropa -4 4a': (AC i a)2 i AC2;4a2: AC2 + 2a AC + a2 + 2AC2 'l 2a . AC - 3a' : 0; sexafi AC x, rogi 2x2 $l::i:*:iiiitii{ii TEOMETPI9 Eees F. l-1. ra in. Ilp 29*
  • 23. xt,z = ' rt,z fE I a a|'l ,2a!2oJ7 _l 2' 2 4 | o a.,17 l-:- *r,-<0. 12 4 -Zotrlio" +Zlo" ornte, AC ="Qvr-r), "* =t +!+i="'li -f +2" =+! L@ -t ellIA ^ 2 4 a'0-l)_3a' *Mc2L68 BiLnoaila: {. 27O. flrnorsqrra:l u, F, y raxi, u1o s ll AD, F 1] nn, y AD, roti M,N, c a' *f 'N, ll An, MrNrcg, MrN,llAD, M"Nscy, MrN" ll AD, orxce, MtN,llM2Nz ll MrN.. OcxitrtxuAMr: MrM": MA[r: MrB,ro zareopeMoro @a.neca DN, : NrNz : N2NB : NsC. Maervro: Mr * cepe- gul;.a AB, N, - cepeguna CD, M2N, * cepegHn liuia rpaneqii -A-BCD, rogr M"N^ = o * b t2- z ' M"N" - cepeAgfi .ninis rpaneqii MrBCN", roni ^-a+bq Sb+a lI{-N^ = 24 MrN, * cepeAn,n .ninis rpanegii AM2N2D, roAi MlNl = Ilpoaegeuo sucory BK rpaueqii BK i MrNr: Ki BK n M"N2: Kr; BK r' M"Nr: K". 3a reopeuoro (Da.neca KKr: KrK": KzKs: KrB' To4i: a) s.^^^ =o'b . BKz K.K. =BK.-1-mrD.2".-.--1..3-2, 3a+b 3b+a .s.-.-.... .BK =4a.+:g.u{ ="ru."*,"v,vrNs^r224.224 n o*2.nrc Dv,v.v.v, 4 I . s* 2 s*" =f,zlc-nc, a+b a+- t 2 3a'+ b 4 6) KK1 BK 4 xra = ff; 3a+b A +o BK 7a+b Jev.r',o= 2 4 - g BK 4 7o *b .BK. - 3b+a * =r+___', BK _7b'a.FK=!!J".py.*u38.^J249432 so",,., _ (7a +b . BK,(7bn o ax) =7"*b . sr,".u, 32 / 32 t Ib-a Bilnoai}t: a) L : 2;6 (7a + b) : (7b + a). 9oo iii: TEOMETP!9 Eeeg F. fl. ra ix. Ilpo$inuxui piBeHb i#ilii;:*iiiiii;i 271. Ocnins<n ,BC ll AD, ro BC ll (ADM). flpoeegeuo uepea npamy BC i rovxy P n.norqzuy (B Ocnirsicr BC (ADM'1, ro (BCP) . (ADMI: PK, Ae K € DN, i PK 11 BC, a orxre, PK ll AD. 272. M € (ADB), N e (ADB), M i N nanetnarr ci nnorillrni. Ocxiar,xlr q,fi rrJrorqrana napare.nrHa BC e Hifi e np.rrMa, .sfta [apaJrerbHa BC. flpoee4elto BDC NK l) BC, K e DC, KN Jrer{rrrb y civniii n.n ToAi qs nroquxa neperuHae tpant ABC no MK. LMNK - uyraunfi nepepra. Ia pinnocrbponnroro ACDB: Vpaxoayrcvz, uro DM : MB -- uaerro: DN : DB :1 : 4, KNDNA BC DB '1:3. .KN 1 ---- a rofi n = i, ** = : , ocximxu ADNI{ N LDBa Ia pirnocroponnroro AADC maouo: lD : 60", MD ls AMDK sa reopeMoro xoornycin: MK'z : MD'z + D MK'-t *o'-2.a.dcos6oo: MK'z=5o: -{.4 16 2 4 16 8 MK = yF. LMDK:AMDN ea 4noua croponalru i 4 na, DK: DN, ZMDK: IMDN). {vur: NK+2MK. P*,*=t-, +=tn."f lo -!*oJB " - 2.w^( 8' 4.-^vLK '!t e' a Jtgn 4 2 -u64 oh - z,li a' Ju _Bidnoqidb: n t On 273. Ilpone4euo n rpani CDB NK il BC, K e DC. To4i pea roqrcr M i N i uaparrer'rbna 8C. Illyxanzfi nepe TpuxyruuxuADC,ADB, CDB pinni, pinHo6egpeu moryrni. AD : DC : DB : b, rofi AC - AB = BC nn -r oc. DN =2= DK. rorj NK=b {2. 4""' "" 4 ""' 4"-' ls LMDN (ZMDN:90") DN ro'i MN = g;* =i,ro LMDN : LMDK sa ABoMa cropoHaur.r i Hyrom uixr 41* r.!^1* o*r-=# ='fi *lJro. illiiitliii;iiii{.::11 TEOMETPI9 6eeo F. !-1. ra ix. Ilp =i'*'=?i (""ls "( oJs a a' t-+_il -_-t.- 4 8/ 4 A) I' az a' 16 -64
  • 24. LAA.C M - cepe4fiHaAAr, - Mo ll A,c, roii (ttnn) e,c. Mo = ! 7..C, Mo = !; so . oJ2, a" a - cropoH 2' 2 Fof* o,trt, u{ro u,fzt tu,tif =1/t +'s-ll q--a Jlnl= uJz Ew' ufz u"E.Jtg = e ! o+ = a.s ArC2 : 3a2, a2 = ={.'ooiBo=!!Z.s/^. .--- -- t^ !J !t d3 3' 2b"hg b'^lG 64 32 ^ ldJtd a',la Duao=i'-Js'z- tT' Bidnouidot d'JG . ,12 S 8. flapanenunicru nfloullrH 274, Ciqxa nrroulqHa neperunae (ABC) no CM . Hepes AD i DE ' ilIo neperuHalorbcfi, trpoBeAeMo nnor4uuy ADE, sxa treperue ciuny naorqzny s touqi O' CM aAE : O' ITtorslnuaADE trpoBeAeHa qepeo npflMy D-8, napa.nellny ciunifi n;rolluiri, ro4i eoHa neperuHae ii no fiaparerbHiii npamift. flpone4emo OK ll DE, K e AD. Tofi LCKM - uvxaHuii nepepie. (CKM ll DE, C e (CKM), M e (cKM). A 275. Ilpone4eMo qepec BrD i M l:.ttottluny, srta neperne civ- Hy nJTonIuHy no npamifi, napa,relrniii BLD, MO ll Bp, O e AC.Illyxaunfi nepepie - LAMC; ABCD (BaApar' flpalra, ,fiHa jreJfiurb o o4uifi s ,qBox napaJreJrbHux nr nrorquni. ,fLcqo Ani npami, xxi fieperfiHarorbcfl, o4niei filaoqz 11wui, ro qi n.rorqzHn napa,re;rrHi. ,{ri naorqzHu, rapa.ne.nrni rperifi nnoiquHi, naparen gepea 4ri naparersHi npsvi moxua npoBecrr'I ffJro OAHY. t{epeo aai unmo6jr+tsi npnui uoxHa npoBecrn trapy n roro:x ri,rsnn o4ny, Ilaparemui nJrorquHn, axi neperuHarorr gani npRui nopqifiHi eiApisnr.. 288. Yepee npoueai S.4 i SAr npoBeAeMo trroulllHy ( ffxa neper[Hae o, B roqnax AiAt, a orxe, [o npxv a B - n'rouqi B no npauifi, u{o trpoxoAurb.repea B lli neperzHy ,4Box ilapaJaeJrbHr{x nJrorqun Tperbolo r Horo napaJrelabHi, rolty [poBeAeMo n B vepes r. B naparerr,Hy -4'4.r i anafi4eIvro roqr{y Bt ii neperzuy veuev SA,. 289. V n;rorqzni o npoeegemo npxuy a' rafiy, ulo a'I a, flHy trp.rrMy b', rqo neperunae a'. flpone4emo upauy To4i.uepea npalti a i b MolfiHa fipoBecrtr nnorquny napaJ'IeJr6Ha d Ba oaHaxoto napa,le,lurocri nJouthH. 29O. flpunycruMo, ulo n n.norquui o( e rrpaMa cl, sl{a H nae B, ocxinrnrz cr i p piaHi. ToAi npsrra a uae cni.n Ha cx, Mae cninruy tovy a B, ro4i cr i B nepetzuarorr Orxte, Ho:xua [pgMa [JIorqIzHu tI trapaJreJrr,Ha n;rorq 291 . Ba MaJrroHKoM 107 ni4pyunr.rxa Br - cepe4vHa O pe1usa OC. Yepea OB i OA nponeAeuo ruroqrHy (O.4 AB i AtBt raxi, rqo.4-B llArBr. Aaa.roriuHo nrorquna AC i APl rar{, rqo AC AlCr V n,roiqzui (,4-BC) mae perrrnarorrcfi i aignoai4no naparelsni npsmurr -4rB treperuuarorbcs, Ba oanaxoro napa,re.nrHocri nrloqrlH 6y,ro 4orecrz. 292, Hexafr. O - cnirssa cepe,qnua nigpisxis AAb BB n.norquHi. flpone4euo vepes ni4pisxu AA1 i BBr iilii:iiii:::iiiri::$ TEOMETPI9 Eees F. fl. ra in. Ilp AC = b8. te LBMO (tMBO:90'): *" - ;, no =$; l;l z* J" %' LAM| - piu"o6"4p"""ii*o'loi q -- 2 r (AM : CM), MO - nucora, ue4iana. S*r, = : AC'MO, t - ,[7 . zt' oJio +u' -hl"r. - "wtqa - - b,l2a' + 4b" Ridnoaidu: 4 A, ,.;.-:al.1i!l 276, rlpoee,qeuo n (.4-DD,) I'{K ll DD,, e (AA,B,) ME JI DDb K e AD, -E e /3. Ilrvna- nuii nepepis - napaJrerorpav EMNK (EM ll NK' MN ll EK), NK :"ME : a' Ia LAEK: 14 - 60', dz=lan. AE=a. ax=li;4--' 4 4 EK2: AE2 + AK2 - 2AE AK'cos 60o; EK" =L*no' -2 I ?" !, 16 16 442 uu" =tgr:u" (, "*, =#. "" =* A t a{7i ,(rc+"?) ToaiP."nn=2lo-16J= 8 Bihnoeiva: o(ro * "[) . A 8 277. Illy:rta:eufi nepepia - LMBD' l flpose4euo uepea ArC i rovxy M nnor4tny,;rra rre- perruae (MBD) rto MO, le O roqna nepetrzHy BD i AC. 902 i:i:j: |-EgyE1pl9 Eees f. l-1. ra ix. npo$insxufi piBeHb ::iii#:i:i:i:i;i:i:i:::i;i:::::
  • 25. i LA.OB1. Bouu pinni Ba ABoMa croponaMr{ i xyrou uiuc nramu (AO :40, BO : BQ Ba yMoBoro, ZBOA: ZBtOAl ax neprnra;rr'ni). "loyi ZBAO: IBA,O i aoHn eayrpiurri piauocroponni npu npar,rrx r4-El i ArBr i ciunifi r4Ar, or){te, 3a o3HaKoro napa,re.nrnocri nparvlzx .4-B ll -d.rBr. tlepes si,4pis(rr BBr i CCl npoBeAeMo n,rouluuy i poa- rJlfiHeMo LBOC i LBTOC. Bonz pinni Ba ABoMa cro- ponaMr{ i nyrou rrirrt nranlr (BO : Bp, CO : CQ, ZBOC - ZBtOCl ,nK Bepruxarbni),rogi IBCO: lBtCtO i nonu rny'rpiurni pieHocroponni npw BC i BrCt i ciuuift CCr, orxe, BC ll Bpt. Maemo: e n,rorqzni ABC npani AB i BC flepertrHarorLc,n i ni4noai4uo napa.ne.nr,ni upr- vzvArB, i BrCr, axi neperrrHarorbc,a i lerxatr y n,ro- t4uwiArBrCr. To4i aa oaHaxoro napa.ne,rrHocri rrJToIr{uH @BC) ll (AP{), qo fi rpe6a 6y,ro 4orecr:u. 293. 3a sracrunicrrc rpauszrrBnocri, axllo o ll V, 0 i] V, ro cr ]i p. Otrre, cr i B ne uorxyrr neperrrnarrrcs, 6o eourl napa.ne;mni. 294. AoeeAenfio, rqo srrqo rJrorqr,IHr4 B i 1 napa-re;rrni i naorilzHa o nepertrHae nJrorqu- ny B, ro BoEa neperrzuae i nnorquny 1. flpranycrrnrro, qo o( ftepernuae 0, a.ne ne fieperrrHae Y, a orxce, o ll y, 3a n.nac- rrznicrro rpaustrrr{Buocri: arrqo cx ll y, a y I B aa yrroeoro, to a ll 9, rqo HeMorrcrruBo. Togi cr neper:vnae Y. IIIo ii rpe6a 6y:ro AOBeCTh. ,[ore4errno, rqo fixqo [p,fiMa o neperrruae rrJrorqr4uy o(, To Borra reperr{Hae i n.nor4nny B, axulo cr ll p. a a a : A. Ilpone4euo e llorql'ni cx, qepe3 tovny A npamy b. r{epea npa*ri a i b, rqo neperuHarorbcs, npoBeAeMo trJlo-, rrlr{Hy y. I{s n:rorqnna nepe'rl4Hae o(, a or}r{e, aa AoBe,qeHIrM rreperr,ruae i F. y n ct : b, y ^ B : c, b ll c. Ilpami a,.b, c ne'. Eiarr, y nJ'roquui y i napare,rrni. flpaua a neperruae oArry g ABox [apaJreJrr,Hux npfiMrrx, a or]rie, neperrraae i Apyry a at c : B, Maeuo: trpsMa a fieperr4Hae n;rorquny I, IIIo ft rpe6a 6y;ro gorecrr, 296. Hexafr aagani Aei unmo6irxni nparri a i b, a c a, b r-, a : r{epes rouxy A n n.norqzni o npoBeAeMo npsMy c, c ll a. He' pee npnlri b i c mo:rcna eAfiHI,rM qr4HoM rrpoBecrtr nJrorqrauy F, F lL ". {epes nparry o Mo}fiHa eAr4rrr4M qI,rHoM trpoBecrrr rrJrorr{rlHy y, flapaJreJrr,r{y B. Orxce, vepea 6y4r-sxi rrrnmo- 6ixtui npsmi Mor+cua npoBecrr,r e,qrrHy rrapy [apaJreJrbHr4x nJroIqrrH, 297. fl.norquna, flrda npoBe4eua qepea r'psrwi CA i CAt, neperr.rnae u no AAr, B - no BBr i AALII BBt. Topi LACA, - ABCB, Ba .4BoMa lcyraMr.r (lC - cnintlruir, tcAA, - zcBB, I -?-4 =+ = 44 . 1ua;,aeuo,4A,: CB CB, BB, a) AC - 2 cw, AB: 6 cu, BB1 : 10 ciu, ro4i BC : AC + A' A4 AC.BBI t AB- Ba - 8 cu. :::1 ='":1. asiarcu &L = "- ;=+:;; ,*i- ','." ""^"""u BC 2BB. 2.lAzsitxu 4 =-=--:=4 (cm): v) AC : 2 cw, BBr: 8 cu, CB : AAr; + = +, AAr:4cMi ' CB BB,' r)AC:2cm, BB,:24cm, BA: AAt. Hexafi gcia AB - tccu, CB * (x+ 2)c.,i*. *,=+, 2 .=-'-"- CB BBr' x+2 x(x + 2) : 48, x2 + 2x - 48 : 0, sniAHn.r : -8 < Bi0noai0u:.a) 2,5 cw; 6) 4 clr; s) 4 cm; r) 6 cu. 298. Hexarl nprma d flepervHae n.llorutruy o n ro.rq npcnna b neperr,rHae rrJroirlr4ny o n rotqiA, fiJrorqr,ru rrpoxoArrrb uepes npsui a i b, rp neperrrHarorrrclr, rreperr,rEae nJ'rou{r4ny o'no AAr, rrJrorqr.ruy F - no BB1 i AAt ll .B.ar. To,qi AACA, - LBCB, aa ABoMa i{y- rawu (ZACA, : BCB. rn neprranan.uni, IAAC : ZCBB, ax snyrpiurri pianocroponni npz AArll BB, i civuifi,43). .AAACACOT}}ce. MaeMo: ' -::- = s. SHaiineuo AA,: BB, BC B,C a)AC:2 cvr, .BC : 5 cu, B,B : I5 cm; 49 = M=AC#, M=L#=6 ("'),BC AI- AA ' A 6)AC : BC : 2 :3, BB,:9cM, ===, : ='"'BC BBr',3 n) AC : 1 cm, BrB : 6 crvr, AA, : AB, BC : AB - A Ac=4- 1 =44- aa-"-A-4-=6- BC BB1' A,4" - 1 6 Hexaft gotxranaAA, - tc cvr, tc > 0, ro4i x2 - sc - 6 x : -2 < 0, orxce, /A, : 3 "*. r) AC :3 crra, B,B : !2 c;ln, AAr: g3' ,4,4' - 6 "*. A4 BB,, Ac _4 BC BB,, !ii t':t{.i}: Bidnoeida: a) 6 cM; 6) 6 cm; e) 3 ciu; r) 6 cnn. 299. AB ll cn, nc c a, BD c 0, cr ll B, ro,qi AB : CD rparr. Bnafigerro AD i BC - giaroirani napaJreJrorpaM a) DC : 25 c*r, AC : 7 cut, ICAD: 90', roAi AACD nnorcyrnlrft, CD - rinorenyaa, AD = tED" - Ae, AD = tE|" 1" = 24 (cM)i to =!en. Ao: L2 c.,r*. 2 Poarrsrel,ro LACO (ZOAC: 90') OC = J AC'z + eO' oc = 'liltz" = JrsS (cnr), rosi BC : 2ac, BC = 2l[93 clr. 6) AC : 5 cn, CD : 8 cu, /ACD : 12O" . Is LACD sa reopeMoro rocnnycin ADz : ACz + CD2 :illiiii:*;iiiiii# TEOMETPI9 6eas F. Il. ra ix. Ilpo904 ii:l reoMerpls EeB3 F. I'!- ra ix. IlpoQin*rrfi piBeHb i:iiiiii{iiii:fi:ii*
  • 26. At)'=25+64-2 a t;. AD2:r2g, AD-u1729 ct.. ,flxu1o y traparlerorpaMa ABDC /ACD: L20", ro ICDB :60'' Is ABCD (BD : AA : : 5 cvt, CD:8 cu) ea reopeMolo xocunycin BC2 : BD' + aD2 - 2BD DC 'coslBDC' BC'=25+64 2 5 8'!, BC2- 49' BC - 7 cut' 2' Bilnoai}a: a) 24 cw i 2^/i93 c:u; O J72g cu i 7 cu. 3OO. y rerpae4pi,4-BCD DM = MN = NB= '!DB tpo"uoeuon'rorqrzHyANC, aHane- ;J perr Hae rpaHs BDC no CN, rpaHr'4-DB - ttSalf. Ilpone4euo s ryaui BDC MM,llCN ' M, e DC, a erpauiADB - MMrllAN, M" e ,4r. Ocrcinbriu M - cepeArrrra DN' ro M, i M, - cepegunz ni4loni4Ho DC i DA i MMr - cepeAH,fl ,risis ACDN, MM, - ce- pegHa riuia AA-DN, I1 roil MM. ='CN=1.25 12,5 (cm). 22 nru,=!dN, **,-t.25=r2,5 (cv). A Y LADC M rM, - cePegHa ,rinia i M,M- =l oc, M,M" 1-. ta = z ("*).q2 '| Pr,rr,, =7 ' 2 12'5 = 32 (cn); i Pn, ,u = l6 ciu: s*,*,,=.llEo6 lloe-z,sf , s,,,*"=f6 ,€J3 =42 (cm2). Bilnoailu: 32 cv-; 42 cxr2. 3O1 . IlponeseMo qepe: r. A' e rpaniAPC Ap, ll AC, C' e PC's rpani -A?B ArB, ll AB, B, e PB. To4i AArBrCr - iuyxauufi nepepio. Y AA?C PAt : ArA - 2 : 3, ro4i aa reoPemoo @areca PCt:CtA:2:3, AA1PC' - LAPC (lP - cnfurr'nufi, zPArC, - ,/ PAC ;t.r" si4nosiaHi nPn ArC, AC i ciunifi A?). A A-P A^C, A,P 2 "loti * = AC' , ocRlrrbKn 4= B, A-P 2 ro ' =-.AP5 Ocri:rsxu LABC - npasum'nuft, to MrBtCt - TeJfi trpaBfirbHfir4' t LAtBtCt. LABC aHoetpiuieatovnoai6Hocri k "-. 302- Ilpone4en4o n (ABC) XY ll AC, Y e BC; n (AA:B) YY. AA.,Y.e 8.C.. To4i rpamoxyrauu XXTYTY - uryrcairnii nepepia, XXt:YYl: a. XB= nV =!a.D Is LXBY (IXBY :9O") P"&, = ,(".**)= Bionoaior,: !(s*sJi). .XY = XBE,, "" =?+ !(s.s"E). 306. Hexafi,4BCDA.B)C.D. - qacrr{Ha goutcu i AAT lpaHi-4ArBrB i DDP,C Jrer+carb y naparrefibur4 x nrro Touy nronlzna (AArD) neperfiHae rx no napaleltH muxAA, i DDr. Ana;roriqnoAD ]ArDr. Orme, y qor uut<a AATD TD nporurexui cropoHz relr{arb Ha trapaJ trpsMux, rogi AATDTD - lapaJreJrorpaM. 3O9. ABDC - napaJreJrorpam, ocHi.nrxn a.ll B, AB ll C CB : 7 cm, AD - ,h2g cv, DC : AC + 3 (cu). Hexafr AC - ,r cM, BD - x cm, ro4i AB (r: + CD (r + 3) cu. Is LBAD AD2 : AB2 + BD2 - 2AB . BD . coslABD ADz : (x -f 32 'l x2 - 2x(x + 3) coslABD, 129: x2 i 6xi9i x2 2x(x+ 3)cos1A-BD, cosZAtsD 2x' + 6x - L2O . iz LBCD BC': BD' + CD' 2BD CD cosZBDC, 49: x2 + (r + 3)'z- 2x(x + 3) cos(180'- LABD), ,.o, P,rr'", = p. Pnu", -Z' ' Pou" P*" 5 sr,r,", - ,r. So,Aq, 4 s* -"' sr"" 25' Binoeilo:24 cm i 16r,8 cu'?. 906 ::i::: TEOMETPIS Eees f. fl. ra ix. flpoQinuxnft piBeHb ii{;ii:i:iliiiiiiiiiii;i;iii 2P._^ 2.3 .20 . p^ .,. = -' 4B-!- = - --" - !{ (gx{). 55 .20'rE .q = 4s*r =o' --4 - 4ooJ5 = 16J3 (cm,). 25 25 25 ' 49: v2 + x2 + 6tc. + g +3x@ i 3)cos1ABD, cos "ro..i 2*"^+6x -^l2o - o^-zx'z lu*, ,*, + 6x -' 2r(x + 3) 2x(x + 3) 4x2 +12x 160:0; i + 3x 4O: O; x : 5, x : Orxe, BD : AC : 5 cm,.4B : 8 cwr. Bi1noei1a: BD : 5 cu, /B : 8 cu. 31O. flpone4eno n rpairiADC MN I DC, N e AC,n rp Toai LMNK - r-uyxaurfi nepepi;. AMNK .! ADCB. Ocni.nrru AM : MD : 1 : 3, ro AM : AD : L: 4. Togi xoe(birlierir noAi6- _1Hocrl rp[]ryrHuxtn h = i. ^ o'JB A Src.o. 4 On, Aea-cTopoHaplBHoCTo- ponuboro LBDC, roxi ot -25! . !5' 16 _ -16 48 a --. P.-.^ '3V3 VB Vs [{ili}::r*t;t;$i{i TEOMETPI9 Eees F. Il. ra ix. l-lp
  • 27. ls MBD (ZBAD :90") BD -- JAu AD' , BD = p*o* =ir*", = fr, o"nin * P6vNr i Paooc: k' Sourr i Saarc : h'i S*** 64=4. Bidnoeidu: cm;4 cuz, 311. no6y.qyeuo nepepio rerpae"4pa rrrlolqrrHolo (ADP)' ueft nepepia - MDP. Ilpooe- D AeMo r{epeo roury M y ',pati ADC MK ll AD' K e CD, y rpairi CDB KN ll DP, N e CB. To,4i LMKN - uyxanrii nePePia. MKMCMKl or*u ,]LCAD i o"= "c. l? =E' A **=i =5t (cm). 3a yuonoro MBC * Pinno6e4Penuft (AC : AB) i P -cepe4rana BC, 'toPi AP - rrne4iaua i nucora, CP : 8 cm. ls MPC (IAPC :90') uaeno: AP = lW - cP' , At' = lo' - g" = ar&T (cu). AcMN- rcnPi W=H H=]. ."oro =!!4 "*. ocxi.nrrcrE DC : DB, ryo LCDB - piano6e4pearafi, to4i ueAiaaa DP e i ftoro nucotoro. ts LCDP (tcPD :90') DP ='Eo" - cp'' DP = ^hr' - a' = 15 (mr)' ft'N =1. KN=lq=5 (crvr).ACKN - LCDP. roai *' , iJ 4J2r -2Bidnosidb: 5cli., cM, b - cM. 312. Ba ynonoro A'JVI : t1N, roAi MN ll Apt. Ciuna unoulrlna (MNK) neperrruae na- pa.,re.,rini rrrrolqulrn (AA.D1) i (BCC) no ilaparrerbrll4x npfiMr{x' V rpani BBtCtC npo- neger,no K? ll MN, T e BBt. MNKT - npsMotyrur'rx i uyxauuft nepqpio' MN : KT: a. B, C, flponegeuo a rpani DDTCTC qepes roq-r{rl N i K npsrri' na- pa.ne,rr'rriDC,roAiNEKF-'p"*o*y'""*iNK-iioroo,ffi.,''|u giarona,rr,. *F = ,+, *ffi" E"FR : a, "n = i? - a'. NK = i Jro r."r. P,.,", = 21a. 3'nnl = l,(s * .,{ot. 2t- Bidnoai}a: f, a(3 + V10l c:r. 313. flo6y4yeMo nrrolqrruy (BBp). Ilponegeuo n rpani A,BrCrD, MN ll BrDt, N e ArDt; r rpani AATBTB ME ll BBv E e AB; n rpani,4SCD EK ll BD, K e AD' Ilpauoxyrnnx MNKE - ulyxannfi nepepia, y fiKoro ME : NK : AAt: c. C 31 4, Ciqua nrrorqrrHa nepernnae rpaur AATBTB no AK y LABD.E,K - cepe4ra ;rinia, r:omy nX - nn, 2 ( "['-bP) /" "T^ri D -ct^L',l=2c+rla"+b..SMNKE-ME 2, Bibnoaia: z"*nE *u"; , JO:* . 2 rrlr,rHa rreperfirrae napareJrbHy rpan:o ArBrCrD, no Xn KE ll ArCu E e BrCr,.E - cepe4rana BrCr. Topi AK -yrru""ii .repepis, AI{EC - pinno6iuHa rpaneqi 3a ymoaoro AB : BC : a, rogi lC = ali, 4C, = KE - cepeAng ,rinir AArBrCr, ,o*V XA = $. flponegeuo srpar'e4iiAI{EC eracory KH, H e AC, I 16 KH LAc. Arr =AC -KE , AE = 2 E o"lid,tz - o 2 - u, 12 6 Ia MKH (1AI{K :9O') KH : AH tgtKAH, s^*={{! ru, E oJi'a.Jz + t; - , utz s*,,-----i- 4 tsa=t+ frr*=4 flxnlo o : B,z8.qM, a : 7s',nnaeuo: s -3'3'78'z- Bilnoeida: =20 Awz. 3'15. Ilo6yayeuo nepepis rcy6a n.norquHow (BMD). V rpani DD.CTC npoBeAeMo Cfi )l DM, E e DD,, E - cepeguua DDr; n rpani BBTCTC npone4euo Cf 1 FeBBr,F*cepegtnaBBr. flpama, [aparrerrr,Ha OM (O - cepe.4fiHa BD], C,A. To4i vorr,rpuxyrwraxAFCrE - myrcarruii nepepiz, A.FCT pom6, ocni.nr,r<r.r AE - ECt : FCt : AI sttt rinoreHyau nnx npsMorcyrHlrx rpzrcyruraxie M,ED, LCIED' LC, AAFB (narorr pinni ri4norigxi rcareru). :1tI.91..'"..1:.: ,if$iiiili ,il lii;i::11:::::: DA rs MED (}ADE : so") AE = JAD ED". AE = Toni P*","=^ +=zal|. s-c,,= !7,c,.rr, pe Ac!-8a2, ACr=oJs, rn=nD=ali; s*",, Bi|noeiiu: 2aJ8: o" JB 2 908 iii: TEOMETPIS 6eee F. fl. ra ix. rlpoQinuxui piBeHb i:.*;1i!ii;iiiiiii!iiii tliiiiliiilii..".*li: TEOMETPIS sees F. Il. ra in. Ilpo
  • 28. lld,'.r....r: ' : lf!,1':.:.i:.1 i;;I'iilii: ':;li::i{::::: 316. Ilo6y,4yeMo nepepis rcy6a trrorlunovr (AB$). Hexafi O roqna treperr{Hy 4iaro- Hareit ocHoBtr Ry6a. flpoBeAeMo B rlJrorrluHi BBID|D Do,ll OBt, Ae Or - roqrta neperrluy AtCt i BrDr. IIIyxa- a Hufi nepepis npoxoAurb qepeo r. Or. Ilpaua ArCr myxa- -^1 noro nepepisy rapa,lerrbHa npauiii AC nepepisy (-A-BtC). ToAi nvHanufi nepepia AArCrD, cropoHr{ sroro Aia- roHa,:ri xy6a. Toai D = DCr = 4C, = oJi, P^,",o = 3aJ2. Ocni.nsxu LA.CD lPanu.nlHuii, A ' -2 - ^ Grh)',h a'J3 TO b.,..r= 4 =- ^z lo Bidnoeidu: zorli: " f . $ 9. Ilapanenuxe npoerryBaHHc i froro enacrlreocri cl 332. a)aei npami, rqo fleperunarorbcff, a6o ogua npaua; 6) Aei napare,'rrHi np.aui, a6o o4Ha fipsMa, a6o '4ei rouxu; n) npama i roqxa, a6o 4ni npnmi, qo treperuHarorrcs, a6o 4si napaae'nr'ni npeui' 333. flapane;rbne npoercryBanHfi He s6epirae Belauqrrnr nyrin y npoeropi, roMy rrpo- eKqieIo trp.ffMoro Kyra Mo)+(e 6yrw 6yatattuii xyt, a6o npouinr, a6o npaua' 334. a)Tpu rorrxu rextarb ua o4Hiii npxmiii; 6) rpu rouru He Jrexcarb ua o4Hifi npauiit; n) rpz rovxlr, u{o JretxaTb na o4Hiii nparrifi, a6o rpu Totl}tl1, lqo He IeJIcaTb na ogHifi npauifi; r) rpr rounu He ,re)*curb na o4nifi npeltill. 335. Pos6zeur.r MHororcyTHtrlr Ha rprrnyTHr{Kr4, AOBO,qnrn na OCsOni AOBeAeHIIs' nOAa- noro s aa4aui I sa c. 90 niapyvHnxa. 336. Ocxi.nrHrz npoerqii trp.flMrrx napa.nemni (AB1ll Cp), rc AB i CD - uumo6i:+tni npaul, orrxe, BoHn He neperrrHarorbcfi. 337. flapale,rbHe rrpoeKryBaHHn a6epirac ei.nHolueuHa AoBlr{r4H napa,re,rsultx si4pia- xin, torvry napaJlerr'HI4MZ npoerqialru cepeAZH cropiu rpunyrnfi:r'a ABC 6y4yr:r ce- pennHtt cropiu iioro napareJrr,Hoi npoexqii LAlBrCr. Iloasaqre cepeguuu cropiu LAlBCr i nponegirs cepegHi .niHii i ueAianu. Boiur 6yAyrr napaJIeJIbHI{Mtr llpoel{- qinuz cepe4nix nisifi i meAian AA-BC. 910 ::l::i: rEOMETPIS Seee f. Il. ra ix. tlpoQinunui piBeHb ;i;i:i::;:i;iiii::i;iiiilii;i:i 339. rtnagpari ABCD 4 oci czuerpii, nci npoxo4arr roHarei;i. .II,si oci npoxo4ffTr, qepe3 giaroua,ri KBaAp rreHAunyJTspu Ao cropiH rcBaA- para. I{i gei oci naparemni cro- poHaM r{Ba,Apara. OrrHe, e napare,rorparriArBrCrD, npoBoAuMo AB1 np,fiMr qepes aiaroHa"ri naparelorpama i ani npsmi vepea rovxy O, - Toqrry neperuuy 4iaroua.nefi naparem- HO CTOpOHaM [apaJreJrorpaMa. Oaep;HaHi npnmi - napa.ne.nrHi npoexuii ocefi cuu 34O. IIi npxmi npoxogrrb rrepe: roqxy Or, e nny trpoenTyerbcfi ro*rHa O ne- peruuy aiaroHaleri i napaaersri cro- poHaM napaJrerorpaMa. 341 . ilapa,re,rsuoo npoexqicto ve4iaur BM 6yrc ue4iana BrM,. 342. 343. Hexafi y pon6i ABCD M.e BC, MN I BD, BD LAC, MK LAC. BO IAC, MK LAC, , roti Bo ll MK. AHaroriqno MN ) BD, .. rogi MN il AC. tlepeo rouxy M npoeegemo nprvi, napa,re,rsai 4iaroHarsnr naparTeJro- fpaMa. 344. aY xnasparaABCD PN LCD, PK LAD. V npoexqii xea4pa- B Ta - napaJrer'rorpalraArBrCrD, MNt BIC), MKllCpt. 6) PT LAC, PS LBD. V npoenrlii HBaApara trapare- ;rorpamiA,B,C, Dr M S1 | Apt, Mr1llBp1. A 346. y pieno6iunoi rpaneqii Brcorr, npoeegeui a BepuHH rynrx xyrin, Bi.[TlrHatorb eia ocHoeu pieHi siApisxu. Topi y ArBrCrD sttcoru BrH t C$ (BtH CrK ni4rnnarors piaHi eigpiaxu A1H i D1K, Or:ne, B,If - uyr{aua Bucora. flapa.ne.nrnoro npoenqiero .npauoi, He napa!'reJrbuoi uanpsuHy rrpoexTyBaHus' e trp,flMa. flapaneasHoro npoeHqiero eigpiarca, He napaJrerbnoro HarrpsMKy trpoeHTyBaHHfi' e nigpiaox. flaparersnoro npoexqiero uoilItrHu e rr,aou{uHa a6o npsMa. Ilapane.nsHoro npoex4ieio npauoi e npxlra a6o roura. IIapare.nsHoro npoeHqierc ei4piara e ni4piaox a6o roqna. laianenrnoro npoexqiero ABox trapaJrerbHrlx [pflMr'rx e 4ni napa'relsni npani a6o o4na npama, a6o gei rovxn. flapa,reruntto npoexqiero ABox trpsMlzx' arci neperr'rxaror:rca, e gni lpaui, aHi treperfiHarorBcs, a6o o4ua nPaua. fla-pa,rersHoro rpoerqiero 4sox nzuo6irxHux rrpflMux e 4ni npami, axi neperr- Harorbc.n, a6o npaua i rovra, flrta He Jlextrrrb Ha Hta. M 7D iil{iil::{:::::ir::::il: TEOMETPI9 Eees f. Il. ra ix. Ilp
  • 29. S4E,ynpflMoKyrHoroMBC(lC:90')qeHrporrr{caHoroKoJra-cepeAr4Harrrrore- Hy3q - ro.rna M. Pagiyc nnr'rcanoro 2S KoJra r= ua.AC+AB' BC:6crvr,AC:8cu, toAi AB : 10 cu, 4A r=-=2 (cMl. 24 Pa4iyc nnlrcaxoro r{oJra craHo- r"t" lBc ,u Loc 34 i pa.qiyctr, nponegeni B rorrxy .qorr{Ky ai croponauu rpr4KyrHrrHa' rrepreu,qurcyJrspni Ao c'lopin. M, - cepeguaa AtBt. 352. a)flpoexqis Ha (,4-BC) n uanpnui PAr. B - r. B A r:. P * r. A, roAi npoerqia LPBC - LABC. Ssc : @ 6) flpoen4ia u,a(ABC) y nanpanni PM (M - cepeArrH AB) r. B * T. B, r. C * r. C, r. P - r. M, roAi np enqis APBC - LMBC. Ocxiruxu nre4iana tp"*yr"" xa airuru r?ono na 4ea pinnone.nr-rxi rplrxyrnrznlr, r 1 St"r" = o @' -1 Bidnoaida: a) @; 6) :Q.2 IIa npoercqii A1B1C' qenTp orrr4caHoro RoJra r crx, = tcrnu cru, = lcr'q, KptllA{' Horll Brcl, or - 4eurp BrrtrcaHoro Korra' 349, Hexafi'rouxuK, M,N lentarrnanpsuifi aiM ten<]f.;rtuixl( i N'KM:4cM' MN :6 cm. ix napa,renrui nPo- erdqii K' Mr, Nr .nexarr Y l.no- quni cr na npartift b i Mt :le- rxtrrr, uirrt K1 i Ni. rr, ll MM, ll NN,. 3a yaara.nrneHoro reopeMoro KM MN @a'neca &l4= MrN, Vpaxonyroun, rqo NrK, : KrMt+ MrNb rvraeuo: KtMt: Nt(t - MtNt' roAi 46 = -, aergxu 4MrNr: 30 - 6MrNr , lOMrNl: 30, MrNr : 3 crra' 5 - MrN, MrN, rogiKrMr:5 - 3:2 (cu). flxnlo ro.mu K, M , N poaMiqeui na npamift 4 rarc, rqo K 'nexcurr rvrirx M i N' ro 364.,il,iaroua.ni napa,re.norpama treperuHatorbcs i roqK TOMy OAHy 13 3aAaHU X TOqOK BBaJfiaeMO TOqKOTO nepe ny 4iarona.nefi, EanplnJraA, rouxy Ar, BHaxoAuMo ro Kr.r, crrMerprrrrui 4norvr iirururr rotrr{aM eignocno Ar. T xa An - craMerplrqna r. A' r. A" - rovqi A3, A4 i Ab Beprfir,rnrr [apaJreJrorpaMa. AAAA, - tryxaue eo6 lr(eunfi napaJreJ'rorpaMa. 3a4ava uae 3 poae'aa ocriffiru xoxcsy a :rovox Ar, 421.A3 MoxHa BBaxarrI T rcorc nepernHy 4iarona.nefi . 365. Hexafi A, - npoexrliff Beplrrruri rprrxyrurrxa. rovqi A, ri4nocno cepe4ran cropiH A, i A, - ue niAnon uo rovrcu A. i Au. To4i MAA" - rnyrcaHe ao6parnen TprrlcyrH[Ka. Sagaua mae 3'posn'saxu, ocni:rr,xa xorx s roqoR A, ,4.2, Aa MolrcHa BEalrcarr,{ Beprrlr4Horo rpr4 MtKt:10 cu; M,Nr : MrK, + If1Nl, MtNt:10 + 5: 15 (cu)' Binoeia: MrNt : 3 cw, MrKt: 2 cv a6o MrNt: Lb ct.r'' MrKt: 10 crt' 351 . a) flpoenuiero M-B,C tta (AtsCl n Hanpsmi BtB e LABC' 1" 5..^^=-a". 2 5) IIpu npoexryranni na (AArBr) n nanpsrvri ArC, uaeivro: npoexqis neprunuuA - t. A, nepurunu B, - r.8,, neprura- sm C - r. A. Orrlce, npoexqix MBP - rigpieox.4Bt' n) Ilpoerqia ria (BB'C,) r xanparri ACrrouxu A - r. Cr, t. Br - r. Br, r' C - ,- Cr, LABP * LBICP' 1" S*,"," = 2a ' MK - . rl,':MN-MK,KN: MrK, K,N, 42 6 - 4 - 2 (cu). roai L,TE= b, H[RA. 366. y pinnocroponnroro rprrxyrHtrra rrre4iaur e Brrco uu, rogi 6yAr-sxi nep[eHArznyr'rfipz 4o cropin qro- ro rpr4KyrHHKa 6y4yrr naparerbHrzMn me.qiauam. Ha oo6pancensi L,ABC - n L,ArBrC, npoBeAeMo uegianu AF, Cp, BrIl. {epes nnyrpiurHro rouny M LArBrCt [poBeAeMo npaui, napa.nentui CrE, ArF, B1K. IIi npsui - uyxani ao6pa:rceuna rip,nMrzx. cr A1 967. y pinnocroponnroro rpr4xyrur4na qeHTpoM Brrr{ca rca neperr,rny iioro rregian. I-da rouxa no4irre xoxuy paxyroqu ni4 neprrranr.r. Toruy Ha napa,re.nr,niit npoex urz, Toqra ix uepeuaNy - rIeHTp Brrr4cauoro r{oJra. A A 368. Cepe.qunni nepnen4rzxyrspu Ao narerin npa- MOKyTHOTO Tpr,rnyTHr4Ka napa;rear,ni xarerav i e cepe4nimu aiirislru rpr,rxyrufiKa, BoHu neperu- Harorbc.r n rouqi, sxa e cepe,qnHoro rinorenyar.r.Bi}noaida: aj D1 a2 -. a2 _: tll -22 j I f i I g 10. 3o6paxeHuq Oiryp y crepeoMerpn- So6paxeana dliryp y npocropi: o rpnxyrHlrn ao6parxyet ca rprrr(yrHr,rxoM; . napaJreJrorpaM, KBa,qpar, np,aMoriyrHurc, pou6 so6 MOM; r rpaneui.s ao6parxyemca rpaneaiem: . rco.no so6parr(yerrca e.nincom; . rrae,4ianz rprrxyrnr4ra ao6parxyrorrcn ue4iauauu fio g12 iii;l reOrvrerplg Eees F. tl. ra ix. Ilpo$inuxufi piBeHb ii;i*:':i:fi;:i{iir:+ i,:Lii::i{iii:*ii;:: TEOMETPIS Eees f. tl. ra in. Ilpo
  • 30. M - cepeguuu C' B', N - cepenuna A' C', O - cepe4uua A' B'. NO, MO - cepeguuni neptrenAlrr(yJrspu go A'C' i C?'. Ocxilrnrz aa yuororo A'C' : C'B' , To CO - megiaua i nucora, ro6ro cepegrannufi nepneH4urynsrp Ao {B'- 369. IleHrpom onrrcaHoto noJra e roqrca nepertruy cepeAlruH[x nepnen4zxy,'Iapin 4o cropiu rpnxyrnzxa. flnrqo BrKtiCPr - ao6parxeHHa Bucor rpzr(yrHrarca, eHail4elro cepeArrHrr Mt i Nt vlg' noei4uo cropiH ArC, i AB1 i trpoBeAeMo rrepel Hux nps- rri, napare.nr,ni ni4noni4no BjKl i Cf 1^ Toqna O ix ne- peTuHy e rIeHTpoM otrzcaHoro r(oJla. 37O. I]enrpou BnlacaHoro RoJra e roqxa neperuHy 6icexr- puc rp[KyrHIKa. Itorxsa 6icenrprrca rp[nyruzxa 4i.nutr nporuJTelrcny cropoHy ira ri4piawr, npolopqifini 1'. 375. a)I{eurp onficanoro HaBKoJro [paBrzJrbnoro rpfi]ryrH MeAraH Tpur{yTH[rta, fifi4 noAlJrfle Ko]r{Hy MeAlaHy y messi 2 : 1, paxyroqr ei4 nepmzuz. Ha ao6pameHn canoro lcoJra B qenrpoM O, npose4euo giauerp BrDr; OrD, nogiazmo Haenir roqnoro -Ilr, gepee fl, npon np.ffMy, sna so6panrye nep[eHAr4xyJrspHy np,flMy go Boua neperuuae rcoro B rounax ,4., i C, nxi e so6pan Mu BepuuH Btr[caHoro rpltnyruzna. LAIB.C! - ao6p H.fi npaBLJ'IbHO|O Tp[KyTHUKa, BnZCaHOTO B KOJTO. 6) Ha ao6pa:xeHui ArBrC, npaBuJrbHoro rpnr{yrH[Ra npoBeAeMo megianra BlMr j,ArNr, rrprrJTerr[M cropoHaM. flrclqo n npfiMol{yrHono - LABC BC :AB: 4: 5,roAC : BC :AB: K Toqna ix nCiperuny O, - qenrp Bnvcanoro rcoJra. 376. I{eHrp KoJraf onncanoro HaBxor'ro npsMoxyrno- 4 ro rplxyrHr4na, - cepeAlHa rinorenyez. Ha so6- ; ft;tiili:;iii: : 3 : 4: 5. BiAuirumo sa C'B' rouxY -L raxY, qo C'L : LB' : 3 i 5, a Ha A'C' rouxY K ranY, n1o C'K : KA' :4 : 5. flponegeMo Bo6paficeul{s 6icenrpxc B'K i /L. Toqxa O ix neperrny e ao6paxennalr qeHrpa BtrucaHoro Ko,aa. 37 1, Hexaft ,4 B' C' - zo6patxeuHfi rpzltyrHrrxa ABC. Kiueur 6icenrpucu AL noginae cropony n raromy ni4- nonesni: CL : LB : 3 t 2, ocHi.nrxn CA : AB : 3 : 2. Touy ni4rtirurr o na C'B' rovxy -L' raxy, rqo C'L' : L'B' : : 3 : 2, ron A'L' - eo6pa:*ceuHa 6icercr:pucu AI. 372, Hexaft A'B'C'D' - Bo6pa)fieun,r pieno6e4peHoi rpaneqii ABCD, y arcoi AD - 2BC, roti A'D' : 2B'C', K' - cepepuna A'D'. Ocnint't<t rpaneqin ABCD pinnoleApena, ro AB : CD: : CK, roti CH - nmcora i uegiana LKCD. Bi'p- vrirr.rmo rovny fI'ramy, Iro K'H' : H'D'. rop,i C'H' so6par*cennanucorzrpaneqii.Ilpoeegeuo B'M'llC'H', B'M' - ao6paxerua 4pyroi srcoru rpaneqii. Bi4uirramo cepeAr4r{r{ 6iunux cropin A'B' i C'D' ei,qnoeigHo roqfir4 L' i E', toai L'E' - so6- pa?+ceHHfl cepe,arsoi rinii rpaneuii. 374. a) Bepurv nu ArBrC rD, Bo6palxenHs BnI{caHo- ro KBaApara - Hisqi BBaeMHo neptreHAunyJrfip- Hux giamerpin AlC1 i B1D, otrucanoto Rora 3 IIeH- rpov O,. A' 6)a) 6) Ha eo6paxtenuiArBrCrD, RBa"4para cepeAanrz ctopiH Mr ND L, K1 xixqi ssaeu- Ho rrepneuAuldyrflpHnx 4iamerpis BnzcaHoro HoJTa 3 qeHTpoM O,. Tovnu My Nv Lv Kr - rovxra aorilIty nora 4o cropiu. s14 ii{: reovrerple Eees r. rl. ra in. flpoQinsxnfi piBeHb lifi*i:{fiitiiiiii paxceHHi RoJra s rleuapom O, npone4el,ro ,qiaMerp, H sxoro A, i B, - eepuuu[ rpunyrHu(a. By4r-axa r C, na xo.ni (ei4uinua ni4 A, i Br) - rpers Bepuuna rcyrHura; lcl:90'. 377, IJ.a Bo6paE{eHHi Hora nlonegirr 6y4r-sxi gna gia ix xiuqi. O4epxhHufi qorlpuxyrxtrK - eo6pa:xeuu B KOJ'IO. 379, Hexaii y rp,rrMoxyrHzi4 LABC (lC : 90") BntrcaH CEDF, CD - 4iarona.nr, KBaApar, CD - 6icexrpuca l . rprrca no4inae rinoreuyay na ni4pisHra, nponopqifin " JrrrM ftareraM. IIIo6 nnzcarz KBaApar, Tpe6a BHafiT[ rinotenysi. rxa nogirnc rinoreHyuy B TaKoMy sigHo nplr:rer,ri Karern, i vepes aHaft4eHy rorrKy rrpoBecrrr n nyJrspu Ao xarerie, Rxi napa-ne.nuri nporrz,rerxnzlr K a) Ocniasxu xarerfi TpuxyrHuna pinni, 6icexrpr{ca nps .nogi;ise rinorenyay sasnir. Ha eo6paxeHHt LABlC1 a. M, - cepeAr.ruy ArCr. {€pea M, rpone4emo MN1 MJt ] AS1. CtNlMlLr so6par+cenHa BnncaHoro r{B cl Ll Br a) c1 6) Hexafi CrAr: CrBr: 3 : 5, rogr ArEr: ErBr: 3 CrFEtK - 3o6pa;fieuHfl BlucaHoro fiBaApara. n) Ocxi.nsxlr rinorenyaa ni4uocurrca x 13 : 5, ro r{a flxrqo CrB, i CAr: 72: 5, ro B1E : AtEt: L2: CrFrErK Bo6pax{eHH,fl BnzcaHoro KBaApara. 381. flrrqo y pou6a rocrpzii rcyr 60", ro Menrrra giaron poui. TpuryrnzH, yreopenufi ABoMa cropoHalru i ue Pouniit, roAi: iiii;:liii{ilil.:;i{j TEOMETPI9 Eeee r. l-1. ra ix. Ilpo 6)
  • 31. a) noAirirb nporrrJrerxHy AJrs ryfioro r{yra cropouy poM6a HaBnir i cno.nyuirr ii cepeAr{Hy a BepilrlHoro ryrroro rcyra. Ana.nori.ruo no6y4yftte eo6parxenna Bttcortr, nporegenoi a 4pyroi BeprlrrHrz ryrroro Kyra; 6) no6y,qyiire ao6pancennr Brrcoru, npone4euoi 3 Beprxrrull ryloro Kyra, IlposeAirr rrepeg ilport4Jlelrcny cropoHy nparvry fi onycrirr na nei nucory 3 Beprfllrnr'I locrporo Kyra rrapaJreJrbno nrcori, npone4enift s BeprrrftIrI4 Tyr]oto Kyra. 382. Ha ao6paxenni LAlBrC r rpaxyrnuna ABC BrCr: ArBr; M, - cepeguaaocnonnArCr, BrM, - ineAiana i 6i- ce(rpr4ca; N e ArB, raxa, rqoArNr : NrB, : 2 : 3; CrN, ..._ 6icenrpr,rca; BrM, a CtNt: Ot - qesrp BIIfiCAHOTO KOJIA. 383. Y MBC AB: BC. Ha ao6paxexni ABLC, AtH, - Bracora, AtHl IClBt.I{eutp onrzcauoro HoJra - Toqra ne- peru Hy cepe.rr HH trx neprreH.4l4 xy,rapin 4o cropi H rprzlcyrHrrna. M, - cepe4Iana ArCr, BrM, - ce- pegunnuft nepneIrAr4KyJrsP) Lt - cepeAr4lra CrB1. rlepea -L, [poBeAeMo np.fiMy, rrapaJrer's:r'y ArHr, Toqxa ii [eperuHy s MrB, - tovxa Or - qeHTp onvcanoto noJra' 385. 3nafi.qeMo cepeAr.rny ArC, - rouxy Mr; OrM, - cepe4Lrnuiaft Ilpoee4emo nucory Brflr ll OtMr, SnafiAeuo cepe,qr{Hy BrC, - rouxy Kr; OrKt - cepe4nnnrafi nepnenAr{r(yJrsp. flpone4eno nzcory ArN, ll OrK,. gepee roqry neperr.rHy ao6paxennn orzcor i repurrany Cr npore4emo nucory CrEr. At Bo6paxenna nr.rcor ArNr, C.ED B.HL. 389. flrcrqo r - rcoe@iilierr nponop4iftuocri, no ctopo- Hr4 npflMor(yrH v:rca x i: .fsr. r:ogi giarona.ri J*+ s*" = z*, a troJroBr,rHa Alaro- 81 CrHaJtr - f . To4i B*4, : BrOt, LATBTOL - pirno6e,4penzft, BrM, * irne4iana i nzcor:a, onxe, BtMt L AtCr. 394. y Brrrrcasoro raxrrM trr,rHoM potl6a Aei cropoHla JreJ*arb Ha cropouax rpr4rcyrHl.rra, 4ni innri - na- A, pa.ne.nr,ui iu. To4i: na B,C, snaxogrirno rourcy Mr - xineqr 6icerrpzcr.r xytaAr. ArMt 6y,qe giarona-n.rro pom6a, ocxi;rxz ftoro giaroua;ri e 6icenrpucauz ftoro rcyrie . rlepee 'rovxy Mr npooogramo npaui M$1i]A$l (K, e ArBr)i MlLtllAlBt (Ll e Ap). ALKTMTL. -' snucauuir pom6. flo.noxcelrH,E rotlxr{ M, ruanavaeuo rari: a) AB: AC, rcaiAB: AC: l: l, orxe, BrM, : MtCt: | : l; a 6) AB:ZAC,rotiAB:AC- 2:1, "r orxte,B,M,:MrCr:2:li n) AB : AC : 3 : 5, rogi BrMr t MrCt : 3 : 5. 395. Hexaft A lBtC D lE rF r - so6paxenna nparuJrbnoro [recrzrcyrrlr{Ha ABCDE F. a) Ocxi.rrrru giaroua.ni CF iAE [eprren.qrrny,rlfipni, ro npone4euo ArE, i CtFt, arci neperuyrbc,fl n rouqi Kb ArKt - rrepireHArixyr,trp 3 rotlx.r4 A, ua CrFr. 916 ii:l reOuErpts sees F. Il. ra ix. IlpoQinuxxr piBeHb iiilfi:ii::;:ii*iltiii 6) Ocxi;rucra AC LCD, ro nponeAeuo A1C1, A1C1 DCl n) Ocxilrxn FB L BC, ro qepea rouxy A, npoBeAeMo neperr,ruy a rrpoAoBJneHHsM cropoHn BtCv ALHrli Fr BrH, - uepuerrArrxyJr,fip a -4t go BrCr. S 11. Merogn no6yAoBl,r nepepi3iB MHororpaHH A.nnopurn no6y4orrn nepepisy MHororpaHHrarca rrroulr{H 3 roqnra Ha norepxni MHororpaHHI,rRa: 1)O6upaemo nafi6i.nrur spyrrHy rpaub MsororpaHHr.r(a,q.n c.niga ciunoi [,'rorlr4Hrr. Haft6i.nr,m spyquorc 6yAe ra rpa onycrlrrr,r npoenqiio rtox:roi is ea4anux rovorc. 2)Ilpoercryemo xoxny ia Ba,qacnx rotlot{, HafipuHJraA, H Icnye 4ea Br4Aa fipoexTyBaHHs: qeHTpaJ'rrue i napa.nenu .Barrna BacrocoByrorb BasBuqafi A.[a no6yAosrz nepepis .TyBaHH,fi - aepuraHa niparri4x, flapalenrne npoeKTy no6y4onu nepepiain npnart. 3)BnaxoAr.r:uo roqrcy nepernHy npartoi, qo npoxoArlrb s rrpfMoro, qo fipoxo,4nrr,.{epes Asi rouru npoerrlii. O c.niAy ciqrloi nJrorqrrHr.r Ha nJrorrlulry ocrioBz. 4)Briaxo,qrzuo roqKr4 fieperrlHy pe6ep Muororpanrrr4Ka B .repes gaAagi rourcu rpanefi, i c:riA ci.rsoi nJrorq[El{ na 5)3'equyer,ro flofiapHo gnaft&ini rorr(rr Ha pe6pax rano cl Br .'._ [epnerrAr4rcynap e A, 4o C KeAD. ,,ti,.ja>:.1!i:.: 402. Hexaii M e BBL, Ne CCr , fJpone4err.ro npnmy MN 4o ne- perrauy s npsMoro BC, rraeuo ,ro.rrey X. V n.nouluui (ABC) e 2 to'rxra ciuaoi n.noulnuu K i X, ropi qi u,roquHu fieperrrr{alortcE no npsnifr KX; KX^DC: E, XK^DA: K, XK^BA:Y. flposeAervro YM, YM ^AAt: L. LMNEK - uryxanlEii nepepia. 4O3. Hexafi Me AAr, Ke EEb Ne CCr. MK ^AE : X; MN AAC : Y; iii:i:::i:iii:i;ii:i:iiii:i: TEOMETPI9 Eees f. I-1. ra in. IlpoS IIepIIeHAnrcyJr,trp.
  • 32. XYnED:F; XY^ C, : S. SN aCrDr: Z; MK aArEr: W; WZ ^ ArBr: Ri WZn CrBr:7. KMRTNSF -llyxanufi nepepie . 4O4. MN ^.AC : X. s 405. a) MN ^ CtD : X; RX n CrB, : K; RX a YN^AA- E. Orxe, ERKMN - t:tyxauufr nepepia. 406. n) Cnpoercryeuo rouny 1( y HatrpflMRy CC, Ha Huxuro i nepxHro ocHoBu npusMr,r. Kt ii npoex4i,s na (ABC), KreCD; K, - Ii npoexqi.a na (ArBrCr), K, e (A,BrCr), K1K, ll C1C. Cnpoexryeuo rovny -L y Hanpnmxy ,4.4,: 4O7. Cnpoerctyeuo rouHy 7 Ha Hr.rr+cEro ocHoBy y HanpgMKy S7. Maeuo ro.nty K, K e DC, A roulcy R y nanpamny S.B, oAeprfiuMo rovHy A. AK ^ RT - Y. Cnpoerryc- Mo roqrty P y sanp,sn{Hy SP, oAep?ffhMo rouxy M. MAoPR: X. XY^AD: :E,XYaCD:F; FToSC:Q;QPaSB:O. Orxe, ROQFE - trynanzfi nepepia. Ilpoee4euo qepeo rorrxy P npsuy SO, AeED; MPoAO:Y; XYnCD: K, XY.ED - F. Ilpone4emo qepe3 roqlcy P npamy FP, FP ^ SE : T; XY aAB : Z; MZ nSB : @, Toti MTFKNQ urynaunfi nepepie. AtDt AD y z7- Lt - ii npoerqix ua (ABC), LreAD; L2 ii npoexilis ua (ArBrCr), L"eA,D,, LJ2)lAAr. Ilpama IN relrcurb y uon4zni (LrLrBr) ine- peruHae Bepxnrc ocHoBy npn3Mu n rouqi X, X : LzBl^ LN. Ilpaua ,LK Jre)+(urb y n.norquui (LrLrKr) i ne- pernHae BepxHro ocHoBy nprraMu e rovqi Y, Y : L2K2^ LK. XY ^ AlDt: F, XY ^ C1D: E. LFr-t AAr:S,EK^CC:7. Toti SFEI"N - uyrcawuir nepepia. ra iH. npo0inbHrr piBeHb l*iti{ftill:li:{::* 3a rra.nrosxom 159 6). y xy6aAtsCDAtBrCrDr nparvra MN neperunaernca e CD n rouqi X; XP a BC : E, XP ^ AD : F; MN o DDt : Y, FY n AtD' : S. Onxe, V ny6a ABCDAT Haerbcs a CC, n MX^A1Bt: S, PE o DDr: Z; Y Orxte, OSMEPF CnpoeHryemo routty M y Hanpaltxy SM, oAepJ+(uMo rouxy 7, T eAD, BT a MP : X; XK aAD : E; EMaAS:F. Orxe, EFPK - uyHaazfi nePePia. 3a rlra;rronronn 160 6). Y nipaivri,quS- BCDE Z MN aAE : X; MPaEC:Y; XY aED:F; XY oCD :K; EC aAB : Zt 4O8. 3a ua;rrouHou 159 a). FSNME - ruyxanuii nepepis. 918 iii: TEOMETP|g Eees F. n. ll{itiiiifiiliiiititi rEoMETPle Eees I- Il. ra ix. IlpoQ
  • 33. ZNaSB-T. Orlreie, MNTPKF - nryranufi 409, Cnpoercryemo tourcy P na HrrrfiHro ocHoBy y HarrpsM- xy SP, o,qepxl,rMo roqry P' PreAB; crpoeftryeMo rovxy -E y r:arrpnmrcy S-R na (ll3C), o,qeprrcnuo rovry -E, Pp nPR: X; XQAED: K, XQABC:M,XQABA:Y; PY^SB:T,PY^SA:O. Ort<e, KROTM * ruyxanrii nePePis. 41O. CnpoexryeMo roqxy .L y nalpamxy.4At Ha Hrr)$rlro ocuoBy, oAept*tl{Mo tovriy -Lt, LreAD; ctrpoer{ryeMo rouxy K y sa[pJrMxy AA, na nuxnro ocnoBy, oAeprcr,IMo rouny Kr, KreAB; LrK,-,CD: X; XM^CCI: P' XM ^DDl: E; ELAAA,: Q; ' OK^BBr:T. Orxe, O?PE - lryxauuft nepepia. 412, flposegeNlo n (BCS) ONIIBK' C 3a[aqi 3a roroBr4Mu MantoHKaMt,t Ao po3Airry 3 BapiaHr A 1. v LABC MN llAC, MB : AB:8 : 14. L,MBN * aABC, ro,qi Mx =12'8 = a! = g9. Bionoeioa: MN =B!.'14 t4 7 7 2, ABBAT * tpaneqia B octroBaMra AAti BBy flposeAirr, BAzll BA, Are AA1, rogi ABBTA, - rrapaJreJlorpau, y aroro-4A, : BBr: 4. BrA, a MMr: Mr. V napa;relo- 92o {ii$ TEOMETPIe 6ees r. I'1. ra ix. IlpoQinuxrft piBeHb iiitttl:.t{,rtitiiil'r rpava M BB rM, M M " : 4. PosrrgHeMo AA2B 41 i LM ZM|MtBt: )AABI grc ni,4noniAni rlp]/ AArl) tWrlW ro4i LArBrA, a A,MrBrM, i M'M' = M'B' . ArA" 4Br' 3a ymoooro -Al'11 : MB : 3 : 2, rogi MrA, : MrB, - 3 A,Ar: AA, A$.t, AtA) : 14 - 4- 10. Orxte, ^[# tvtw,--MMrfMrMr;'MMr-414=8. 10 Bi4noei0a: 8. Ilpznycrz*ro, u1o c i d ae uuuo6ixni, ro4i noHz a6o ra i uepes npalri b, c, dlwoxtlr.a upoBecrr{ eAuuy rrJ'rorrl cL, To Bci cni.nrni Toqr{r4 rlux flpaMrrx B rrJrorqr,ruoro cL a"ire ga yMoBoro.Toqxn neperrzHy npsMrrx b, c i d, s n nifi npnmiit, orlrce. Harte nputryuleuHff HeflpaBr,rlbHe ft rpe6a 6y,ro 4oaecrz. PK, PN, MN, KM - cepe4ni ,riuii ni4nori4uo AAB KM llADll PN i KM = PN = +, MN )) BC ll KP I22 Toai KPMN - raparrerrorpau, P."*" : 2(NP + MN NP + MN: 13. lup- twN =s, lzNp=i6, lNp=s,lvlg€Mo: { i { INP + MN = 13; lzMN = 10; IMN = 5. To,qi BC : 2MN, BC : LO; AD: 2BC, AD : 16. .. Bi1noai1a: AD : L6, BC : lO. 5. Bi4ouo, E{o rrJrorqfiHa, fiHa npoxo,qr4Tr, qepes [prrMy, Hae qro [Jrorqn ny no nprmift , napa,re.nr,uifi 4ariift upar CD ll AB.3a yuonoro AC Il BD, rofi AnnC - napa.ire. AC: BD:17. To,qi PBD.:2(CD + BD), SBDC:2.( .paJrerorpaMa poa6ranae fiono na 2 pionone,runi Tpr4r( v LACD AC - t7, AD - 2(i" CD - 25, l. p^.^ = 94, 2 ^'" ro4i S*," = 2$4@4-i6,ltsa- 2ilGa-17) = 2h4. Bidnoeilu: PBN: 84, Srrr" : 408. 6. flpz napa;reJrbHoMy npoetrynanni y nanparri.4A, napa B rrapaJ'reJrorpa*r ArBrCrDr, ocxi.nrxr.r nigpiexrl EapaJ cs B rrapaJreJrbHi ei4pisrci.r. 3uafiAirr, roury O fleperrru ABCDI rouxyO, neperrairygiarona.neftnapa.re,rorpam si4rori4nux giarona"reft, ro4i ni4pieon OO, - cepegn neLti A,ACC,.Ia rpanerlii.A,A CCr: oo, = * -nt"' , BIBCC. oo,="u'loot, ani4xra.BB, :2o;,- DD,2 BidnoeiOa: 11. 7, Ocxinsuu ABll DC, ro ABll (SrC), rogi civna n,ronlu rreperuuae (SDC) no npauifi, lapa"re.nrnift-4-8, orr+ce, [ my KM, M e SD. Tpaneqin ABKM * urynanrafi nepe SC, ro.qi M * cepe4ana SD i MK - cepeAun .ninis iiiilii:i;:{iiii:i;:i: TEOMETPIS Eeee r. l-1. ra ix. IlpoQ 3. 4, r ;il,:.:.:.!t:.: OeCS; ON aBC : X; MX^BA: T. ' To4i MONI - uryxanr.rft nePe- pia, y nlolluni snoro Jlencarb rosxuMiNinpamaON, oN ll BK, a orrfie, i nrorqnna nepepiay na- paJreJrbua nparrifi BK. MNMBMNS AC AB. 72 14
  • 34. KM : !DC.KM:6. 2 Hexafi -L ..._ cepeArHa AB, Nt cepepuua DC; SN, nepernuae MK e rot4i L, i L, cepe.qraua MK. Posr.nstsemo A SON', SO eucora nipaui- Av, O - Toqr{a treperr.rny 4iaronaaefi ii ocHoeu. Ilpone4eno LN I(ABC), rosi SOll LrN i N -1 cepeAilHa ON' LrN - ; "O, oN: I oN,aeoru,- ] rlu,-tt.2'"'2 rs A osc ( t soc: 90') so : J sc' - oc' , sc -t2,oc - r ac: MJ2 -oJz;so,lT44J2 -6Jr.22 To4i,L,N : snlr.h L LL.N (lLNLI: go") LLt: ,lrN" + 4N2 ,LN : g, LLr: ^l8r+72:.[bB. s*o,: ! "rr(*+KM),ssrw:9Jib3 -..os. Bidnoeida: g JtSe .". oq. 8. flpoee4euo qepea rorrxy M npxuy, napaaelrHy AC; BoHa rreperr4nae CrC n ro.rqi K, rK - cepe4una CCr. Ciuna n.nortruHa neperuHae rpaws AArN D no N M, a rpanr D.ly'CrC no NK. DC n NI{: X, AD a MN : Y, XY nepernHae rpaus ABCD n rovqi B. Orxce, MNKB - uynawnil nepepie. MNll BK, BMllNK, orx<e, MNKB - traparreJrorpaM. MN, NK, BK, MB - rinoreuysz rp,fiMoxyr- Hrzx rprtyrHur{in ei4nori4no MAN, NC1K, BCK, AMB a pinHzmn (areraMrz, ro4i MN : : NK : BK : MB i MNKB - pou6. Pvura: 4 MN.lz LMATN ( t MA.N : gO"') MN : llM 4N . MN: ^lrb+roo : bJi.P*,uu:20J8. Bidnoaida: 2O J5. BapiaHT E 1.MNllAD. MN BM . MN =!. ntu =t2 =s"' AD AB. 12 4 4 Bl1noeilu: MN : 3. 2. Ilpoee4irr qepes roqr{y.4 npamy, fiapaJrer'rr,Hy ArB' go [epear4Hy s npauoro BB,; roaHaqre rovxy ix neperrruy Br.flpawaAB" fieperHe npamy MM, e rouqi Mr.Pos- rJa.sHeMo LABB2i MMMr: LABB2* LAMM2Ba ABoMa rcyraMu, ,o* ffi = W 3a yuonoro -4M i MB : 3 : 1, ro4i AM : AB : 3 : 4; AA1B1B2 i AA.M tM 2 - napa- rerorpaMu, ro1i ArA: MtMz: BtBz: 2, szig:r-u MMr: MM1+ 2. BBr: BBt + t BtB2- 8 r2- 10. or*", 1 =*Y*', MM, -z=?9, utt, - b,b. Bidnoeidu:5,s. 4lo4 3,Ae u,O e u,rogiAO c 0t', ocxiarxz C e AO,roC e a, B e a,O e oc, ro4i BOca; ocxinrxu D e BO, ro D e o. 3a goae4enuu roqru C i D na.nentarr a, ro4i CD c a. II{o fi rpe6a 6y.no 4onecr:u. 922 la:l: rEoMETPts EeBs r. n. ra in. t-tpoOinunur piBeHb ljiii:l:i{::ti{ii;:i:i 4. (D1SCl) a a: pC, (D1SC1) n B : D,C, i DC Il DCi (/.1SBr) n B:A,-B, i,ABl A$1. AlBr D:CD rogi AB ll DrCr. Ocni.nrxv DC 1 Dpf io ' Asaroriqso 4oee4irr, qo BC ll AD. To4i uorupzxyrsut< ABCD fiapaJreJrorpaM aa oBHa 5. flrorqnHa (-4-BC) neperuHae o tro MN, MN ll AD. flpose4euo BK ]) CD, BoHa'neperr,rHae MN i roqui L, AD - e ro.rqi K, KBCD - napa,'rerrorpaM, LN : BC : KD :4. Posrrsseuo LMBL i LABK: nolrra floAi6Hi 3a 4BOMa KyTaMrr. Ilpone4euo BH L AK, ML ^ BH : M, H e AK; 2Y - *!. ML: MN - LN, ML - 2, BH A]{. AK AD KD, AK 5. ro.ui "# =:. Hexai r - LJil;i.tt !,1. roli BM - 2x, BH - 5:r, eei4xn MH - 3*. ro4i s*uo = **;M .ttn, sq. u;n ."r. arisHn x:f. ",,"o BClAD.BH, sacD= BiOnoei}u 78. 6. ,flrrqo,4C ll a, ro AC Jl ArCr, rogiA-'ACC, - napaae,rorp Ha a roqr(u B s AABC, - BM, a rvre4iana, npoBeAena . MMrllAAr,MMr:.AAr.Posr.nnnemorpaneqiroMrMB ' MO: OB: I:2, MO: MB: l:3. Ilpore4emo u rpaneqii M.MBBL MB2ll M$v B2e BBt, MB"n, O0r- Or. Hexail OO, x, , BB" 3x, OOr: MMri x, BBr: BtB2+ 3r, ae IM x = =(BB. B.B"r. 'Ypaxonyrovn, rqo BrB, : MMr: AAr, nraeno: OO, : AA1 + r, Ae )t -!r"", o41, rori oo. AA. t I BB. ! AIL =? dn - ! sn.. oooo Bidnoeidu: ? o+ - t^ uu,. O.f 7. Ilpoee4euo MN I DCll AB, roai ABNM - myxaHufi Posr.rrggeilro A SMN i A SDC: lC - cninsuuir, ZSMN : ISDC aH ni4nonigHi tpu MNIIDC i ciqxifi SD, roAi A sMlr N A SDC i Y! - E! . DC SM ^SKzSK2Ua vMoBoIo - - -.To.[I.KM|'SM3 g.*". {{ -2.uN - 2'8 - 16 .'DC 3 3 3 Hexafi L cepegi.rna AB, O - roqna neperuHy I 4iaroHareft naparerorpama IBCD, ii!*:il:iji::f:;*ifl: TEOMETPIS Eees f. Il. ra ix. IlpoQ
  • 35. Ilpooe4euo KE LLM, E e LM, KE ll SO.ASOM - LEKM, + = y: = "+ 3uar?4euo So a A BoS, t "o",:90"): Bo : *U*,"o : n ff,"rt'r, !{o, rto, so: 'lEs'- Ba',so: r{oo-e2 :znltz.n'rx: MS:1 :3, rosi EK: So : I : 3, EK : '{ ,r* : oM : t : s. am : t!, orxe, -L-O : LM - EM, LE: S - f] : Ol. oo ra LLKE (tLEK:9O') zr : .[IE EK""*: F'r4$' ,LK:2,1rc. rl-16+0vi3ssryu: ; LK(AB+ NMt,S*," -- : 2,113(8- ;),S*"r- rs Bihnoaiu: nOf *". uo. 3 8. Cnpoexryeuo 4iarona-nr AC na n,rorqrny r:r'rxcrrroi ocnorr r<y6a y nanpauxy -{A' ne At * xiueqr 4iarona,ri iri.rxcnroi ocuorr.r C,4r. Ciuna n.norqzHa neperrrHae H[]rcHro ,o, ToniM= I = =1-"^' us adi -t "ti -t' J' Bilnoaia:1: (rD-f) 9. fl;rorqzua nepepiay i naoiqi.rna ocrroBr,r napaJ'relsui. T noli6nuir rptrnyrHtrKy, xto rexflrs e ocuoei, ix rcoe nogi6nrzx (piryp nignocarbcs sr{ KBa"qpar xoe$iqienra .1peprSy cTaHoBrrTb - rrJrorqr ocHoBIl, flnorqa ocHonr.r " =+ , " =ryf : r44JB &oo rosi n.nou.4a nepepiay s'= + '144'fs :9 ',ft 1cu') BidnosiAb:9 uE cv'. 10. M - cepeAnna BD, N cepeArzua CD, MN ll BCt NKllAD, KeAC, MEll AD, EeAB, ro,ai K.Ell BC. MN : KE: ! sc: ?.xu - ME -!tn - ".22'22 Punre:+.|:za. ; Bi)noai?a:2a. ocxoBy rro MN. MN ^AP: F. 9epea rourcy P nponege*ro npfiMy, fiapaJreruuyAC, - nps- nty FK, KeAA' FKIAC, FK.nexzrr y civ- niit n;rol1zui. C:rigz ciunoi nJIoulzE[ Ea nro- rquiri nuxnsoi ocuoBrr Jre*carr, na npsuifi MN, X e MN , Y e MN, XK i YI( neperlrnarorr 6i'r- ni pe6pa BB, i DD, niAnosi,qHo n roqnax -L i E. fl'rruxyruux LKENM - uryxannft nepepis. A1IN ll Bpl i MN - ! B,D,: B.D.: o Ji. *r : oJZ. 2'2 :lj",xr,: KE; LM: I___- l( a" ( a" aJ13 llzl ls] 6 Tranoai sagavi ;1rq KoHTponbroi'po6oru 4o poa.qiny 3 1, fl.norqzna (SBC) neperunae nJ'rorrtrr4ny a no BC, nrrorqfiHy 0 - .ro BtCp BCll BrCr, rogi M e SMb M,eSMr. Ea +2'12 oJrs n _ a(6rE + 13 + 2JrA 6 Il.norqnna (SlM) ueperuxad cr no AM, B - no AMv AMllArMr, rosi AASM N AA'SM,, rosi AM ^S,4:- =-_:-:.SA:.AA,:3:2, 4M, s'4t ro.qiSA:SAr:3'5. otr*u. 6 =?.o,u,:10 cn.- 4M, 5. Bi1noei1u LO c'ttt. Bidnoaiou: a'rcJt + rc + 2^r!18) 6 Tecrogi 3aBf,aHHe,qo posginy 3',illc.ii:l.ii 7, MPI AC; BM : BA: 3 : 4, ro4i MP : O,75AC, MP : 6 c;ltt. A, MFII BD, MNII BB,, FPII DD,; MNPF - xBaApar, MN : a, MF - a, BD - a,[2, AM : AB _ MF : BD, M = _o=-,a11a : o_, a arl2 '!2 MB-a-:: - a(f,2-t. J2 J2 2. Hi, ne Mornyrr,. Tprz nepnrunz qorupr,{r{yrHrrrca Jre}rcar Ta Eeprrr.IHa He Jrerl![Tb y uifi. To,qi ogna 4iarona.nr Jre neperuHa€ nJroxlr,rHy y nepruuni rroruprlnyrnHKa,,sHa na.ni. Orxre, giarona,ri r:arbro vorzprafi yrnr?rna uzmo6 3. ,flxrqo y rpaneqii, snucanoi e xo.no (a rara rpaneqia pinno6ona), giarona-ni fieprreuAr,rr{yJrspni 6iunru crop To r{euTp ofirrcanoro rcoJra - cepeAr{Ha 6i.nruroi ocnoBr To4i nepneugrryJrapt't, npone4eni 4o 6ivnrax cropiu ia pa xora, napane.nrui giaronaacu tpaneqii, O}lllBD, oMllcD. 1 3 4 i) 6 7 8 I 10 B f B b B A B B B B 924' t::t TEOMETPIS Eees F. Il. ra is. IlpoQinsnr,ri piBeHb tliilliltf;i;h:ii; :;:liiiii:illiiiiii:: rEoMETPlg Eees F. Il. ra ix. Ilpog
  • 36. 4. MN - cepeAgn nilr.ist LADB, MNllAB, I MN - ,AB: KP cepe.IlHt riHia AABC, '1 KPtIA-8. KP - ;AB. Orxe, MNll KP, MN: KP,to4iKMNP na- A paJreJrorpaM. fl n u{o y trapareJ orpaMa AiaroHari . piani, ro qefi napa.neJrorpaM - npsMonyrnnrc, LKMP - npauoxyr:nlrii, I MKP : 90", I MPK 3o , roAi KM - 1^ MP, KM - 5 c,v. 2 MN1h- i:-.Suur:S*" l:4. S*"": Ssc S.r", ro4i #}; : s,-_.. .l :.$::,sxi4xn 4Srr": 24 + SMBN, Srr":8 cu2; ?/rq L S-..- 4 Srr". S-. - 32 cmt. Bi}noai0a: 32 cm2. KM - cepepvs niuia LADC, DC :2KM, DC : lO cv. Is LKMP KP : MPsin I PMK,I{P : 10sin 60" : 5 rE ("*). xp-llB.,4B-toJB cm. 2 OcHimnz KM ll DC, ro DC naparemua nnorqtni (MNK).II{o fi rpe6a 6yJro AoBecru. 5. Hexafi M - cepeguuaAB, N - cepeAuHa BC. allAC, rcaiMNllAC,AMBN-M-BCarcoeQiqienrouno4i6nocti B To4i n.norquua Q4rAN), sxa rpoxoArrrE, -repea npsuy treperrrHae ii no nparvriit OO' napanenrniitNrl/. NN ii rpe6a 6y,ro 4onecru. E,Hexaii M cepeguua AD, N cepeguna DC, ro4iMN cepe4na.iriHiaAACD, MNllAC. SM -ue4iana LASD, OteSM, Ot - roqxa neperuHy megiaH, OrM : OrS : I : 2; SN - ue4iana ASCD, O, e SN, O, - rouxa treperrny ue4ian, O"N : O"S : I : 2.Y A,l4Si/SOl : SM: SO, : SN : - 2 : 3, roai OrO, li MN, a orxe. OrO"ll AC . ro4i OlOr,ll (ASC)..Iqo ii rpe6a 6y.no gonecrra. 9. Ocrci,rrnu rxo,4ni Ani np,nui ne rrzrro6imui, to rorr O a6o napa.ne.nrni, b6o neperranarorrc.s. Hexafi : . c A)npauiaibtapanetr- Hr, TOAI qepeS H[x Mo]fiHa npoBecru n.norqzHy c. He- xafi npaua c neperrruae rrpsMy HeMo)rcJrr.rBo. To,qi cll a, a ontte E) npsui g i b neperuuarorrca, Becru nroquHy cr. f,Ixrqo c ll a, c uxmo6i:xna a b, rqo Heuorx,rzeo ,fIrculo c neperHHae a e tovrli, ei uisuiit BiA ro.rxr.r [eperr4Hy o. i b, To c rrrzmo6ixna a rqo HeMoJrcJrrrBo. Orrne, nci rpn nparvri a, b t c a6o nap -lerrni, a6o neperzHarorsca n o4uifi rouqi. II{o ii rpe6 6yro 4oeecru. 1O. Hexafi cepeArHa pe6pa SC - roqrca N. Ilpone4emo qepea roqxy N trpsMy, trapaJreJrbHy SM, NE ll SM, E e DC, E - cepeatxa MC. rlepea rovxy.E npone AeMo flpffMy, napa:;enslry AM , EK ll MA, K e AB. rlepea rouny K npone4eirno rpffMy, fiapaJre:muy AS, KFllAS, FeBS. 9orupuxyrrrun KFNE - uyrcaHzfi nepepie . V ocsoni nipami4z relrcurb RBaApar ai croponoro 12 crv KE - All4, All'l - tqO' t nW, AM : $44+ s6 : oG(cm). KE : 6Jb cv. fpaHi ripaui4u - npaezrsHi TpznyrHtrr{x ia cropoHo L2 cvt. SM sr.cora ADSC, t* : o"l" , SM : 6 J3 cru. NE . cepeAHfl .nini,s ASMC, NE : KFrt .4s. N( - !AB. LKFB^ AA,SB i KF =!.xr -"4AS44 y AFSN ZFSN :60', FS : i U", FS : 3 cu, SN : cvnycieFN':FS'f SN'? 2FS.SNcosZFSN, FN': IN:3Jd cu.Orxe, Puouu:KE+NE +FN+ KF,P + s : 3(2"1i + zJd + s) (clr). Bidnoeida:3(2 J5 + Z uE + g) cv. iii:ilitlriltiillliliff TEOMETPI9 Eees F. Il. ra ix. IlpoQ 1 Z, 6. flpone4emo s (ABC) uepea roqxy O npsuy MN ll BC, M e AC, N e AB. Hexaft AF - uegiana LABC, O e AF. V nto-. D qnni ADF trpoBeAeMo Of ll AD, T e DF . MN ll BC, roai MN ll (CDF), ornte, rrrrou{uHa, qo trpoxo"qrrrb vepea MN i O?, neperunae rpaHr CDB no KP, KeCD, PeBD. To,qi KPI)BC, MKPN urynaxufi trepepia, A (MKP)llAD, (MKPllCB. flxrqo E cepeArHa AD, ro EF reperrrHaerscs e (MKP) e ro'rqi S. EF ^OT : S. Poar,rsnervro LAEF i LOSF: sonu nogi6Hi, ocxlmrcz,4-0 ll OS. F.S FO FS:-:- l:S,ocxirsmnO roqxa FE FA FE neperllHy rue,qias A.4BC, a orlr{e, OF : AO : I : 2, OF : To4iS-F:FE:l:3. Bi}noei0t: I : 3. 7. Ilpore4emo NNI ll.4Ar, Nre DrCr. flapale.norpam AArNrN - nepepia napa.ne.nenine4a nJrox{unoro (AA.N); napaJreJrorpaM DD.MtM - nepepia naparenenine4a trJronluHoro (MDD), Mte AtBr. AN a MD : O, A1N t ^ M tDr : Or, rogi OOr :rinia neperu- Hy nror{rrn (MDD) i (A/N). OcHinsxia Nrl/ll DD', ro N1N ll (MDD). A FA: I:3 I 926 il:l: TEOMETPT9 5eB3 F. Il. ra ix. IlpoQinsnufi piBeHb li;:tililltil{;:,:i:::i:]
  • 37. PO3Alfl 4. S 1 2. KW ruix npnurrur,r. Ilepnex4rarynnpnicru npe Mllx . V pieuocroponuboMy rpurdyrur.rxy xytz no 60'. . V npamoxyruury 4iarona.ni pinni i Toqrcoro fieperrrHy giaRrlcr nanni.n. . flxrqo B npgMoltyrHoMy rpur{yrnraxy Karer y 2 paau ueulrrait ni4 rinotenyaz' ro4i nin Jre)firrrr, nporz rcyra 30'. 424. Ocxlnsxn Kotnua rpanb ry6a - HBa,4par, ro4i giarona.ni rpaNeft pinni: AB, : ADt: 81D1, ro4i M3rD, - pinnoc'roponniit, roAi ZBtADt: 69". Bi1noei}u: 60". 425. a)Ityr witxAD, 81 C1 "^m" i BC -- r1e xyr uix B$ i BC1, ro6ro IBOC, ro4i ABOC - pinso6eApenuir, IBOC: 180" - 50' ' 2 : 80'. Bi}noai}u:8O". 6) Illyxanraii xyr - ZBOC, LBCC. - npnrtoNyrnuft, sayMoBoro' tBcp:30", ocxi,rrxr.r BC = a =! sC. =l .zo. 2'2 C Mu"*o LCrOC - pisHo6eApeHr.rft, lctoc : L9o" 2. 30" : 120't tBoc:180" - 120":60'. II cnoci6: LBOC - pinnocropounifi, rogi IBOC : 64". Bi1noai0u:60". 426. a) Illyxairrafi nyr DOC, Br Cl ocxi.nrxv DDTC.C - nna4iiar, DCt i DS - fioro 4iaroua-ni, rcfi IDOC:9O'. Bidnaeida:9O". 6)Illyxannii xyr - IA'CD, rouy rqo AB ll CD. LArDC - npsrroxyruzfi, romy qo Ag4 L (ABC), c An t DC, aa reopemoro npo rpr.r flepfierrAr{nyJr,ffpu, A D AP LDC. Hexafi DC - a, roAi 4D = ali: rs/-cD =# -'Q= ^{e, ACD = arctgJi. Bilnoaidu: arctgJ2. II cnoci6. LA.DC: .C = o{A, OC : a, .D = ali; (o16)'* o'-(onlz)t Ba'+a" -2a' 2a2 , J5. aosZACl): ',--= - =-= -_-; ' 2. aJB ' a 2'l3a' 2'l3a' :J t; IAPD = u."co" !" . 928 i;:li TEOMETPIS Eeeg F. Il. ra in. IlpoQinsnufi piBeHb ;;iiiiiii:.rii:ii:!i;:i n) DrD ll C,C, roAi myxanraii xyr * lBrDrD. Ocrci,'rr,nn Dp L (A$p), BrD, e (AIBP), rogi lBrDrD : 9A'. Bidnoei1t: 9O'. r) TTpose4emo DDrllAC, ro4i ruyranzft xyr - lBrDD* flosHaquMo pe6po ry6a -48 : o, Togi BrM : D"Mi LBTCTM: BrM = 427, aBC ll AD, uryrcanuii xyr * ZB,CB. B1 ct AB$C (tBpC :9O"|, ZBBQ :30", ro,qi ZB9B : 60 rcai A DDr= D" BrD = TeroM f -' E"'.la"+-=,1-= 4 /4 LBTDD2 Ba reopeMoro nocunycin: 4D; = 4D' + DD B.D2 + DD: - B.D: 3a2 + 2a2 - 5a2 cos IB,DD" = -)::-----::J-----:tz 2BtD . DD2 2. aJB . a,l2 Bi0noaida: 9O". Bi)noeiOo: 64". 6) Illyralrnfi xyr - 1AOB, LAOB - pisHo6eApegufi. c M,fr (44:so")' 48 = JLa'? + o' = o'f|: no=oB=aG: 2 / -2 - 2 coslAoB= I ls€]" . f "gl' -, l,( rsii,JB I 2i 2t ) 2 2t2 ZAOB - arccos 0,6. s) CrD Jl ,48r, ro4i lA1CtD * uyxauui. MrCrD: ACr= onli, CrD = ArD = '[+" o" = o"B; cosL4-c.D - 4c: + c)D'z - ,4D' - 24Ct.CtD 2a2 1 "[o 2.a,12-aJ Jlo 10 4cP= ur""o.f . Bihnoeihu: ur""o" @. .10 ::i;:iiliii:i{iiii;ti; rEoMETPlg 6ees F. l'1. ra ix. Ilpo 30 v"i rAq 10 u., c. I
  • 38. 428. CnopucraeMocb reopeMoro: EnIqo 4ri npaui, nxi neperzuaorrca, napa.ne.nr'ui iH- orxte, a, I br. 429. Ta:g.., axrqo A e (BOC), trpoBeAeMo AF L(OBC), Ba yMoBoro ZAOB : IAOC : 60". AM LOC, AK LOB, AAOM: AAAK, rolri OM : OK, FM L OC, FK L OB, roxi IBOC:90". 43O. Hexafi xyr mirx a ib - 30', ro4i ar ll a; l(b; at): 30', oa yMoBoIo xyr mirx a 1c - 4o", ro4i nyr mirx a, i cL * 4O". Maevo l(b; cr) : 30" + 40' : 70" I 90'. Bi}noeila: rpami b i c ire uorxyrr' 6ytu nepneHAuKyrflpHuMu, 431. Tax, HanpuftraA, maemo xy6, rcai AOBBAA - BaMKHeHa rraMaua, Ae O - Torrna neperruy 4iarona.neii rpani ny6a, AO LOB; B& L OB, ToMy rrlo Ko)fiHa rpaub xy6a nepneu4uryrflpna Ao inuroi. g, C.B 432. Hexaii ABCDAIB.C.D, 1ty6, xyr rvri:x ABriCDt 4opienrce ryry rrirx ABltA1B, Al roMy r-lo A,B l] Dp. Macmo: ABt i ArB - 4iarouali rpani xsa4para AArBrB, ABj I A1B, rogi A.B, L CDr. 433. AL LOC : LBAC : LAOB Ba ABoMa r{areraMr4, roti AC : BC : AB, onxe, LABC - pinnocroponuifi, Kyrr4 rlboro rpzxyrHuta no 60o, BiOnoei1t:6O'. a 434. aLABC - pirnocropogHifi; LAOC (/AOC :9O"): AC=oc Ji=+Jz, rosi P*" = S'4J, = ntE km). Bilnoai|a: L2Ji ctr. 6) p=s.arli=zal2. BiTnoailu: SaJi c*. s) AC : BC : 5 cw (LOBC - erunercr,Huii). 930 il:ij reOUerpre Eees F. fl. ra ix. npoQinsnrfi piBeHb itti:ti{liliitll;:i:l:'iii /NA^LA-" ll ^ LA" LAOB (1AOB:9O"): AB = oB. Jz = sJr; p*" = Bnli + b + S =10+ gJ BiTnoaia: 10 + 3rD gu. 435. Ilposegemo BrN )l np, aa reopeMoro <Dareca: BB ro4i BrM - cepeAnfi .riuia AA-BC, ea nracrusicrro Mhll AC, orwe, IMB.N - xyr mint BD i AC. e.Ir =! nc = B cnn: a.N = so 4 cu, '22 ro4i n AMB'N croponu 3 cm, 4 cm, 5 cin, ro4i AMB',V - rpfiMoryrHufi, IMB.N : gO". Bidnoaida: 9O". 436. Ocxinrnu ftorfiHa rpanb xy6a - KBa,qpar, a giaro 4raHy.napni, orrxe, DCr I DtC. flporurexni rpani na parerrHi, ro4i ABrll Dc' ArB ll DrC. Maeuo: 4iaro na.ni nporu:rerxnux rpaHefi xy6a a6o napaae.nrHi, a6 nepnen4uxyrapni. flttqo ABC DA.B lC 1D L - npauorcyrHu fi napa.ne.neni neA, to.ui fioro rpaHi - rrpfiMoxyrHhnr, ale 4iarona.n npsMorcyrnr{Ha T{e neperrrnarorrca ni4 ryrou 90o rogi giaroua,ri fiporr{Jre}r(Hux rpaneii npslroxyraoro napa.ne.nenine4a Molrcyrb 6yru a6o napaJre JIbHtr Mr a6o yreoprcnaru xyr, eigrtinniii si,a 90". 437. OcHirucr rerpaeAp npannttnrz,fr., ro4i nci fioro xyrHzHu. MN L AB ea yuonoro, M cepe4zaa MN napaae,rrHa Ao Br4corr4 MBD, axa nponegena roxi MN gopinHroe ii noronzni. Bucory nparznrn - ""13. Tpr4rcyrHu(a sHaxoAr4Mo 3a QopMyJrolo ^ , d -. ' ' : -2,--, 'r aJ3 aJs poHa rpr,rxyrHuna. MN =:'224 AnanoriqHo BuaxoArrMo AoBlr{uHy nepnen4uxy;lapa ^t;(MG L BDl. MG = u',u . 4 AL : DL - Brcorrz rpunyruzxin -4-BC LM L AD. IAML (ZAML : 9O"): ML=,tqj - AM" = MLrBc. *"=+ 3a ymororo K - cepeguna BD, AK L BD, AK ll MG xG=lBD: MF=KG-la. 44 . ^ o Js o"l3 o,li a IJldnoaidb:-: 4: 2t 4. ,:i:ii:i:i;i:i:ii:t:::i:iiii: fEOMETPI9 Sees f' Il. ra ix. Ilp 30* i BDC, ro -.2 aJ3 I 2)
  • 39. 438. aArB JJ DrC, ro,ai ZAOB - xyr rraix A8r i D,C. AAB1B - [p,rrMor{yrrrzrn, AABBT(ZABB1 : 90'), tst4AB=H=YE=Ja, IB.AB:6c,. LAOB * pinnocroponnifi, IAOB :60". Bi}noai?o:6O'. 5o.' 2a- I a' ts tB,oo1 = "# = * = + t tB,oo, - u,"ru@ Bihnoaiv: ur"tSf. 6) ACllAPt, lBpC _ ryr uirx OBtiAtCt. tBro srcrqo -AB : a; BrC = sJr; LBTBO (lBrBO : 9O"): c! "JB C qAC ll A{1, ro4i ZBrAC - xyr ivrixc ABt i ArCt. MBBI (1A84:90"): ,43, = ^tF - n' = Jaa' = 2q'; nC = atE; BP: BA:2a. A,BTAC B reopeMrr Kocrl- ABI+AC2-Brcz _4a'+2a2-4a'_ 2a' - L _Jr.Hv"iB: coslB.AC = Lg = - 2 ABt AC - 2 .2a . ali +a' ,li 2J, 4 ' t; IB.AC =."""o"Y.'4 ,t; Bidnoci]u: arccosY. 4 s) SHafiAeMo rorrny rreperrrry giarona;reft AP i BtD - O. t{epea O npoBeAeMo BrDrll BD. Toai ICOD" * ryt uilrc A& i- BD. Ouci.nrxr.r O * cepeguna AP i BrD, rogi D, - cepeAr,rua DDr, DDr=+. a,D"DC (lDrDC :9O"): E al t q' oD"=!sp=",12='9-,"222 coslB.oc - oBl + oc2 - B'cz - ' 2.Oq.oc tBroc :90". Bi4noei4t: 9O'. n) OI{ ll BDr, roAi ll - cepeAuna DDr; IBIOK - xyr uirx OBt i BDr. B.o=o#,'2 F;;A =;rtj = B1O = A oJ5. 2'ov =Lpp. =2'2 A LBtDtK (lBtDtK :90": B,K = JAuT xo1 = bltf .(i)' = coslB.oK _ Bro2 + oR2 - BrK2 _ ' 2.B9'OK tBpK:90. Bi1noaida: 9O". r) B,C, ll BC, f " "'! 4 &E t{b:oc =L;'c =!2' 2 rogi ZOBTCT- ryr ui:m OB, ra BC. ArO = CrO = { B,cl+oBi-oci _ B,cl cos IOB.C. = "2 BrCr OBl 2-BtC,-OBl _ BrC, _ a.2 -J6,2.O81- ZaJ6 6' loB1cl =u"."o. f . r: Bidnoaidt: o.""o"9. 6 44O. V rpani (,4-BC) npoBe,qeMo MMrll BC, a rpani (,4-BD) trpoBeAeMo MtN ll AD, B to4i INM.M - ryr nrixc BC i AD. L,MMTN: MN : 13 cM Ba yMoBoro; ( o,fa' I oJs I z )*l z-) ---;nG222 cos ICOD, = ,l;n:)(*) 4 Bi}noei1a:9o . t) AD llLDb rogi IBtAD - xyr *tirx AB L i ArDr. r:jjlt :l.li: LAB,D: AD : a: da, = ,[a1 sa' = 2o' i;lilrti:iif B,D = lio 2o' = "J8: (o,ls' = a2 I 4a2: r:liliii:!:::.:.1;: 5a2 : 5a2, roti LAB,D - npanoxyrnraft, i,:til;::i::':'J roz.i ZB,AD - 9o'. ,::iil:::i:::$ Bi}noaida: 9o". . ;r!.:':.:.].1.: 439. a)O, - rorrxa fleperrrHy Aiarona.xeft A rct i B.DD rosi. OOrll DDy lBrOO, - Hyr mi)Ir OBt i DDt flnqo pe6po sySa a, ro6ro AB: a, _ -.t;roai BrQ - a,l2: B,O, = 2r' : OOr: DDt: a. LB'OtO (ZBrOtO : 9O'): = 4 + 4 =UzcaDz:9l". 29 JLo B. Ar A A A cr E)'*(oJz)' =;J;" = I a c" (oJz' ( oJi' Itu'J *lz]-l.z.J gg2 ;i::::: |EOMETPIS Sees f. Il. ra in. flpoQinunr,ril piBeHb ii::i:i:i;iti::i::::::* fiiiti:ii:i:iiilliiiii: TEOMETPI9 Eeeg F. Il. ra in. flp
  • 40. IMM.N: LzO", rogi uryxanrfi nyr - 600. BiOnoeida:6O". 441. flpone4emo MK ll BD, MK - cepeAHr{ nini'. LBCD. HexailBD-a,rogi UX=1; ^, =+ @M - ","o,u?*"r. ZAMK - uyr mi:xt AM i BD: cos IAMK = AM2+MK2_AK2 2.AM.MK ^AMM1a LACB,,opi #-!, **,=L#= 2 1-0'5 =7 1"*;; ^BMN - LBAD, H =+, M,N = 24.] = s t.*1. cosIMM.N _ MM: + NMl - MN' _7' +8' -73' _49+64-t69=_ 59, =_1, ' 2.MM1 .NMt 2.7.8 2 7 8 2.7.8 2 saBlnAlr r'rerxarb r oAnift naorqrzni) KD I roMy rrro K M : AB, =: DC, = 11g, 2'2 i KMOD - flapaJlerorpaM. Orx<e, IC.KD - xyr uir* KC1 i MO. Hexafi pe6po xy6a a, roai AK = i, ^KAD (1KAD- eo'): KD = F-^ = E =; Dq = a"JZ; LC,B,K 12Crn,K :90"): BrK : KD; c,K = .!E,c1 + a,r" = Y LC,KD: Ed Ba V+ z KCI + KD'z - C,D' _e.os IC.KD ' 2.KCt.KD E Vb LCTKLI = arccos-. r: Ridnocidn: o.""."9. 5 n) Ocrirrniz M, N - cepegwuw ArB, i BtCr, ropi MN - cepeAufi tiuia LArBtCt, ro,qi MN ll ArC,. LD{,C, - pinirocropouuift, ACr: CtD - A,D arc 4iarona,ri rpaueii rcy6a. ArO - nzcora LDA.C' aa eracrunicrm ArO - 6icextpilca:r"yra ZDAP. ZOATC.:30" - rcyr mir+t MN iAP. ' Bi0noeida:30". ., A 444, 3a yMoBoro nyt uirx A.BrArD - ct,, ocrirrHz ArD ll -arC, ro,ai xyr nixt -43r i BrC - a. Bi1noaiOt: a. a 445. LBAp - pienocropouniii, rouy rslo AtB, nefi ry6a. O - roqxa treperlluy BD iAC, ro4i A BDh o,li.'[s ,"6 40 = 2- = --r'= - (MarI' a))' a) i:i:i:i:i:i:i:i:i:i:i:::i:j::.* FEOMETPI9 Eees f. Il. ra in. Ilp r-l" 5a" .la" + -V4 ga2 5a2 - + -Za' 44 ^1. 2.54 .A'lO 22 t; IAMK = -""o"9.t) t; Bidnoeida: r.""o"9. 6 442. flposesemo s rpani (ABC) MAt l] ,4-B, rosi ZATMD - Hyr minc DM i AB. 3a yvo rwn. = 1 an. '2 LABD (IADB :9rJ"l: AB = oJZ: .a.m - L^"li '2 Y LBDC (IBDC : 90": DM : MC : MB - patiyc onucaHoro HanftoJ'ro LBDC r<o:ra, il rori DM=!nc-'oJz.'22 Auaroriuno, y LADC (IADC: SO'): D4 = A4 = 4C =oJi. Or*", LDATM - plBHocropoHniit, roAi ZA'MD : 60".,2 eor) M - cepeAr.rua BC, 'topi MA, - cepe4n.a .ninia A LABC. lloa*aqramo DB : o., ropi DC : AD - a, Bi1noei4u: 6O'. 443, a)V rpani DDTCTC npoBeAeMo OtO2ll AA, rogi AArOrO, - npsMonyrHr{x, ArK : KA, O1O : OO2, A ropi oK ll 1,,o,, nrorll ttcr. Maeuo: t4o MC,ll KO, ro6ro xyr uix nunra 0o. Bi)noei7o: MCll KO. K 6) AB,ll DCv KM 1J AB!, ocrcinrru K, M - cepeAr.rur'I AA, i A,BD rogi r n,rorquui (KMC) (4ai napa.ne.nrni npaui c1 D I o,5 j' a' I oJs l' a' I 2 J I 4 -l z ) _ 4 _ r _ J5. - ---;i6: -;lt - tF 6 ' 222 934 ii:ir rEoMETPlg 6ees F. Il. ra ix. tlpoginuxrr piBeHb fiiit:lil]ifiimii
  • 41. AAAP (ZAA.P : 90"): AtP = =16' xr =f,no =!""t2' AA.DD -* rcBaApar, ftoro 4iarona.ni BoaeMHo neprreuAltr{yJlspni, ro4i AJ L AD1, I' - roqxa repernHy 4iaroHa-neit HBa,4para. +e =l= +o =!"A (na.n. 6))' '22 (ruar. n)). ArK : ArP (MAK : LA.AP 3a ,4BoMa xareraiuz), LALKP - pieno6e4pe- uui1 ArT ..._ ftoro Bncora, ro4i AtT I KP. yMoBoro M - cepegr.rxa CD, roai BM - me4iana no6eApenuit, aa yMoBoIo K - cepeguna AB, rogi M Or*e, MK 1,4.B, iqo fi norpi6no 6y.no goeecrz. Ananori.{Ho, LKCD - pinno6e4penzii, roai KM L ulo ft rpe6a 6y,ro 4onecru, 3a ynnonoro AB = a, ,oot "M = +', rosi LBKM (1BKM :9O'): MK =.,IEM" . BK, = afinoeioi: a& . .2 447, Hexain AB, CD, EF - crep'fifii, *r.ci sa ynoron e cepe4rani O i AB L CD, CD L EF, EF -L.A-8. flxr HyrrzD Ba AB, roAi tpirypa uepee JIIoB He npofi4e, AB : CD : EF : 2 tt, a d: 1,8 m. MOE (IAOE :90": AE = ,[Ao' * Eo' = ,lf+7; = Jz (*) = 1,4 u. Bpa ToBrrlr,ruy crep'fiHlr, Bce oAHo rle Meu[re, nix 1,8 l BilnoaiOo: ra&, Motr{Ha. 448. HexaitABCDArBrCtDr * napa.ire,renineA, yci rpan sroro - pom6u e rytorvr 60o. Snafr4eIrno ryr uirx BD i:43r. Ocxi.nrxu CtD ll AB' roAi lryrcanz* xyr - IBDCr LBC.D - pinnocroponnift, roMy n{o fioro cropoun - AiaroHaJri rpaHeit (pou6ir), arci;rerxarr' npor:z 60o ro*ry ZBDCT: 60". Bi1noei1r':60". ;r" lo'* n = rr=|re=f; LA,TP (ZATTP :9O": 4r = w -rP, = tr$=,ffi =l"a =*" LA,C1D * pinuocroponwiit (ArCr, ClD, AD - 4iaroxa.ni rpaneii ry6a), rogi nep- renAraxyJrsp eAr4o CrD - Br,rcora ArG LAlCrD. G - cepe4una DCr, G - rowna treperrr'y 4iarona.neft rpawl. DDrCrC. '* = 4";J' = 4t' = * P r) (ivra;r. r)). LDArK - npsMoxyrHrnfi, rouy nqo ArD, L (AAP), ro4i AlDr I AtIr. AIM I KDf ^AA.K (ZAAK: e0"):,4,K = FC =iJi; xn = ^,* = + LDpK (ZDpK: 90"): D,K = 3 KM : xi MC, -_ " a - xi LA,MK (1A,MK - 90"): A,M' -- ,4K" - MKz =1o' - t; LAtMDt (lAMDt: 9O"): ,M' - ArD? - MDI = a'z -(2" - -)' , 5" 9" 10" ^ 1Oa2 ^ ]^}a', 5a 4o'-*'=a' 40'+3ax-x"| 4a'=3aJci Jc= 4 tSai x= +8"= 6 .^ oJa o,[2 BJz o,[o "JbBidnosidb:-i Z; 4oi 2t B 446. Ocxinsru rerpaeAp -43C npanrz,rrn:t,fi, ro1i lcorrcsa fioro rpaHb - npaen.nr'uufi rprrnyrur4rc. Bi4ouo, qo B piBHI.rx rpunyrrrrrnax pinni nrcotu, roli BM - AM (aa 936 fli: rEoMETPlg 6eee F. Il. ra in. IlpoQinunnr piBeHb llli*ii:i;i:;f: r::::* 449. a) Bnafi.qeMo nyr Irnix me,qiana*rra, ari npone4en Horo pe6pa rpaneft reTpaeApa. Hexafr CD : BM =+, AM =+, LAI(M -pinno6e4penuft , lX = 'a'U = $; xm =|sc=i; D r) a" s"' g .la" + - = -a'!42 !.,i,il!i:.il:;: I lLlii'il.l;l:il AKz+AM2_KMz cosIKAM = 2.AK I aJa ]- I aJs I' a' I t+t-_t - 2) 2) 4 ---:-;6-;Tz. 22 AM 5,a- 4a 3a2 6 -2- I I ri (o,la' loJ coszA_MB=eu|+ly,_4t _l z ).lT)_ 2.AM-BM ^ oJa "J2-222 6) Bnaft4erao rcyr mi:x irne4iauarrra, Rni npone4eni i:i:i:'::ti:i:11:i:i:i:i:::li TEOMETPI9 6ees F. fl. ra ix. flp
  • 42. ot =!pt =!.o =9,2 22 4' ,,' ,a' , a a -2.AL.OL.cosl20' *4, ta 2 z Z nU,O: Al=f,; AO2 = AL2 +OLz e) SirafiAeNro rcyr minc nzno6irxHNun npsMuMr DK, AM. FM ll BD, LM llAD, roli @LM) | @BD). V naorquni (FLM) tponesewo MO ll DK, O - cepegrExa F-L. Orxe, ZAMO - nyr nirx DK iAM. t; AAM: AM =016 : 2 MO - weniasa LFLM, annii pinnocroponuiii i noAi6nufi B ^ABD. FM-ML=r' ;,*o-"f . Snaii,4eno AO: LFLC - pinnoctopoxnifi, IFLC : 6O", ./FLA : 120". f 1t- z)- 10.10.16 4t1 = -. 4R. - 10.10. 16 _ {.25 _ 25 ^ 1 - E - (cMl. 4a.4 t2 3 3 $ 1 3. IlepnexguKynepHicrb npcMoT i nnou4llH 459. Hexafi M - cepe4uHa pe6pa ry6aA-BCDATBTC flponegeMo ML ll CD, FM 11 Bc, roai (FML) lL tac Ocxirrxiz CCr L (BCD), rosi CCt L (FML). Ilepepie KFML - xnagpar,.fioro nnorla- a2. Bi1noai}a: a2. nh S =; - trrotrra npsMolcyrHoro rpnrcyrumqa s xa s=lon flroula rpffKyrHtrKa ai ctoponoro a i z resa, - h. 46O. Ocxinrxu MB L AB, MB L BC, rogi aa oeHar innorsqww MB L(ABC):,ryofi MB LBK.MaeUoAM rroryrnrzit (IMBK: 90'). 3a ymosow MBC - pin penzfi, rogi BK LAC i MB L AC, uaemo: AC L roai MK L AC, LMKA - npsMonyrrrr.rft (IMKA LAKB (1AI<B: 90'): -4-B : lO, AI{ : 6, BK : 8; 1 S*un=t158=60(cm'z). LMBK (LMBK . 9O": MK = '/15 8' = J2% + 64 = Jzes = fi; son"o = !-n a= 51 (crra') Bi6noai0a:60 cryr'?; 51 (cm'z). 461. 3a yMoBoro MO L AC' MO L BD, roqi MO L (ABC), roai MO )- OK, LMOK npamoxyrHni' MO = ,a ; tgcI xaY= q . Bilnoai}a: -a | -: . sins tg0 srn0, a' a' o' 4a' r a2 + 2a' '7a2 7a2 = -+-r =-=- : AU =- : lO=arll . 4 16 8 16 16', 16 4 i oG l' , 1 oJ51' _zo' aoslAltlo_ AMz +Mo2 - Ao2 _ 2 I ' q l_ t6 _ 2. Arl4 .MO - aJ3 aJ3 z. "243a" 3a" '( a' a' ' 1.4 2 _4 t6 16_2 - 3a2 3a" 2-3 3' 4 r) SnafiAeuo Hyr uix CK i AM. V rpani (BCD) npoee4evo MK I CK, mr = lcx 1 aE = o6 ; ,-atv t- ryr uirx cK i AM. 2224 LALD: IADL:6O"; AD : a; DL =loX -1", B 24 AL2 = AD' + DL2 - 2. AD. LD. cos60" = o' * 4 - z' I 164 t"=o",lrg.4 LAML: cosIAML= | ,Jd l" , I of,ll' -tso'AM'-ML"-AL' I z I ' a t 16 _ 24,M.ML -=j6T- 24 'i 1 " 0,' ct" 2-" '16 4 . ,' r..:.:;l.it at.4 1 8.3a2 6 3a" 3a2 LBa2 -,+_4 16 16 3a2 4 Bilnoaior,: *t *t ?t 451, Hexafi F e (ACD), npoBeAeMo Cp'llCD, C$tllAB, DJ ll AB, ro4i n.norqr.rna nepepisy BrCtDrL naparrerrbua AB i CD. 938 ii::: rEoMETPlg Eees F. t't. ra ix. flpoQinsnxi piBeHb tiiiiiii;iiiiL;:i;ii:iii iil{it{il:iiitlftiii rEoMETPle 6ese r. Il. ra ix' l
  • 43. 462. 3a yMoBoro MB L (ABC), rosi MB L AD, AD I AB (ABCD - Ksagpar), roai A-D L (MABI, LMAD - np,nMor(yrrrlrilt (ZMAD : 9O'). LMAD: MD = ^tMA" * AD' = tliT *7 = o"lg. Btdnoai}a: ali. 463. MB L (ABC), MB L BD, MB LAC. AC L BD wx ,4iarona.ni KBaApara' AC I (MBO), rcaiAC L MO, A rqo fi rpe6a 6y.no 4ooecru. Bidnoei0a: A 464. MC L (ACB), rosl MC L CL. LACB (ZACB :9O): AB : LO, CZ - tvleAiana, rori c-L =L AB = 1.to = r.'. 2 2 AMCL (ZMCL:9O"Z ML = JMc cE = tf44- %= 18 (cm). Bi1noaila: 13 cvt. 465. ABCD - rpsMonyrHrrt<, ro4i AD L Da, AATDID - npfiMoxyrl{r{+ roti AD I DDr. Maeuo: AD I (DPC), roai AD I DCt, ropi ABTCTD - np,fiMoryrurrr{, rqo ft rpe6a 6y.ro gonectra. 466. Ba ylroBoro ni4crani ai4 M ao BepruuH nra4paraABCD pinni, ro6ro MA- MB: MC - MD, ropj LBMD - pinno- 6eApenr.rii, MO - fioro ^aegiada (O - qeurp r{Ba,qpa- ra - roqna neperuny giaronalefi), roli MO L BD. Ananori.rno AoBoArrMo, ulo MO LAC. Maemo: MO L (ABC sa osHarcoro ilepnexAr.rxyrlsp- nocri npauoi i n.norqunz. Orxe, M O nepneH.rlrrxyJlspaa nlor:lrHi KBaApara, u1o fi rpe6a 6y.no gonecriz, ,flnrqo rouxa pirnori44a.neua ri4 nepurzn KBaApara, ro rrepnenAvrcyJlsp, npoBe- geuzft ia qiei rovxz 40 rrJrorquHu (Ba,4para, trpoxoAlarb qepeg rotrxy neperr4Hy aiarona.reft KBa,4para. 467. 3a yMoBoro MA: MB: MC - MD: lO cttt, toxi MO L (ABC, AC = AB.,ti = a"ti.J2 = 12 (cu); AO : 12: 2 : 6 (crvr). LAOM {ZAOM :90"): MO : 8 cM, roMy np LAOM noAi6nzfi Ao ernlnerc .rplrnyrulrna (ei croponauz 3 cm, 4 cu, 5 cu). Bilnoai0a:8 cv. 468. Ifenrp KoJra, onftcanoro HaBKoJro npgMorcyrHrrra, Jrexdr{Tr, na neperuni fioro 4iaronarefi, rogi OB : OC : OD : OA: .R - pa,qiyc q6oro no ,4or:rcuxli raoJra BHaxoAnMo ea (popuy.noro C : 2nR rori -R= I =4"" =2o."2n2n 3a yuonoro MO L (ABC), rosi IMOB : 9O", LMIB: MB = JMo oB" = Jsi = 2". ,flrcn1o noxuri marors pinni npoexuii, roAi noxr.ili pisHi, rorty MB - MC : MD : MA: 3a. Bi}noai1o:a. 469. flxiqo PB: PC : PD : PA: L4 cm, roAi PO I Hexair AB .: {2x) q}t, BC = (3r) cM, roAi P: 2. (AB + BC): 2(2x + 3r) : 10r (cu), Ae .r - AoBrrcr4na 1 qacrranr,r s rux, Ha axi no4i,re nzft uepramerp np,sMorcyrlrrana. Maemo: lOx : 4O x - 4. AB : 2. 4 : 8 (cu); BC : 3' 4 : 12 (cu). LABCZABC : sD")l Ac =,le';L2' = JGf +144 =./Eo8 = = J+n = zrEi @w1. ^pOB (.POB:9O"): po=JBp" -Bo'=JlF -nz =fie6-b2 ="Ett Bi1noai4v: 12 c;lr. 47O. 3a yMoBoF M - pinironig4a;rena ni4 nepurran A ro6ro MA: MB : MC. MO r (ABC), roti MO L MO LCO, MO LOB. Maeuo: LMOA: LMOB: LM ea rinorenyaoro ra RareroM, ro,qi OA: OC : OB. O O - qenrp KoJra, otrrcanoro nasxoro AABC. ,flrcrqo ro.rxa piruonig4a-nena ni4 neprrran rprlxyrunx ne4eirrzft ia qiei ro.*cu Ao nJrorqr,rHr,r rprrxyrHar(a, np orrrcaHoro HaBKoJro TprrnyTHlr(4, 3a ylrosoro LAMC * pianocroponnifi' Mo - izoroBrrcora' l,to = MZ# ; MC=AC=aJZ; " *o=e*=* "Jo2 :.[N,-^.... iiilii:::*i: ':'.r;th}}s 471, Ocni,'rrrcu M pissosiAAatena ei4 neplruH rptrrc rogi nepneugzrcynap MO 4o ilJrorrtrt4nrr rprrxyrnnrca xoAr{r}, qepec qenrp AA-BC, LM,A (lMoA: 90"): Mo = JAM' - Ao't A pa,qiycKora, onrrcauotoHalrcorr oLABC; OO = # = ,,5 940 l{:i; reoruerplg Eeeg f. I'1. ra ix. flpoQinrnerft piBenb ii:"ltiititiil;*:{ii;ii: [:ir:ii:i:i:i:i:iii:i:ii:l:r.:; FEOMETPI9 Seeg f. Il. ra ix. Ilp
  • 44. Mo = "8", - "' = ^ltlo - o"/rl - o'6s . !'- s ! s Jd g ,"EdtiIJfionftlbt -. t 472. 3a yMoBoro SA: SB: SC, rogi SO I(ABC), O - qeHrp fioJa, ofllrcauoro naexoro 4,4-BC. AO - BO - CO - R, R - pa.riyc onucaroro Kora. R = o.b: , a, b, c cropoHh rpilKyrHuKa, 4S I so = ,[sn' - 7io' = .lrcs -922 = Y 16 Bidnoaidt: 3'/231 . 4 473. a)Ba yvrororo A.ABC pinnocropoHHifi. Ilpoeegeuo AK I BC. ABIE - 36. n,AK = 2 =, z MA I (AtsC,roti MALBC. Maemo: BC L (MKA), roti BC L KM. Powac: MB + MC + BC : 2MB + BC (MB : MC). LMAB - npamonyrnrzfi, erzuercrxNfi, MB : 5 cxc. Pouac: 2. 5 1 3 -- 13 (cv). t*r" - trMK-BC. LMAI{ (tMAK :9O'): fry fa+i = S - n.noula rp[rcyrnurda. B S= Jptp at(p-bl(p-", =,m=,rl 2 = Jro. o . G. 4 = 4. 6. 2 = 48 (cm2). ti _ 10.10 12 _ 61. 4.44 4 LSOB (ISOB:90": 474. AC L a, AC I BC, roni A,K ll nc 6 e l?). 3a ymouoro AAr: AtC : m i n, rogtAl{: KB: m: n. Bi}noai1u: m: n, 475, 3a vMoBoIo AA, L a,8.B, L a, rogi AA, ll BB,, C e ArBr. , MA|C - A-BB1C. AAL _ 4C. BB; B,C' B.a:x.tooi E=3*t,'' 3 x 4r-3+xi3x:3;x:1. Bi)noeiOt I cttr. 476. a)Ba yuonoro.4,4r L a, BBt L u, MM' L a, rosi AA, 1l BB, ll MM, i M, e ArBr. BtBAAt rpaneuis. MM, - ii cepe4Ha ,linia, B,B+A.A l0r6roni MM, =L =8 (cu). '22 Bi)noai1t': 8 cw, 6) flponeAenro AI L BBb rogt MrMr: AA, : 6 c LBFA^ LM,MA. - BF = "o ,""'t'"t" MtMz MrA' BF - BBt - B,F -I0 6 - 4 (cn'1' !=!, - - MMt: MrM2+ M"M:6 + 1:7 (cn)' Ri.1noei}t: 7 cv. 477. a)Ouinr,xu AA, L a, BB1 L a, CCt L a, DD' AACCT - rpaneqia, fiiBDDt - rpaneqia. O - roqxa neperilHy giarona.neii naparerorparra.4-BCD, OO, 1o. OOrllAA,, O, - rovHa neperuny giaro- na.refi napane.norpama ArBrCrD!, OO1 - cepe.4n,r .ninia rpaneqiii AA.+CC. 9+13 AanA,ACC,OO, -=ll (cM). '22 BB. + DD. P,tt B'BDD, OOt = -: ^riz 20 - -DDt = 1r; 2o ' DD, - 22: DD, - 2. 2 Bi1noai1u: 2 cv. 6) Ma;rroHorc i poan'saanna auaroriuui a). AA. + CC. 15 + 13 IhrA,ACC, OO. - = 14 (cM '22 [,na B,BDD' oo, = "t!+!t,']+2t = 14: Bidnoaida: 3 cvt, Fiiitii:iii;ii:i:ti:t TEOMETPI9 Eeee r. Il. ra ix. l- /txwC p11j _al2'r Vro 4 ct; 2 I -zI igv3irwrc =,,lntrd, + AK2 =l/ro r 1 """ i = s -l Jgr ,=@ "*,.224 BiOnoeiha: f "*'; rS c*. ",lt6 AK = - f: tuax (zMAK : 9o"t: Je1 2 1:r$!:ili{ :.:1:'Tl.L!i: ''i;r+i!:* , i ;:;!:.:,:; MK= LMAB (tMAB - 9O"): M B = cJi: P*u" = 2' MB + BC = 2cnli + " = r(zJi + t); s.^.--=1 MK.BC ! tJr.r="' J7. 2 22 4 Bionoaiot: "(2"!i +1)t 9;h. (4 . 13 - 25)(4 .13 + 25 942 l:iii reoruerplg Eees F. Il. ra ix. IlpoQinuxufi piBeHb iliti:li:*i':l{*iiilliri
  • 45. 478. ABCD{B,C,D, * ry6, ro4i roxaa fioro rpanr - rcBagpar. BD .]-AC, B& L (ABC), rosi BBt L AC. Al AC I (BBP), n4o ft rpe6a 6y.no goaecrr. 47g. M -cepeArznaz43, A rci rpani rerpaeApa * npauunrni pirui rpar.yrnr,rrcra. CM : DM - Bracorra rpaneft, CM L AB, DM I AB, rozi AB L (MCD). LMCD - nepepia rerpaeApa, anra* uepnen4rlrcyrfipfluii Ao pe6pa,43. LMK} (lMKc: 90"): Cr = ^{Mc' - MK" = u[ . 3a CD : 2. CK : t2 cv AC =r2Ji rl.n; CO = 6.tE crv. LMOC (ZMOC : g0"): L -,2 , -,2Mo=lb"El)" -(a"lz)" =J4.s4-s6'z= = ^Es6 12 - rGZ = 8 (crvr): Binoaiv:8 cvt, 48g. ABCD - xnaApar, -AC L BD sx 4iaronari KBaApa- ra, AP L (ABC), AP I BD. Maeuo: BD I AC, BD L AP, roai BD I (APC), AK e (APC), toai BD L AI{, r4o ft rpe6a 6yno lonecr 484, LACB - trpaMorcyrurair, roti BC I AC, AP L (AC ro,qi -AP I BC. MaeMo Ba osHaroro nepnergzrcy.napxo npauoi ra rrrorrlr4urz BC L (APC), rozi BC L PC. 1 s.^^^ = :Pc .BC. LACB: 2 AC=cB=4=ry=rJz (cru); 22 LPAC (ZPAC :9O): AC s"tz.z roJ, loG '"-cos3o'* .rE - '6 - 3 ' ^ I 1o'6 -r 25rE 5oJ5 Suca=2. B .o!Z = g =- (cM'). 50Js ,6rdnogrdb: 3 485. Ocxiltrcu O - rovra neperuny ue4ian pinnocropo O - qenrp rioJra, orrrrcanoro HaBnoJro AABC, ro4 'OC : OA: OB - ft - paaiyc qboro KoJra. LMOC: LMOB: LMOAia ABoMa r(areraMr.r (zMoc: zMoB: zMoA:90", oM r (ABC)), rol MC : MB : MA, orwe, M - pisgosig4a.nena nig Beprlrr?ru LABC, ryo fi rpe6a 6y;ro 4onecrz. o'Ji a'Es^.". =+, a =AB, " l" =zzJa; a2 : 27' 4; 44 a.= 2.8. Ja = oJs (*r); oc = 6f = o t"*1. J3 LMOC (IMOC :90"): MC: 10 cm ar( croporrr4 npaM HOIO AO erfineTCbrcoro Tpr{rcyTHr,rRa. Bidnoaida: IO ctr,, 486. SA : SC : SB i SO L (ABC'), roai OA: OC : OB -E - pa,qiyc r(oJra, onficanoro nasxoro AABC. - abc 8.15 17 _ 4s 4../81,11111.(20 - 8x20 - 15 )(2o -17) 2 8.15.17 8.15.1 7 17 = - = d.D{civll. 4J20'72.5-3 4'2'5.2.3 2 I So"to = -MK'CD: LBMK (IBMK :9O"): BM = "*=E9S=i"i9=oJF (cu); c ,, MK = ,lBK" - BM'= J(oJs)' -6' = J6" .2 = oJ2 (cm). 1- so",, = i.uJ, 12=86,[, @M,). Bidnoaida: 36J2 ciu'. 48O. ABCD - porvr6, aa yNrosoo AC - BD : 4 cvt. LMBA(IMBA:9O"): - MB=,,tm'- AB' = J(zJse)'-1s'? = J4€9-100 =.6b6-100 =,1%6 =16 (cna). Ia sracrusocri 4iarona;refi pou6a: ABOA - np,fiMoxyrgr{fr, BO2 + AO2 : AB2; /-2 r*,t2 BD:x;AC:xra; lil +l'l"l =100;' '2) 2 ) -r2 + x2 + 8r + 16 _ ,nn. 4 2x2 +9.x+16 = loo; x, + 4.r + g - 2oo: o;. 4 x2 + 4x - 192: 0; Dr:4+ 192: 196; *=-2!'n, xr< o, x": t2. BD:l2cw. I LMBD (IMBD: 90') no4i6nuft 4o eruuercrxoro tpr4- Kyrulrxa (3 cu, 4 cru, 5 cu). MD : 5. 4 : 20 (cm). BidnoeiOa: 2O cw. 481 , AM L(ABC), AM L BD, AM LAC. AC L BD aa sracrusicrro 4iarona-neft pou6a. Maeuo: BD LAC, BD LAM, roaiBD I(MAC),rosi"BD LMC, rqo ft rpe6a 6y:ro 4orecru. 442. MO L(ABC), MO LOK, MO rCD. LDOC * piego6eApenrfi, OD : OC, OK L CD, ronty qo K - cepeAr4Ha CD. Or:ne, CD I OM, CD L OK, no4i QD L (MOK). A f,t" = t "*, #fitt:r.ff . rr.i:r-J.,_+r: M D 944 {da:; TEOMETPIS Eeas r. fl. ra in. IlpoQinunufi piBeHb i{*rt*-1tl{tiitl:i! ii:':ilii:lfi:::i:iitli rEoMETPlg 6ees f. Il. ra in. flpoQ
  • 46. AO:8,5 cnt. LSOA (ZSOA: 9O"): sA=JsoAo" = Bi1noei}a: L2,5 ctt. 487, F - cepeAr{Ha rirorenyeu,4B, roli A-F : FC : FB' LAMB - pinuo6e4peuafi, roxi MF - ue4iana, sucora; A MF LAB, LMFA: L,MFC sa rpboMa croponalrnu, togi IMFC : IMFA:90o. Ornte, MF L AB, MF I FC' ro1i MF I (ABC). 488. 3a yMoBoIo PA: PC : PB : 26 ctt, PO L (ABC), roAi O - cepeAr.rHa rinoreHyara l-B. I,ac-nc=a; l1ic_ sc=+: llac ac=sa: 1ac BC=te2; l9 lec = a+ nc;.l l(4+ BCTBC =192. BC2 + 4BC - L92 : O; D' : 4 + 1'92 : 196; BC:-2+74;BC:72c;lr, AC:12+4:16(cu). LACB (/ACB: 90"):.4-B : 5 4:20 (mr), AO - l0 (cu). LP,A (tPoA: 90"): Po = JN" - Ao'z = Jr6'z -1o' = {6?6 - 100 =,24 (cv). Ridnocida: 24 cv. 48g. MA: MB : MC i MO I (-4-BC). ro4i O - cepeArrsa rinorenyaa AB' CK - 6i cercrpr.rca, roAi 3a BJacrrnicrro 6icenrprzcr rprrr{yrHr.r*a + = ry = :AK AC .24 LMOB (tMOB: 9O"): l2h' E% 4oo 12 no = JuB" - Mo" ' i!" 100!/-e-" =; = 7,5 (cM). M AB:7,5.2:15(ctu). BC:x;AC:y. lx' + y' = 225' lx'? + u2 = 225: t-t-1J I I I l._= -: lx= Ai Ala24l24 9a'* a2 .22b: 6!S zzs, b/b o{o ,,2 _57G 22b . u =24:]1 =72, _la,+, "625255 7 24.3 2t r=- 24.5 5 P*": AC + BC + AB : L4,4 + 4,2 + 15 : 33'6 (cnn). Bi1noai)a:33,6 cu. 946 i::* fsOuefplg Eees l-. Il. ra ix. IlpoQinunufi 49O. QB: QC:QD: QA,8O I(ABC), O - qeulp r neal ABCD, rosi OB : OC : OD : OA: R - pa4iy qr,oro KoJra, npr4 qr,oMy rpaneqia pinuo6egpena, AB : CD: 10 cu. O e MN, MN - ni,qpiaox, rqo a'egnye cepe,qr,rHrl ocHoB rpareqii. MN : AK : 6 cwt (LAI{B noAi6nr.rfi erzrrer- cbxoMy rpaxyr uur<y. BK = t"ry = 8 (cy), oN:x. 2 LANO (IANO :9O"): R2:4+12- ABMO (ZBMO: 90"): R' : : 100 + (6 - x)'. 4 + x2 :1OO '136 - 72x'l x2; l2x - 132: r: 11 (O aHa- xoAlrrbcfi zoslr.i ABCD). Ao=J4+t2r=Jrx=sJ| LAOQ VAOQ:9O"): AA - ^[Ao oA' = .lf2b + 2* n = !Eb6 + 11) = ^lr Bi}noei}a: 20 cw. 491 . a)AA, L a, BB, I ct, MM, I cx, rosi AA,ll as,I rw u,, rogi,4,, M., B, netxart Ha o4Hift npauifi. Ilpoeeseno ArB l1 Bv rogi ArAr: M,Mz - 11 cu. MM, - cepeAHs :liwia AABAT M M, = l^ A4 = l(11 - b) = 8" 2 " 2' C MM, : lMttW" - M,M"l : lS BiOnoeiOa: 3 cw. 6) LAA2B * A,MMzB, A4 _AB. 16 _4. MM" MB' 1L- x 1' 16 4(11 - r); 16: 44: 4x; 4x:44-76; 4x:28; x:7. MrM :7 cm. Bi)noai)a: 7 cm. M2 ri /r?r...-.-...r. .r. n J t :it^...:.Jr..'1.1. 492. MN - cepeAH.n niriis MBC (ZC :9O"), MN I AB tWtl ll a. AA, L a, BB1 I a, CC, I a, ro1i AA, . BB, ll CC,, AB = AFt = aJ+3 cn; romy rqo,4ArBrB - npamoxyrnzrc. BC, r CC,. BC, - 16 cv; ACz L CCD ACr: 12 cru. LCC2B (ICC2B: 90'): CB'? = CCi + CzB2. LccA VCC$,: 9o"): CA' = CCI + C,A". AACB (1ACB:9O'): Afiz : AC2 + CBz. ccl + 16'z + cc: + t z" = (+Jqs)' ; 2CCi + 16' + 12' = 21*!'4 K20 plBeHb :l:i..t!:i..i*l:l:l:l:l:i:l:...l:i ii{:iitili;l{tii:i# TEOMETPI9 Eees F. Il. ra in. IlpoQ
  • 47. cc: = 8. 43 - 8. 16 - 6 . t2 : 8. CB2 : L22 + 162; CB:2O ct$ CA2.: Bilnoeidt: aylaS; 20; nrli. (43 - 16 - 9) : 8 't8:144; CC':12 "*' '122+122; CA=12J2 ar 493. BK L (ABC)' rot! BK L AD, BK L CD. Maeuo: AD LAB (ax croponut xnagpara), AD L BK, ro4i aa oanaxoro rreprreuAI'rxyJr,fipnocti npa- rroi i nnorquura AD L (ABK), ro1i AD I z4l{. Ana.no- rivno 4ono4raIvro, s{o DC L KC. LABK : LCBK (sa ,qBoMa KareraMrz), roAi AK : CK, LKAD : LKCD (sa rinorenysoro i ratetou)' LABK (ZABK : 9O"): AK = ^[* + x, xc = Ja" + "'; LKAD(II{AD:90'): KD = .[KA' + AD" -- Jrz;7 .] = Jra'z +b'z ' Bidnoeilo: JV*u'; Jz.b'. A 494. Hexafi a L a, a J- p. ,{onegeuo, qo cl ll 0. Ocrci.nrru a I cr, roAi a L b, a L m. 3 roro, rqo a I B, uaelrrro: a L bt, a L rnt, rosi o I b, a Lbr, roni b 1l b,' Anaroriuno zl ll rzr, Maeuo: a ll 0 sa oBHaKoIo napa,renr,uocti nJIorqI'rH. 495. a)Ba yMoBoro cr ll 0, o I o i p, b .l- cr i B. A e a, Be c[ - rotrxr,r reperrauy rignoni4xo rrpfiMr]Ix a i b s o, F; aLAB,bLAB. To,qi a l] b, n1o frr rpe6a 6yno AoBecrrn. 6) Hexail a I B, b I cr. npunycrlrMo, rqo a ll ro"qibIp.aIa. Maeuo: a L B, aI o, ro,qi I ll cl, qo cy[epeqr.rrb yruoni aagaui. On*ce, a i b ue morxytl6Ytra napaJreJrbHr4Mrt IIp,fi M'rM!I' 496, a)frcqoA e tx' rogi upoeegeuo rrPaMY 4, Ae a. By.qyelro b I a, ro,qi b I cr. 6) flrr{o A e c[, ro4i npoBoAHMo 6yAr-sry rpsMy .z' 4 € c[. rlepea.A 6yayeuo a' ll a. Eyayeuob,Aeb,bIa, ro.uibIa,bLa' 497. Bitouo, rr1o rorrxu, piaHosiAAareni si.q riHqis siApiara, Jlerlcarb Ha cepe,4I,IHHoMy rrepfleu,qrrxyJrspi, nxuft fipoBeAeHo Ao AaHoro siApiafta. Hexafi A, B * gaAaui rotrxr'r, 4 - AaHa nparrna. Bygyeuo cepegnxnzft fiepflenAr4lcy- nsp Ao eiApisnaAB, na neperuni rrboro fieprreuArrr(yJrapa 3 AaHoro flpsMoro o srra- xo,qrnrbc,s rovra X, AX : BX (ua;1. a, 6). V nznagry, xonvt AB I o i cepeAlrna O ni1pisxa AB Harre,r(rzrb a, lo4i O - firyxa- Ha roqna (rual. o). Sagaua ne rrae poae'asxin, sxuo AB L a i a se fipoxoArirb cepes cepeauuy ni4pioxa ,48 * rouxy O (uaa. r). 948 {ii:ii reorvrrrple sees F. fl. ra ix' l'lpoQinsm}rfi piBeHb iiliiiliilliiiiliiii 499. feouerpravne micqe rouox, pinnoni4AaJreur,rx nig Arrrbcs Ha nepuengurcy.napi Ao nJrorqr.rHn rpr,rRyrHr{Ra, r{oJra, orrzcauoro HaBHoJro rpr,rxyrHr,rfta. a L (ABC), X 500. Hexafi LADC - pinnb6e4pexuft, AC - froro ocrroB a LABC - pirno6egpenrafi., AC - fioro ocsoBa. llposeAervro DK IAC, ro.qi K - cepeArlnaAC, roti BK L Maeuo: AC L DK, AC I BK, ro4i oa osnarcoxr nepnen rcynapnocri npauoi i urorquau AC L (DKB), rogi AC qo fi rpe6a 6y:ro 4onecru. BC, ro4i LABC - pinrio6e4peuuf BK I AB, ro,4i K - cepe4lrnaAC. Ananori.rno, LADC * piono6eg DK IAC. AC L (BKD), roxi AC L BD. 5O2, Hexaft KABC i PABC - nparrnnsni rerpaeApn, O - qenrp MBC (MBC - npanr.r.nrxzft), rogi K - pinnoeig4a.nena nig nepurlrn AAJIC, ro.qi nepnes,quxynep KO (KO I (ABC) npoxoAr,rrb qepeo qeHrp rpzryrnuxa -A-BC. a) 6) 498. Toqxa X, pirHosiaAareHa nig ycix aepurnn n nep[eHAurcyJr,Epi o, sxift [poxoAr,rrb rrepeo roqny aL(ABC,Xea. 501, 3a yuonoro.A.B : B' ^&" A'-+ satp nep Ana,rori.ruo, PO I (Aj,C>, o - uenrp AABC, ' P ro,qi KP L (ABC). ,/ 503. ocxi,rrxu PABC _ npanr.r.nrnrafi rerpaeAp, ^(rosiPA:PB:PC,POL(ABC),'- lO ro4iO-qexrprpfiKyrrruxaABC. Y 504. Hexafi LAIIC - nparu.nrnraft, AAl L (ABC), B& L (ABC, ABBAI - r{Ba.qpar. LAA!: A.BB,C (aa 4noua rcareramlr), rogi A,C : B,C. Orxte, LCAtBt - pinno6erpeH PMc: zp,ro4i BC ='!, ",c = BC.J, =lA. fiill{l{iill$t{ilit}i TEOMETPIn Eeeg r. fl. ra in. IlpoQ /[ / B l'l--- 7
  • 48. n - n 2p 6 , 2p -2 obJi. 1): cM _ A,8,, sM,B,r 'u,Bf- " .q "- 3 -g' LCBrM (zCMB, :90'): cM = gn:'- nrni = {+")" -[*)' = ffi=euz, ^ l2ppn p'{z 5-,q.=rTiu'= 9; . . +n! , B,c' - A,c' ,c.oslB. =coslA. = _ L- - 2 4B,.BS zP "!i 'p Ji3 xw=rE4*nN'= 8"" oJ6 V+ 2 506. ABCD - pisHo6iuHa rpaneqir. 3a yrvronoro A-BC LACD - nlllacauuir y re )r{ r{oJro. 3a xac.ni4xom e reo .ACDeMu c[HVclB: , = 2R, R - pa,qiyc fiora, EHe sin 1AD( onucano lg,arxono ABCD. d =2R: n= d sinn 2sino Bidnoaidt': 2sins $ 14. I-lepne*AuKyncp i noxnna Ao nnorqrHlr 512, AC L cx, AB - rroXntrra, BC - ii npoexqia, ec = JAB' - Ac' = lio' - n' = 30 (cM). Bi1noeiOu: 3O cttt. 513. MO I rt. Hexail MA: 13"r (cm), MB - 5x (cv),4e -r cM - aoB)lchHa o4Hici uacrnHn MA> MB, rosi OA> OB, OA= 4^|10 cu, OB : 4 L,MA: Mo2 - MA2 - oA'z - (Llx)'z- (a"[o)' . LMOBI MO2 : MBz _ OB, : (bxz _ 4r; ; 769x2 - 160 : 25x2 - l6i 144x'z : 14,4; x2-t;x-1. Mo=Js, 4, =B (cm). Bidnoai1a: 3 cvt. Ie gnox noxnnnx, [poBeAenr4x a ogniei Toqxrr Ao flJrorq uesmufi Kyr Hax[Jy Ao trroqrrnu. 514. LMOA (LMOA : 9O'): IMAO : 3O'. LMOB (IMOB:90"): ZMBO : 45', roai OA > OB, MA > MB. OB : a, twB = ali; LMOA (IMOA: 9O'): MA: 2MO : 2a. BiTnoeiTa:2a; ali. 515. LPOA(IPOA:90"): PO:OA, PA=sJz; OA: LPOB (ZPOB: 90"): PB : 2a, OB = actgS}' = aJ3; AAOB (1AOB: 90'): -4,B = JOA'z + OB' = ^,8 * U' D^2 + PB2 - PB2 2a2 r 4a2 . 4a PAB: cos z APB" 2.pA.pB - z.ali.2a 2a2 I J2 .l; : zAPB = arccos Y. 4J2a' 2J2 4 4 .t; Bidnoeidt: arccosY. 4 L,tn. 2" CM, ?zJz2 D o I 2." n. 4" oP' l aE 8 ^z,lq 2^12 4 gn "" l2 ^.6' -(2p .a' -l zp 1' costc = Bp2 ) Ap2 - A1e,' _ls/"] _-lT':l I e ] = 2 BP'A,c , ?pJ, +J, 4 (Jz' o' , a (Jzl' o' .4P' 72P' 9,"-.r 9,,-." 9 g _12-B -- tq-, - t6F - 76 A' gt' 9 Bidnoeida: ? rQrE *1, !'fi , 9, fr t !. 3f-!- ^'r 9' 4' 4'4' 5O5. MN - cepeAHn ninia AA.B,C,, rosi MN ll lrc,, l,tN ll ac; 'qsco - HBaApar' roal BD L AC, BtDrll Bn; twN AtCr, ropi MN L BD. ArCrll AC, ropi lArC'K - xyr uixc KC i 'AC. Hexafi pe6po xy6a - a, roai A,K =1; a,C, = onli. tKa,Cr(zKA,C, - 9o'1: 2 " tplAC.K=4K , o =. t-=!', A,C, 2.oJ2 2,12 4 ,t; z4C6 - arctg f,; M N 1l A,Cr: ArCr ll AC. ,NMK -xyr vir+t MK i AC. MN = KM "{, LKA,N (lKArN :90"): F" r'- .l- + a' + - = !4 4 cosIKMN = n 2o' 6a' 2a2 _ 4 4_ 4= a' a' ,42 422 2, KM.MN 1 -i IKMN: 120". Bi0noaida:0": 90'; ur"rf {Z; rrO". 2 - .2 laJ6l "("1?''-f,"6IK^!__yt_!!'=' z ):l z ) 950 ;ii:i fEOMETPIg 6ees f. Il. ra ix. IlpoQinuxufi pieenu:!;iii;i:iiiii;:iiliiii;!i;: Iitl:iiil$tiii{l*i;;t TEOMETPI9 6ees F. Fl. ra ix. IlpoQ
  • 49. 516,-MSB: ln=^l 'qS' +,BS'z-2 /.s Bs'coslASB= = Jro, +200 - z. 1s .z0 {l = G6e - 208 = J361 = 19 (crvr). MSo (zAoS: 90"): So = 'lls' - % = 12 (cru); LSOB (1SOB: 90"): oB -- J4OO -144 = J256 = 16 (cm). Ao2+Bo2-AB' 5" +76'-lqt 2.5.16 s A ^BNC(tBNC - e0"): cN =,84'4 =fis€0 =v Bilnoai7a: 6fi5 cn; 26 cm. 2.AO.BO 25+256-361 80 1 - 160 - 160 2' zAoB: L20'. Bi1noei1t: I2O', 517. LMAB (IMAB - 9O"): MB = ali, MA L AC, LMAC - nParroxYrnzfi. LABC: ACz = AB2 + BC2 -2AB BC cos120" = 1 =a'na'+2.a2 1=3a';2 AC = aly. LMAC: MC -- ^tMA'; Ae = ^rF + w' = za. Bilnoaia: aJ}; 2a. 518, AC - npoexqia noxutro MC, rosi MA I (ABC), MA L AC. LMAC: AC : aclg a; AC L BD; AO:OC,BO:OD, toAD=+' 2 LAOD: AD= AO - actgo. d-c{' cos 2 Zcost .^ octgd BtOnOALOb: 2 "os'12 519. .4-B -L ct, roAi BD, BC - noxr'uri, AD i AC - ix si.qnoeiAgi npoercqii, ABAC: ABAD. fxqo BD - BC (sa rinorenyaoro i Hareronn), ropiAD: AC' 520. 3a yr4oBoro C e a, ABlla. flponegeno AM L a, BN I s, rogi MC - npoex4in AC na a' NC - npoenqia BC Ea cL, MN - npoerqia AB sa a. MN : AB, roMy Irlo A B AM ll BN, AMNB - rpsMoxyrnl{n. 3a yuonoro BC - AC: 4 crrl, to4i nexaft AC : x (cw), BC:14+t(cM). tc2 + (14 r x)z : 2U'' )c2 + lgi + 28tc + x2 - 676 : 0; 2x2 't 28r - 480 : 0; rc2 + l4tc - 24O :0; Dr:49 + 24O :289; x :'7 tL7; x, < O, x": !0. AC : LO cw, BC : 24 cw. 521. Hexafi ABCD * pon6, /D e a, BB, L a, CC, I BB, - CC,. LAC,C (lACrC : 90"): Ac, = ^,lW * cci = fre'? - 8'z =,[-s.z+= vE .a € = = 8r,6 (cu) - npoexqir AC Ha c. LDB,B (ZDB,B :9Ol: DB, = ,!on'- en1 = ,ll't - e' = J{. n = 4JB (cm) npoexqir BD na g, 3a n.nacrnricr:ro 4iarona.neft port6a AC L BD, LAOB: cos/AOB= LA2D (zAoD: 90") /, =,tAa Do'= VE'z + 6'z = ^ABIB (1ABp: e0.) /Bl = @rE{ = fio'? - na a. LDC.C : MB1B,togi DC, : ABr:6 crra * np rra cr - BtCb BtCt: BC : 10 crr, Bilnoeio: 8J5; lG; 10 cm; 6 cnr. 522. 3a yMoBoro cr 11 0, ro,qi, srcrqo AIl I B, CKIF,ToAH:CK. LBHA: AH =,11,s" *% = 12 (cu); LCKD (ICKD:9O")l KD =,lsz' -Lz' =,lrs. 49 = 85 (cu). .. Bi0noai0u: 35 at 523. 3 rourtu B, B e a, npore4euo BC, BC L a, roli CA * npoer Ilpone4euo NA L CA (N,4 rogi ea reopelrorc npo rpr{ iiiihl:liiH:ir : :rb.:.t ..n tal ., /:F;.-.^r-.^1..-r... ii$iiililrlll ",:Y:ir....;.r...:.:a.r,:l[{Tr..q...;;1; BA L NA. ,KM II NA, KM e a, KM Lcr, ri{o ft tpe6a 6y,ro 4onecr:r. 524. BK LAC, ane BK ne yrnoproe 90'a BC. 525. Hexafi y MBC AB : BC i BM L (ABC'). M Ilpone4ewro BK LAC (K - cepe4unaAC), ro4i ea reope*roro fipo rpr4 nepIIeIrAuRyJrspr4 MK IAC (MK * roxrrrra Ao (.4.8C), BK * 'ii npoercqia na (.43C)). 526. Hexaft ABCD * npamonyrnun, OM L (ABC,), ON L CD, MN I CD Ba reopeMoro fipo rprr ilepneuArafiyJr.spu, Arraaoriqno OK L BC, MK I BC. s52 l*: rEouerplg sees r. ll. ra in. IlpoQinunui piBeHb lirt{iiltttiil{iii;; iiLii;:+i:,:ttiiii:i: TEOMETPI9 5eB3 f. Il. ra ix. Ilpog
  • 50. 527. Hexafi AABC - nPrmoxYrttzfi, lc :90", MB L (ABC), BC - npoerqia MC ua (ABC), BC L AC 3a YMoBoto' rofi, MC L CA sa reopeMoro rlpo rpr{ fiepneHAr'rr{yJrfipr4' ro4iKcninnagaeaC. 528, Hexafi LABC - pinnocropoHniit, M O - ro-rna neperr'rgy fioro rue4ran' ro6ro O - qeHTP TPlrrcYTHuKa, OK LAB, roil MK LAB, roMy nlo OM I (ABC)' MK - noxu,la, OK -'ii npoercqix sa (ABC). 529. ABCDABTCTD, - ttY6, rogi ABCD - KBaAPar' BD L AC sx 4iaroHa,ri rcBaApara. Y LACP npoBeAeMo ON ll AC" 3a reopeMoro npo rpu nepneHAlrnyJlspu NO L BO' ro41 ACt L BD. 53O. Hexafi ABCDA.BSpT - npflMorcyrHnii napa're;renine4' C, roti ABCD - flPflMoKYrHnr(. V' flHqoAC' -L BD, ro.qi ON I BD -, (oN ll AC,). '" to4i npoenqis ON ua (ABC) - CO. CO L BD, C MaeMo ABCD - KBaAPar. AD 531. LABC - PinnocroPonuifi, K - cepe4uua BC, loai AK I BC. -A1{ - npoercqia MK aa (ABC)' Ba reopeMolo npo rprr trepneuArnyJr.spu MK I BC' 532, ABCD - npflMor{YrHr4n, MB L (ABC), AB - nPoenqia MA aa (ABC)' AB IAD, roli MA L AD ea reopeMoro trpo rprr neprrenArnyJr'flptl' Ananoriqno BC - nPoexria MC' BC LCD, MC LCD. Macno: LMAD i LMCD - nPauonYtHi' 533. S, -BC - reTpaeAp, MCB npsmoxyruuil, /ACB :90" AS L (ABC), roai AC - upoen4ia SC ua (ABC), to,qi SC L BC sa reopeMor) npo rpu trepneH,qr.rnyrspr. Orxce, 6ivHi rpani npRuorcyrHi Tpr4tyrHr.rxr,r. 534. ABCD - IIpaBr.rJrbHr.rrr TeTpaeAp, A A1 nepnen4lrxy.napni. 535. Hexaii. SCDArBrCrD, - t<y6, rogi xorxHa fioro rpanr, HBa"qpar. oot r (ABC), OM I AD (arc pa4iyc Bnr.rcanoro Hora y HBa,4pat ABC Ba reopeMoro npo rprr nepnerrAarcyJrfipu OM L AD. OD 1l BpD BBt L (ABC), BrO - noxuna ao (ABC), BO1 - 'ii npoerqin aa (ABC), - ro4i BO, I AC (sa sracrzsicrrc 4iarona.nert Hea4para) BtOt L AC., rosi OD I AC, KL ll aC. Orxre, OD L KL. 536. Hexaft,4-BCD - po*r6, AB : BC : CD : AD : 2O ZBAD :60", ro,4i LBAD - pieuocroponniii, BD : 2O cxr. AK: BK, MK L(ABC. 3a eracrnnicrm 4iaroHareif'pou6a AC L BD, rogi rrpo KN ll AC, KN L BD, Ba reopeMorc npo rpn treprTerrA MN I BD. Aga.noriqHo KF ll BD, rosi KF L AC, MF 11 KF---BO = - 10=5 (cu). 22 LMKF (IMKF : 9A'): MF ='tTr' - 5" = ./i69 - 13 rN =!ao = * 6 = rJd (cu). 222 = $44 + ToAi rco'fiHa rpaHb - nparnrlu oA L (BCD), oK t cD, OK - npoerqia AI{ ua (ABC, roai AK L CD. D cD L (ABK, AD LAB. Or*ce, nporn.nexni pe6pa iirkii:i:ll$, tFi..'.,.'+.1..r.. i !:f,$:l-l::::::::::i. ,i !n:I-....44..:.4i r:ei,r.r..'1..i.'c.r 'r;i$fiiiLilif:' k:..r-...i.irr'... riftl:i:iilli LMKN (ZMKN :9O.: MN Bilnoaidv:5 cn; J219 cu. 537. Hexafi y LABC /ABC:9O'. O - cepegrna AC, MO L {ABC). Ilpoeeseno OFllBC, ro4iOF IAB,rosisi' -1,48, S.F ri4craur si4 S 4o AB, Ana.nori-rso OM ) AB, OM L B SM L BC, SM - siAcraur ni4 S 4o BC. or=!nc=9 (cm). 2 tz'+(sntz)" g54 t:::i: TEOMETPIS Eees F. Il. ra ix. IlpoQinuxnft piBeHb i;:ii;iiii;::iii;:::ii;::ii :iiiiit:i:i*llit::1:i rEoMETPlg Eees l-. rl. ra ix. l-lpoQ
  • 51. LS2F (zSoF: 90"): SF = ,172" - r" = JiZ4 n 81 = '12% = 15 (crrr). ol,t =!ln= 5 (cu). 2 LSOM (ZSOM:90'): SM=JSO'* OM" =Jt-44+25 =fi69 = 13 (cu). Bi0noaida:13 cu; 15 cna. 538. Ifenrp Bnr,rcanoro B TplrrcyrHnrc RoJra sHaxoArrrbc,fl Ha neperlrHi 6icexrplrc rpu- nyrrlsra, ro,qi OK L BC, OF L AB, ON I AC, OK : OF : ON : r - pa4iyc nnuca- Horo Korra. Ia (popmy;rr.r s = 1p., lp =As* nS * 'qc . 222 S - n.norqa rpnrcyrrruxa. Bupaaxmo pa,qiyc r. o - L3+ 4+L5 = 21 (cM); s = J21 . (21 - 13X21 - L4)(2r - t5) = =Js'r,8t.z.B=8.4.7 =84 (crvrr); t s84r===- =4 (cM). l p21 Ba reopeuorc flpo rpr,r flepneHArrrcynspa MF L AB, MN LAC, KM I BC, LMOF : LMON : LMOK, rosi MF : MN : MK; LMOF - eranercrrclrt, roruy MF : 5 cu. BiOnoai1u: 5 c;la. 539. Hexafi LABC - Ba,qaHrfi, AB: BC: 10 cM, LABC: flposeAetro BO LAC (O - cepegznaAC), roqi MO -L AC (reoperr,ra rpo rprr nepner4raxy;rapr.r),, MO - si4cran:. ni4 M to AC, 3a reopenorc xocnnycia: ACz : AB2 + BC2 - 2AB . BC .cos.1ABC: AC= '6. 10' = ro.6 (crvr); a,MCK (ZMCK :9O"): yc=JMK'-cK' =fo,-r,o, =./6,4.19,o =o BiOnoai1a: 2,8 cm. 541. ABCD - rerpaeAp, sa yMoBoro nci pe6pa rerpa rpanb - npann.nrnzri rprrxyrrr,rfi. DO I (ABC), O _ qeHrp A.43C, OK L AC, ropi DK L AC. AC L (BDK) LBDK - nepepis rerpaeApa flJroilInHoro, fifta fipoxo Ar4rb qepea DO i nepneu4r,rxyaspna 4o AC. LBKD - piono6e4penzft, BK : DK _ Br,rcorrr rpr.r xyrnrznin, af =$; DA o o,fs "" - G = B - paAlyc ollucaHoro rcoJra HaBrdo_ ao LABC. LDOB (ZDOB:9O.): Do =JDB Bo' =.i", -- - 7V - " o. v - e =/-=tvo; s*,* =.f er.Do=! aE uVa -o'JI8 -Ba',li2223'"L2-n-4. ^2-l;-6nno6tOb: 4 542. 3a yuonorcAM - MB : 7 cv., AO - OB : Il c:v., ro4i MA : AM : B : 5 (MOB no4i6nraft 4o err.rnercrrc - Hexafi AM : 5k (cw), AO : 4h (cvt), MO : Bh (cyt), rosi MB : (5k - 7) cw, BO - (4k - lL) cut. Maemo pinnsurttis LMOB: gh, + (4h - l1X - gh _ 9k2 : (5h - 7)2 - (4k - 1t)2; th, : (5k - 7 : 4h + tl)(Sk _ 7 + 4h _ t7); thr: (k + 4x9e - 18); thz : th2 - L8k + 36h - 4. L8; 18ft:4.18: &: 4,roaiMO: B.4:12 (cu). Bidnoei1t: 12 cv.. 543. 3a yMoBoro ZAOB:90", ro4i AAOB _ npamoxyr C - cepe4zna AB, rop!AC: OC: BC; OC: 10 "*.LPOC (ZPOC :9O.): PO: OC.tgZPCO: 10.0,b: b (cm). APOB (zPoB :90"): Pp = '15'z + 162 = J2b + zsa =.88r 1cm;. LAOB - no4i6uuir erurrercbxoMy, AO: 12 cu. Jlxrqo s o4niei Torrnu Ao nJrorqr,rur.r [poBeAeuo 4ni noxv.ni, ro 6inrura ra, npoenqia sxoi 6ilrura. Maemo: OC < AO < BO, ro4i pB - nafi6inura noxn_ oc = 5J5 (cM). LMOC (IMOC:9O"), rosi Mo = .,ffic' - oc' = ^1125 - 2b.s = ,lzrb J5 = JISI .^ -EItldnoqlob: Dvb cM. 54O, AACB - rpsMoxyrHr,rfr, MC L (ABC), MK - roxnrra Ao rogi iI npoexqis CK, CK -L /*B ga re- opeMox) npo Tpn nepflenAL.KyJr,fiprr Au = JAc Bc' = ,lt2 16" = 2o (cu); CB = JBK,AB; 162 : BK . 2O; BK _ 16 16 _ 4.16 _ 64. 20 5 s', B = ^l%i = 5Jo (cu). (ABC), . iltr:......,..-'... : r i:iil[r:.:1..:.:.:.:.: ! /,{I;.-r..-tr:.i.:. I r,Lit'-r-.-.:1.-r-.:. r,*ifir:t...J:..7 AK=2o-9!=zo-1p!=z! 555 CK = JAI( .BK = 36 l) 6.8 4A - -(lA /nn, bb na, CP - naftmenrra. AP}C (ZPOC: 90.): pC = G-tot = .,[! - 100 = ,E Bidnoeiu: J281 "*; b€ cn. 544. O - rtreurp Brrrlcanoro a AABC rco;ra, ro4i ON L ON : OK : OF : r - pa4iyc Brrr,rcaHoro xo.na. Ba reop tapu MN L BC, MK LAB, MF LAC. itili{il{{if:j{il:j rEOMETpte Eees r. n. ra ix. npoe956 i{::: TEOMETPIS EeB3 f. Il. ra in. ItpoQinuxri ptBeHb ili::;iili:i!tiii:;$:i
  • 52. 3a sracrusicrrc ni4pienin Aorr{qHnx' ari nponegeni e ogniei rouxra: AF: Ak: 5 cu; CF: CN: 12 cu; BK: BN: r' Maeuo pianantts: 172 : (5 + r)' + (I2 + r)2; 28g :25 + 10r * 12 + L44 + 24r + f; 2r2 t 34r F 169 - 289 :0: 2r2t34r-12O:O; f+!7r-60:0; : -2Oi rr: 3, r, < 0' AMON_LMOK= L'MOF: LMON - erunercr,rrzft, roAi MN : 5 cu' Bi1noai)u: 5 cv. 545. 3a yMoBoro o ll 0',q-a r F' flponegemo MK LF, rcfiAB ll MK, ABKM - flPflMor{Yrrrnr' BK :3 cw. AKBN sa teopeMoro nocanYcie: NK'z = BK2 + BN2 -2'BK'BN'cosIKBN = =9+64-2'3'8 NK: 7 cu. Bi1noai4t': 7 cttt. 546. flpore4e vto AM L c,' CN I o; ocxi;rr'xr AO : CO' to AM : CN ' MD - rpoert- ^i--i rnni ix A uig cropourz AD na a. V pou6i eci croponu pinni' lo'4i ix npoercqii ua cr Pieni. I t 'l2x 5x;s*,, = i,AC. BDt 12o = , 3Ox2 : L2O; x2 : 4; x :2 - Aonixuna oAgiei qacrunl4 ' Ha sri nogi.neuo 4iaronali pom6a; AC : 12 ' 2 : 24 (cttt); BD:5'2:10(cu); LAOD - nPanonYrnnft, An = i,qO" * A' = fL2" * "; - 13 (cu): LAMD(1AMD:9O"): MD = ^[AD' - Arrt = $i, - 44 ='E-zB = sJs NM - nPoexqi-sAC na a; NM : 2MO; MMO: MN - 2O clt. Bilnoeila:10 cm; 20 c*r; 5J5 ctna' 547. O - {eHrp icoJra' sxe ,4orr4fta€rbcfl go cropin AP i AK' roriAO - 6icexrPuca IPAK' ZOAK =Z' Bi4piaxra Aorr{qur4x' npoBeAeHrx s oAuiei rotlxrz Ao Ho;ra, pinni, rouY CK : CM i BP: BM, roti PA + AK : BP + AB + AC + CK :2P' ate ^POA: LKOA (ea rinoreHyaoro ra rocrpllM Ky- rou), ro4i AP : AK : P; LOAK (1AOK: 90'): LsoK (zsoK: 90"): 7, = 49; 2 (cu); Mo --'hz' - a"+ = 10 (cM); , Fi. :ii:tr:i::::::i:.:i::, I hbi.);.^.rr.-'....rr . ,.ab),r;1,1.J.1't Ir;liml{*iii:it :r,lr$:ii:*i+::l: oK = AK.@Z= ptsZ, 958 ltir rEoMETPle 6ese r. n. ra in. flpoQinrxyrfi piBeHb lliitl:ili*ili{*iii ,'d, sr< =.oK =0"u, . cos P cos P Ocxi,rrxa OK : OM : OP : -R (sa ymonoro), ro4i Sl( p@: Bidnocidu: z cos B 548. 3a yMoBoTo LABC -npaeulsuufi, BC e o. flposea enqin ,4C ua a, AB - AC, rozi CO . BO - Z"E . d't't -megiana, [poBeAeua ,qo BC, roAi MO - i npoexqir s*"" = 1616 cn2; AC ,/'uJ: 4 = 4.2= 8 (cu). JS LAOC (1AOC :9O"): Ao=,tAC2 -co2 -V';-kv?f =ias-ze= = Jso = G (cv). 4114 = a*= + = 4u[ 1cu). LOMC (toMC :9O': J-_ MO=,ICO, - MC, =ll2,l7) 4, LAOM (1AOM = 9O"): Mo = JAAr' - Ao2 = r/(+,6)' -6' = JZ8 86 - Jl, BK - vegiaaa AA-BC, nposeAemo KN La, KN LAO exqix BKrra o. I{N cepe,qHs niaia LAOC,l(N: alt = Jnx'-KN'= r/(+'6)' 3'= J4s - 9 '6g ( npoBeAeHa s Beplrr,rHr4 C, 6yAe AopissroBar[ BN. Bi|noeit: 2J5 cu; 169 "*; 169 ".. 549. Ba yuosoro -4-BCD - rpaireqix, AB : CD, A-B e a; OM I a.Ilpose4euo CK L a, rosi CK ll Ol't, f e ,A-tt't Asaroriqno BF I cr, ro.qi BF ll otttt, af I CK. BC KF -npaMorc)nHux, ropiEK - npoenqia BC :r.a(J., FK: BC: : 5 cu. Posrrsseuo rpaneqiro,ABCD (mar. 6), CC,: BBr; CC, L AD; Cp : (15 - 5) : 2: 5 (cm); ICDC.: 180" - 120" : 6O"; CD : 2 . CtD: 10 (cm). a) 6) LAM7 * LA-I(c (van. a1: 49 =o* =!,AC CK 4' DK - npoexqis CD s.a a. = J2s t6 - tT, ^u--*i.aKA3 LDKC (IDKC: 90"): DI{ = {ED, - CK'z = ",[0' - 8' iifiiiti;iiifiill* TEOMETPIS Eees r. fl. ra in. IlpoQ
  • 53. S** = ,a A Pieginoxr.uriMaro'rbpiBrrirpoeRqii(AB:CD)'roAAF:Blf:6cM' Bi|noai|a:6 cu; 6 cui 5 crt; 15 clvt' 55O. AK L BD, BK LBD, AK _- BK =$ ,*= ",' KM = ^[AK" - An4' -- s*"=IB.LM (t e cD; B BL : AL a Pionocri LACL : LBCL); LM = Je' - AM'; AL > AK (rinotenY' sa 6irrrga riA rcateta), rcai LM > KM, ro!i a' S*, > S**; S*r = ftZt ro6ro uafiueuurY nJronly Ma€ nepepia, yrnopenuft BncoraMI'I rpanefi (BCD) i (ACD)' 551 . AII L (ABC), AB L BC, roti HB L BC, AD L DC' HD L DC' Mae*ro: LHBC (IHBC :90')t H tt- BC =,[HC, - Ha, = $? -b, = Jo.ro = aG (crnr). LHDC (ZHDC :9O": Dc = ^{Hc" - HD" = 'llf -:o" = 'E n = Jzr (c'")' LHAD (tHAD : S0'l ----' 4 en = rEx - an' = {10' -(aJa)- = Jroo-96 = J4 =2 (cu)' Bilnoeiia: 2 cttt. 552. Hexaft ABCDAP{IDr - eY6, LAB.D. - pisno6elpeuuir' ABt na;ri rpanefi, AO L BPr ate BvD, L AlCv rogi BrDr L (AAP), tog BrD, L AtC' oK ,+K L 40 tr 4,KO N ICKA, *r = iC = t AC' LAA.O (ZAATO :90')z (AAt: a; AO t BK, ro4i AO L (BCD), CM L BD, rozi AM L rreHAr.rryJr,fipr4. Bucoru rplrxyrnr,rlca flepernHarorbcf AL L BC, BC I (ALD, BC L AD. 554, Hexafi ABCDATBTCTD. * xy6, rosi ABCD - rna a rl2 KMi .D : ,4-D, - Aiaro- KareraMlr, AB : CD, glt = nX = 4!; 2 roli ZABM : ZNAD i Z;MA: ZAND. LAKMT IKAM + IKMA:90", ro,qi ZAKM :90o AI( L BM, a6o AN L BM. BBt L (ABC), roai BB, Maerro: A1V I B B, i AN L BH; Ba oaualcoro nepireHAr4 nocri nparioi i nlorqunra AN I (BBM, ro4i -4iy' L rqo it rpe6a 6y;ro gonecru. 555, Hexaii,4BCDA,B.C,D1- npsMonyrHrrfi napa.ne.rre ea yuororo,4-1y' : ND : | : 7, ro1i AN : AD : I : 8; Za:8.=1-,4][. HexafiF-cepe4rana DC,rogi np 4'-2' AABN - ttnr: {=g= l; .o,qi IABN : I ND DF 2' ZANB: lA?D.Tloaua'ruwo F - roirxy reperr4gy BN MEN: ZEAN + ZENA: 90", ro.qi ZAEN : 90., AI L BN, AF ll KM, BB L (ABC), ro4i-BB, -L.A-F. M ,4-t'I (BB'N) Ba ooHaroro nepnenjlrzry.ncpuocri upa urorqrrnr4, onxe, -A-F I BrN, ro4i KM I BN, rqo ft . 6y.no AoBecrr. 556. a)ABCD * pou6, ro4i sa n.nacrzsic,rp O - cep ri9rol;lnefr.. AM f @CD), rogi AO L BD (AABD -.piriro6egpenrzft, -AO - fioro me4iana, EpoBeAeHa Ao ocuo nu), OM - npoexqia AO, OM L BD. Ane LBCD -pinno6e4penuft, BD - ftoro ocnoea, rogi CO L BD 'Ornce, npoexqis AO newusr na npamift CO. 6) aHa.norivuo go a). 557. ABCD * ponn6, O - ro.ma reperr,rxy fioro 4iaro aaneft, O - pinnori4ga.nena ni4 cropin povr6a OK L AB, OK : r - paAiyc BnrrcaHoro n pou6 xo.na Ba reopeMoro npo rpr4 nepfieuAfiKyJr,fipru MK I CB, T M - pinnoni44a.nena ni4 cropir pou6a. AC : 4h, BD (ft (cu) - AoBncr4Ha ogniei a pinuux vacrr.tr, na ari .neni 4iaronani porvr6a), AO : 2k (cwr); DO : L,1h. MDo (ZAOD : 90"): AD =.tW * oD L02 - 4h2 + 2,25k2; 100 : 6,25h2; 100 I ro 'h' = _ = i - | = 4.: h : 4 (cv): AC : 16 cu; 6,25 2,5 ) BD:3.4:12(clr), 11 ro.Ei S*., =;o".UO=;.16'72= 8.12=96 (cu,) ffi I F = "1, 4K 4c W t/ z %=i, oK=Ao"=*=fr, 4x = oJa i= ft, LA,K: A,o2 : A,Kz * Ko2; ( oJi' a' a' a' iaz a2 a" i;l =B*6'T=Gt 2=T' to6to AA'KO - rpaMonyrlrl4i^, /AIKO : g0', rotiAO IA'C. Maeuo: AP L (ABp), rqo i tpe6a 6yno gooecru' 553. Hexaft ABCD - rerpaeAp' rrpoBe,qeMo BK LCD' 3a y'rvroooro CD I AB, ro ai C D L (ABK, rogi C D L AK' - 1 2 = ? 99 =4,8 (cm); oK:smco = jPreca' ri r = V = l,.tO illtllitiiiif;}.i:iii rEoMErPIe 6ees F. It. ra ix. npo 31 vci rlq to s., c. 1 960 l:i{: TEOMETPIf, 6ees r. 11. ra in. IlpoQinsnrr piEeHb lili:i$tiii{iil{ii
  • 54. LMOK (LMOK :9O": MK ="lfiooK' = W4'8' = J4+4,*4,s = =.f4a+!,,.4,8, =2Jt;5J6 =2J6,6J6 =2 2'6 - 5'' ("*]; Bi)noaida: 5,2 cttr. 558, Ilpone4e wo BK L AC, ro.qi SK -L AC' 5 1 S*", = -i AC'SK. S*" = rfptp - oP - aXP -,), AB+BC+AC. Ee p=--- 2 -,AB : a : 15, BC : b : I3, AC : c : 14' s*"=J2t'6 sn =r!'{2'g'2'4:7 =2'3 s*" =f,nr ' AC; BK ='E ='+ =2']2 =n @$' LSBK {tsBK: 90'): sK = ^tsB BK' = Jzs + u+ = 13 (cu): s =1 13.14=7.13=91 (crrn'z). "AscA 2 Bidnoai}u: 9l cttr2' 559.KBL(ABC),BCLAC'ropiKCLACsaTeopeMolonpoTpI,InepneHAI4KyJI'fipI{. .* - = l-cx . AC: LKBC (IKBC:9o"): -MLK 2 BC-h'cLga: KC=-L , sln u " =1. h .t"tno=ft'"o?ovM.K 2 sina - 2sin,n h2 ctgcr aoo sMcx = t"t", "' h2 cte a Bi1noeida: tRA 560. SO L (ABC), rpoBeAeMo MN ll BC, D e MN i e u;iorqzni (ASL':LL I BC' npo- ;;;;.-o; llas . in ll r'rtr, (MDEll BC i (MDDIIAS' MDEN - uryxanraii nepepis' Ocriarnu rerpaeAp rrpa- strnt uuir, roAi AS I BC, togi KO L MN ' snr"" = rrtov ro, ' M* =?, uN =2!LAMN*MCB,rotil-I( 3 U (O - ar roqrca [eperr4Hy ue4ian npaBrrJrbr{oro rpr'rryr- nznaiAO t OL:2: I). MoK-mas:ffi=!' ox=i, 2a MDEN - rpfiMoxyrutrrci Swpar = tttt't 'KO = S 2a2 Bidnoaida: -;. 962 il{i reovrerplg 6ees r. Il' ra in' rlpoQinsnufi piBeHb tltl:t*ti::i:li:i::::i 561 . Hexafi ,4-BC DA.B.C LD | - npfiMorcyrHrrft napare. roli aa yuonoro AB : AD : a, 44 = a.1Q; enCO rtBa,4par, BD - ali, rogi BB.D,D - KBaApar, BrD L BDt Hexafr M - cepepwa AAr, rogl MF ll A Mprll BO, MrOr I MI, sa reopeuorc rpo rpr,r treptr Atznyrr,ffpu, BtOt L MF. Ornce, Bp L MF i B1D L BD' ro+i BrD L (BMD,), r BMDJ - nepepia napane,renine4a, an<wfr treptreHA Hyrxpunft 4o BrD i npoxoAr4Tb vepee O, - cepeAn BLD. MDt: D,F, ro4i MDf B - porrn6. 3nafi4eno"fioro giarona;ri: BD, = 3pJ2 = "Ji.JZ s-*,, : jao, lrr = ! .zo . orli = o" rli. Bidnoaida: a2 rfi. 562, Hexafi ABCDATBTC.D. - xy6, O - cepe4zna itoro MoKyrHrrrc (ua.n. a). BB2 I B1D, roai MF )l BB", MF XO L EL nH 4iaronari rcBaApara; Bp L EL Ba reo pu, B,D I EL, orx$e, BtD L (EFL), SN ll AlCl (SN ll rrrecrrrrcyrnrrx - nepepie, aa4anzii aa yuonoro aa4a SN : l/I : PL : QP : EQ: -SS - cepe4ni .ninii (tC,: 90"): NL = ClJi = BtF 2 7 =84 (cwz). aE. 2 D1 ANO,L (ma.n. 6) - a" ^-z li.^ o4 vo l'tdnoatdb: 4 $ 15, Ilepnex4rKynepHi nfloull4Hlr 572, Bi1noaiOa.'ni, romy rqo ryr mix nrorqunaMu - r xyr uirx nepneu4r4xyrflpaMff, trpoBeaenzmH 4o,rix rreperrrHy rrJrorqrru, togi xyr uirx a i b nouunen 6yr S0",aaayrronoro,aIb. 573. ,flxqo roqrca M pinnoei44anena ei4 nepurun KBa ro4i nepnengrxyJrsp Ao nJrorqrrHrr xnagpara is M npoxoArrb vepee fioro qenrp - rovHy O. MO r (ABC). AC I BD MO LAC, MO L BD, rca1 (MAC L (MBD). iiiiiiiiii;iiillii:i;i TEOMETPI9 Eees r. Il. ra ix. Ilp 3l* /- 2 ,*., = !!fi srsvrpo = 6.s*o, G+! = s 'ffi a 2a2 39
  • 55. 574, BD LAC, MO L (ABC), (AMC) L (BMD) neperiauarorbcr no MO' MO LAC, MO L BD, rcai (AMC') L (BMD). 575. (ABC) r (ABM), NA L AB, ocxi-trsxl,z' ABMN - nP,fiMoKYrHrlK' r:o4i NA L (ABC), NA L CA, tcAN :90'. 576. OM L (ABC), npoBeAeMo OK L DC, roti MK I DC, n) LADB (ZADB:90"): AD = J,AU' - BO' = 1E"' LAID (zAcD: 90"): CD = JM - Ae = &"" - "Bidnoaia: arli. 58O. a I B, rogi BD Lp, AC L a, AC ICB. LBDC (IBDC :9O': BC -- -E+= sf = o ("*), cos30" Ja T LACB (ZACB: 90'): ,48 = OrE (crur). Bilnoaila: 6J2 cnr. 581. cx, L 9, AD I a, AD ! a, BE I a, BE rF. LABC (AC : BC): AB2 : AC2 + BC2 - 2AC . BC. cos 120'; 242 = 2Aa2 + 2Aa' .!: 2' 24, : BAC2; AC, =24 =24.8,'3 AC=.,8.8€=3"6 (cnn). LADC - pinuo6e4peunfi, . AD=DC=s=$&=+Jo. J22 Ocrci.nrrczAC : BC, ro i npoercqii pisni, roMy BE = Bilnoaiv: 4J6. 542. @BC) L {ADC), BM L AC, DM L AC, IBMD :90': AABC (ZABC : 9O'): BIvt = 41'2 LBMD -pirgo6e4peulrfi , BD = BMJ| = $- A = Bi0noeidu: a. 583. ABCD - KBaIpar, rop.i AB L AD; ABCTD, - KBa ZMKO - xyr rrix (MDC)'L (ABC). Hexair AD : a, Sucp: d2l Suco: So*": a2 (sa Yuoeoro)' s-..- = loc . MK: MK -2o' =zoi2a gK=!an=o, 22' LMOK (ZMOK :9O")z coszoiKM = ffi=;*=i, loRM =u.""o*1.4 Bi}noeita: ur""o. l. 4 577. Hexafi (AMC L (ABC) i (BMD) L (ABC, roal MO LAC, MO I BD, a otxce, MO L (ABC). 578. Hexaft tr ll 0, cr I y, roAi o I b (a e a, a e Y, b e Y), cey,ceB,clla, roAi b -L c, 6) a I F; AC L a, AC L BC; LACB (ZACB :9O"): Ac ' JAB' - Bc" = J8' -?'z =.'{'t5 = n{r (crvr); LACD (ZACD :90'): CD = ^[AD' - AC = !t6= = 1 (crvr). 964 l{l: TEOMETPIq 6eea f. rl. ra ix. rlpoSinunui piBeHb fi{Jl{li}iil.:{tlii ,4D, L4.8, uacmo: IDAD - nyr mix rrr' tDAD:60". flponeseuo OKrllAD, O - aenrp ABCD togi IOKO:: 60', ,qe o.K=oK=AB=2^= '22 LOKO * pinnocroponn OOr: m. BiOnoai}u: m. SAa. a)(ABC) I (ABC), CB L C,Bi A,BTBC: CP = BCnE = aJ2. BiAnosi1a: ali. M "4"NAD [iii$ji:iifii:iiit{*i_$ rEoMETPlc Fees F. Il. ra in. Ilpo
  • 56. 6) BC L CD, CtC L CD, TOA1 ACICD: r" C,D = JC,C' r DC2 =r/(oJz)- + a' = all. Bidnoeida: aJ. n) LC.CA - pinnoctoponuift, IC'AC : 6O". Bi0noei0a:6O", 5a5. a)(AMB) I (,4t/B), rcxi MO IAB, NO LAB, lMoN:90" LMoNz MN = MoJ2 = "J2. J2 = 9!e , 2 A Sua,": FM . MK: 16 . 6 : 96 cu2; Prrrv : 2(FM + MK) : 2. (16 + 6) : 44 (cut). Bi1noai1u: 44 cu; 96 cru2. 589. ABCD - pou6, rcai AC L BD, cp L (ABC), Cp L BD, SD r (1A1Cr), ro4i sa oauanoa (ACC.) L (BB$). S-,"," = e-. ArCr; Saaa,a = BBr' BrDr: S*,"," =4 4cr -AC -rns"r,r,, - B4:4D, BD n' 'BiOnoei0a: m : n. 590. ,flxqo DA: DC : DB, roti DO L (ABC), O - cepenusa AB, 4KllAD, KhllDB, ro4i (,4,1{'B,) llfdnq, KolllDo, KO1 I (ABC); ro4i (A,KB,) L (ABC). MLKBT - aap,anraft, nepepia. LADB * LA,KB,: su'r4 = k,: s.. ..^ =( ?' .s = 1s. ' " S*" M^q 3/ I 'Bt}noeiOa: sst. o ll 0, y^a:a, Y^B:b, allb, cLa, dLa, xyr uirx c i d - ryr uix y i o, piruufi <p; crLb, d1 rb' roai c, ll c i a ll a,, xytn uinc c, i d, - tp, 594, AM: CM : BM, rogi MO L (ABC) i O - cepe4una ninorenysrz.A-8, MO LAB. 3a osnaxon (AMD f @BC). 596. (MBC) L (ABC), ro4i IMOA: 9O', MO L (ABC), CK LAB, ct ro.qi oN ll cx i oN t AB; MN I AB Ba reopeMoro npo rpr{ fieprreHAr4xyJr,flprr, IMNO - rcyr milx (ABM) i WC). 6 IMON: 60" - rYr mi:n (AMB) i (ANB); LMON -pirnocroponniii , MLN = *O =+ Binoainu: U *, 6, + 5A6. @J;C) L (ABD), rcai ICOD : 9O". Hexaft AC : a, no4t AD : a; co =oEt2 ,r=+,rz=f, 4s. I AC, D: cos <p = Aa2 + ADz -cD2 2. AA.AD I oJo )- Ba2 a'+a'-l- I za',- 2 a tqt 2ur 2. a. a 2a2 2'2a2 4 ,iL4;r,.:;:{.:i,i '.;:ftfi:l:::liii I Bidnoeidr': cosQ = -. 587. (ABC) I (ADB'), roil ICOD : 9O"; CO = oD = a--; Co= +, r - 4oerxuna AC: J22 2co 2. a zoJa o"t6 -- Js Jz.Js 6 3 zionoatoa: &.3 588. Ilposegeuo FN ll MK, rogi MFNK - nepepia napare .nenine4a, sxufi npoxo4rrb qepes M, N, K. (FMK nepe- runae (ABC no MK, FM I MK, roMy LrIo FM 11 BB), BB, I MK; FM L (ABC), rofi (FMK') L (ABC). LB,BD (IB,BD :9O"): no =,{4o'- eai = lro' -L6'= 12 (crvr), AC : BD aH 4iaroHani npflMorcyrrrulca, AC : 12 cttu n'rx=!zrc=1 rz=o ("*). 22 Hexaft BC : a; oa =f; cr=f; C LMON (IMON :9O"): BidnoaiOa: arctg 2. LezMNo=y9 =u!14=r," oN 2. a,l3 / C /N-7 lWl.A BjN ":m tIl;":'::':,1;/d--n 966 iii: rEoMETPlg 6eeg F. fl. ra in. IlpoOinuxxfi piBeHb i:iiilii;iil:iiiliii:: [:litl:iiiii{itt;;:ii;t TEOMETPI9 Eees F. n. ra ix. IlpoQi
  • 57. 597, a)AP L (-43C), rosi (PAB) L(ABC). Bidnoai0a: 9O'. 6) (APB) ^ (APD : AP, rop-i AB L AP i AD I AP' ro,qi ZBAD - ryt uirr saAaHfiMnr nJrorquuaMll' LBAD:180"-120':60'. Bi7noai|a:6O". n) Ilpone,qervro OK L CD, OM ll AP, toai OM L (ABC) Ba reopeMoro npo {plr trepneHArrxyJr,fipn, MK L DC, ZOKM - xyr uix (ABC) i (CPD)' Hexafi. AB: d, roAi AP : a, UO = 1,'2 -, J,s S*"o = a" sin 120" = t, t ssco:o.2oK; oK=*=+-+, rosi DrF L PD, ZFDp * ryr uixc (PAD) i (PBC). Pl DF : 2oK(,une' n)), Pr = o6 , 2' tp IFD.D = DF =ou6 - f ,LEL' uLu DDt 2'a 2' t^ .zFD.D = arcts9, ' -2 t) LPDC : LPBC Ba rpsoMa ctoponaur,r, ro4i DF I PC, BF L PC, DF : BF, ZBAD : 60" LABD - pirnocroponnift, BD : AB : a' Hexafi CF: r, toAi LDFC: DF2 - a2 - x2; A,DFP: DF2 : PD' - PFz : 2a2 - (2a - x)2; ILABC: ACz = a2 + az + 2a' L = 3a'' 2 L,PAC: PC = nF* U' = Zol; a2 - 12:2a2 - 4a2 t 4ax - x2; 4ar: 3a2;' 3a' 3a , 9a' 7a2 x=4a=-i Dt' =o--16=16t ^MoK (tMoK:eo")t tgtMKo=ry=:+= L ="ltLlvrtrv-*-r"Jq-ufu- a E ,la a6o ctg LMKO = t ztutXO = at""t9f ' t; Bidnoaida: arcctSf. r) Ilpone4ervro P ll AD, ro4i DD, I PDv DDl : AP : a, DF L BC, zAM.MB IAMB =rr - """"o" u@. 1001 ^ 25J154 IJrcJnoautb: tr - arccos - 1001 flpra posb'sexy Hacryrrul4x 3aAaq Morrcua aanpo[onyB r;ra4, risruemo npsruoryrnr,rft Tpul{yrnlarc e xaretamr S = lab, a6o S = !O'r, fr, c * ni4noni4no Bricora 22 1 1. PIBHICTb: rao=-nc. 599. (BAO) L (AoC), ABCA - zasarLwfr, BO I (oAC), CK IOA, CK : t2, BC : AC Ba yMoBoro. 1 Hexaft AC : r. Sooo. = ;CK .OA = 72. OA a6o 1' Sooo, = jOC AC (OC L AC aa reopeMoo npo rpr{ rrepneHArirrcyrr.flpra), orxce, 6 oA = L. oc' *, r2-OA:OC-x ' (1) LBOC (IBOC:90'): Bfe : BO2 + OCLI *" = (snli)' + oC2 = 175 + oCz; LBOA (ZBOA:90'): AB'z : BO2 't OA2; 2x2 : 175 + OA2 ( BCA, ln = x"lz {2 3 pionocri (1): oC -12'oA , t54+r6s-423 1 -t.ii.rc - 2 Ocxi.nrxlr ryr uirx rrJroull4naMra - rocrpufi 1 arccos - , 7- Bidnoaida: ,"""n" 1 . 7 598. o1 p, on B : a, AC L a, BD La, MCD (IACD:9O"): r- AD = ,[Ac cD'= /(alo)' a 15' = u[' 16 + 51 = BD L (t, ZBDA: 9O", LBDA: AB = JAtr * BD" = Js'z .31+tr' = = .,F. (s1 + 4)' = G' Gl + lot = sJn; CM t MD - 2 : l; MD : l5 : 3 : 5 (cIvt), CM:2.5:10(cu); MCM (ZACM :9O"1; AM = ,[Ac cM" = J(sJ6)' + 10' = Jb4 + 100 = J LBDM (IBDM : 90")t: MB'z : BDz * MDz : 122 + MB:13 (cm). LAMB: 3A OS AM2+MBz-AB cos IAMB = iim*i$ tlffiili{:tl iilM#i$.i riin&:li:i:i:* t!:l !7,::aql:4.1 op = 9 "li:4 -7o' , 2a2 LBFD: cosZBF = 2-Et_4a :# = # = + -1616 968 l{i:i rEoMETPlg Eeee r. fl. ra ix. tlpoQinurrft piBeHb liilfltt-t:ii{trtii:*! i:i::.{r:i:i:i:iiii:::::ri::X IEOMETPIS Eeae l-. Il. ra in. Ilpo
  • 58. 744.OA' roAr r-=I/b+ , ; J xa:175x2+I44.OA2. 3 pirnocri (2): OA'z : 2r2 - 175, rogt xa: l75xz + !44(2x2 - 775); xa - 46lxz + L44.r1b -o; D - 46J2 - 4.r44.rzb = BS72r x2 - 4631-337. '2 x' : 4OO a6o 12 :53, arc BC2 > OB2, roni x2 : 53 - He BaAoBoJrbHse yMoBy Ba,4a- qi. s..-- =Lec, =L ,, =1 +oo = 2oo (crvr,). 222 Bi0noeila: 200 cu'z. 6OO, Ba yMoBoro IBAD - 90., ZDCB :90"; AD : BC : 15 cm (ax nporn,rexui cro- poHr4 fipsMoryrurxa) (ua.n. a): C LBLK (ZBLK: 90"): BK'? BL2 + KLz 9,62 5, LBAK (IBKA: 9O": AB2 : 9,62 + 5,62 * 9,62; ,4.B = !6J' .(96' * 9d * 5d) = 0, i.re.i2)' + (8.1"1 = o,1.8^lL4+ 2 + 49 = 0,8.,6sf (cNr). BionosiAb: 0,8J337 cru. 602. y naorqrani (.48C) npoire,4ewo CM L AB' M - ZKMC - Hyr Irnix (ABC) i (AKB), IKMC : 60". MKB (/AKB :9O"): KM : MA: MB : 16 : 2 - PNBC: L7 + 17 + 16 : 50 (cm). s*" = @= Jr5'8'&9 1 = 120 (cu'?); Sw. = =.CM 'AB: cM = 2'r?o ='*:9 =1b (cu). 164 LKMCi KC2: KM2 + MC2:zKM 'MC cos 60"; 1I:AL.BD=1AB.AD (ADAB 22 no4i6:uuir Ao €rulercbrcoro, ro,qi BD : 25 clr), orn<e, AL 25: 15 . 2O; AL : 12 cvr. Toqxu .4 i C piaHoei44aaeui si11BD, rosiAL : CK : 12 (cnt) (AL L BD, CK L BD); LK : 16 9 : 7 (cm); D LALK (ZALK :90"): ,4I{' : AL2 + LK2 : t22 t I7'; LCKA(ICKA: 90"): A[z : CKz + AK2 : !22 + 122 + 20 a) S-", : 48 cuz; 1 5..-^=:AB CK: 2 S*r: 90 cu2; 1 S*o =: DK'ABi Ilexair AC : rc cM, roAi DK2+KC2_DC2 cos IDKC = 2DK Jr,c - 8 $ t = tE4+ 225 - In = J2 cK _2.48 _ AB pK=2'90-teo. AB AB Bidnoei}t: L3 cv. 603, Cf L AB, K - cepesuaa AB, rogi DK L AB, IDK 72:2.I44+49:337; aC = J337 (cv). Bidnoaida: V55? cu. 601 . (AMC) L (BCM), M - cepeslz':rra rinorenysa, AB : 29 "*,AM : CM : BM :10 (cm). LCAM - piexo6eApeur*, - itz- lo - tL{"* ={f- (16-l2xl6 10X16-10) - = r/16 4.6 6 =4.2.6=48 (crvr,). s** =i*.cM; 4a= Ar{=+=e,6 (cu). LAKM (1Ar{M :90"',1: M K -- .[o' - s, e -- J0, { ts, 6 = {6, r- 4 . 0J . 196 = t4. 2. 0, 1 = 2, 8 (cu). Ii6r 10+10 so.,, = i' ' ';*,u (ts-16XI8-10)' = Jl8 2€€ =8.6=48 (cm,) 1a6o S^uc - Soa.v, ToMy Iuo me4iana poali,rae rpnKyrHhK ua 2 pisuone,rizxi rpI,I- xyrnuxvr), s".^," = l cM. BL; !.ru. ut = n", 22 BL : g,6(cu),2{I : BL. LBLM (IBLM - 9O"): ML= fo' -ri' = 2,8 (c:vr), MK : ML, KL : 2,8. 2 : 5,6 (clr). 970 li;l: reOuerple Eees F. fl. ra ix. IlpoQinunnfi piBeHb i:lilt{iliiifii:iii;i 2.1qq. e! xJc Mt<c (IAKC : eo"): KC2 = r* -[;)'. orrxe, IT 4OOx, - xa:4.962i xa 40Ox2 i 4.962:O;Dr:2O x(200 + 2.96: (200- 192)- (200 + 192):8 3 x2 : 2OO + 4 - l4i x2 : 2OO - 56 : L44 a6o x2 : 25 180'z+ 96'z -13' .r44 32400 +9216 coslDKC= zrgo-sr = BAbG 32 400 + 9216 - L}'1 . 256 32 400 + 921 cos zDKL'= 84 b6o = 84 5 103 IDKC = rT - orccos-. l03 Bi}noai}v: 60' a6o ?tr - &rccos -_ 96 AB, r 180 '? ttx) "A *l,99)'-r x/ l- /'x-to, bAK : 48; iJ;:;:;:;:;:;:ij:i:i;1;;i::; FEOMETPI9 Eees F. fl. ra in. Ilpo
  • 59. 604. (ADB) L (ABC), CK L AB, cK L (ADB), DM L AB, DM L (ABC). LADB: LACB ea rpboMa cropogarvrz, rogi DM : CK. s*.={ z I z -")l 2 -o) 2 -c)i 5..^^=!c.CK. 2 g6 -2s* = c4c =@. 2c LDMB (ZDMB :9O'): M B = JDB,, DM, - ^luz - @ + b + c)(b + c - a)(9' + c - b)(a + b - c) . ! 4c2 LCKB (ICKB: 90'): KB = Jb' - CK', ro6ro roqrcra K i M cnsnagarorr (a6o *rorx- Ha AoBecrrr e pinnocti LDMB : LCKB aa rinoreuyaoro i xarerorvr). LDKC (ZDKC: 90'): DC = CKJ, @ib-c)Eronoatob: % , 6Os. (ABC) r(A1Bp), EA LAB i EA LAB, roAi IEAE. : 90'. Ocxi.nrxu rrrecrrrKyrHr{r(r{ npa- aa:rrni, To cropoHa rrrecrunyr- ruxa gopirnroe pagiycy onnca- HOtO EaBr{OJrO rleCTrrKyrHr{rca rco.ira, roai srulo -AB : a, ro AD:2A, DB: D$; DB IAB i D$ LAB, LBCD: BD - ''E - t' - 2". a.cos IBCD I rao"(o - 2l : G =$- = rzo" - 1BCD l; Bo =,iio o' = oJz; t 6 )-- LD4D (ZDLBD : 9o"): DrD2 --(oJs)' 2 = 6a2. L.D.AD: cosID.AD _ AD2 + ADI - DrD2 _ 4a2 + 4a2 -6a" =4 =!,2'AD A4 2 4a2 8a2 4', ZD,AD =u.""o"1, cA = AE = BD = aJB; '4' c1M L AB, ICIMC : 9O" LCTMC" CtC2 : LBMC ()BMC : 9O"): CM = asin60" = of , g,g' =, + = LC,AC: coslc.Aa -c1A2 +cAz -cs2 ="o'*'o' --Y - no"- 2.qA.CA 2.3a2 l2a2 IC.AC =r"""o"1. BiLnoaila:90o: arccosl: u"".o. 1. 444 c,A = aJ; cM L AB, 2CM2: 3a' 2' 3 4' s72 lt{: reonnerels 6eBs r. fl. ra in. rlpoQinrxrft piBeHb,ilil{it{ii1tt{;$ii* I*{t#tililif;tii rEoMETPle Eeas l'. Il. ra in. IlpoQ 6O6. Hexaft.A.B a, sa yMoBoro (ABC) L (ABK), CB I A MBK, KM ll .aN, ro,qi ZNBC : 9O', KM L (ABC), KM L MC; A,KMC: "JB2 rm =o62 , CM=FC= 2a2 + 2a2 - a2 3a2 3 cos,lACK=-=-=-; 2.2a" 4d" 4 IACK =ur"io.1.'4 Bi}noaiOr,: .ru"o.9..4 607. a)flponesemo AM i DN llAM, AMND aa, ABCD - [p'ffMoxyrrrlrr. 'Ioai AD L AB I AD L (AALB), MN ll AD, rosi MN ll (/e,r,), (AMN) L (AA.B,), AMND - rpsMoxyruux, P*ro : 2(AM + MN) : :2.(2o + 13):60 (cu); Swup: 20 . Lg :260 (cu'?). AlBr: AB: 15 (cru); 15:3:5(cw)-A'M; LAA,M: AM = .rhz" + 5' = 13 (cru)' .Bi7noai(iv:66 cu, 260 cu2. ' 6) Ilponeseuo CK L DN, BL ll CK, BCKL - nepepia s = 16 ! .zo = 65'-.2o = 825 (cw2). 44 Bi)noei1a: 52,5 cwt; 325 cw2. Ilpore4errro CDr i BArll CDt, - nepepis n Ct axnfi nepnen,4uxynapnufi go (AMN roai MN .l- DN i MN L CK, KD KC L BC, Maeuo: KC L MN i. K Ba oalra(oro, (BCK) L (AMN) A-I(DC -,aNrrD (aa reoMa xyrann xc = DN,Dc = ilg = 19+= CDDr1244 PBg6 : 2(BC + CK) : 2 - (20 + L6 Cr BCDAT * nepepia xy6a AB1C1D - nepepia xy6a ocninsru rpani xy6a -C CDl L DCI rrJrorqrrHoro (B rrrrorqrrHoro (A icBa,qpaTr,r, Ba oouar(orc (CDA) L (ABP). 609, /rNE - nepepia xy6a, n snouy -4X ll Dlf, AD L(DDP), AD I DN, ADNE - npfiMoxyrurrr(, Anaroriqno nepepie ATDTMK - rrpsMoxyrHrlrt (AlD1 L DIM)'
  • 60. 610. LAI{C - nepepis rerpaeApa, LDMB - repepls TerpaeApa. Ocximnu rerpaeAp npanz;mnrii, rogl DM : MB - Br4corr,r rparleit, MK - r.ucora LDMB i er4cora LAI<C (AK : KC). AC L MK i AC L BM, ro4i AC L (DMB), AC € (AKC), ro4i (,4,I{C) I (DMB), LDMB : MI<C, Sor.s : Ssaq' Pnve : Pwc. 611. a)Hexafi DABC - rerpae,qp (ocnona sHoro AACB, lC : 9O"). 3a ymonoro rou- xa D pisgosiAAaaena rig Beprrr{H LABC, ro!'i DO L (ABC), O - qeHrp orucarroro HaBKoro L,ABC nona, O - cepe4uHa AB. (ADB) f (ABC), ocxiasrx DO L (ABC)' flpoee4emo MK ll DO, roai MK L (ABC).9epea K trpoBe4eMo ,qooi,rrny npauy ZP, fiKa nepernHae cropoul4 AC i BC MBC. (LMP) I (ABC)' LLMP - Aoei;rrnr'rft; 6 MK L DO; LP 1l dn, 4 e -LP, roAi (LMP) L (ABC) i MMP N AADB, LM : MP, MMP - PiBno6e,qpeHr4fi. s) MK r (ABC), NF ll BC, K e NC, MT ll BC, roti NF ll MT, nepepia MNTF - rpaneqis, (MNr r @-BC). A r) TMLK - nepepia, (f MD L (ABC), roMy nro MF r @BC), MF ll DO. ?K - Aoeimua fip,ffMa, Rxa npoxo4rzrr vepes F i neperr,Iuae AC i BC, r) -L - 4oni.nrua roqxa DB. s74 l{lj rEoMETPle 6ees F. Il. ra ix. l-lpo$inunnr piBeHb ittititl:ri{lii:::* 612. By,4L-sHa roura 6icexrpncrz Kyra pienoei4gaaena ni4 ctoPiH xYra, ro,qi trF L (ABC), FD I BC, FE I ABi FD:FEi(LBK)L(ABC). 613. Maelto: cos45' cos45'= =[:- I =1=cosoo., A 222 aofi (ADC) L (ADB). Bidnoai1a: 90". 614. LAOB:OA:OB:a, zAoB - 60", rofi /ABO :6O', AB:OB:A. LAOC - piexocroPonHifi, AC: A; LB?C BC = ali. MBC - nprmoxyrnuii, pieno6e4penraft, AB : AC. Maervro: ZACB: 45", LBCO: 45", ZACO :60", . ro4i cos 45o ' cos 45' : cos 60', ' roai (,A-BC) L (OBC). 615. Ocxinr,xrz (MAB) L (ABC, rofi' coslB : cos crc cos LD.DC a ADCN aa xyrov (ZD : lC :90') i nponop- DMCN2qifinuuu ilpr4JrerJrr4Mrl cropoHaM[: DDr= DC = -, roAr signonigai fiyrx rpl,rnyrHhrcie pinHi, 3 uboro: toDM + loMD :90", DN L DtM. To,qi ArB L AE i (AEN) L (KMD'); ar+t4o DC : a' ro D1 Nc =f;a; DN =,6C CN = ^ o'lla o'JG S*ut=o g ; Bidnoeiou: o'J!8 . 3 LMAB (IMAB:9O"): MB = E; MA: AB tg cx. LBAC (ZBAC - 90"): AC - An EF; nc = I' cos F LMAC (tMAC - 90"): MCz : AB2 tg2 a + ABz tg $ : ABz(tg'o + tg'? F). MC"+bc'-MB' ABAB"(tg'(x + tg- p) * "o",=@coslC = 2.MC'BC I I [si1]a ., "inl I , _ttg'ct1tg' F. "o"rB cos,o _l"o",ot "o"r0t "9---- -;F7;-te_!,E ,, cosp - (sin2 cx, cos2 0 + sin' F cos2 tx + cos2 tx I cos2 9) c.os0 - 2cos' c[.cos' BG' " - tg' B - -"""'BG - "t"' - -cos'crc =@ 2"o iftifiiii;{itit*til rEoMETPle Eees l-. Il. ra ix. IlpoQ 4a' I C
  • 61. cos2 o,(1 - cosz p)fg"'i "dS - 2 sin' I cos' cr = ffi 2cos2 o.cosB.tgFr/tg' cctg'z p+ 1 sinz 0 sinP :- s"p@.ts'O .'/t+tg'rxctg'P MBz + MCz _ Bc2 -( .f;) * *',r*'* * tu'p, - ffi _ cos lM { th.,r' o *'r' u--+')"*o "o'o _ - - 2Jtg'?o + tg'zP (cos'? g+ sin'? ccos'? 0+ sin'z 0cos'z u-cos'z a)cosa - cos' ct cos2 g'2Jtc'0 + tg' F - "."' cx. cos' B + cost B + sin' c cos' P - 2 cos 0, cosz PJtg' tx + tg' B 2 cos2 B sin2 ct 2cos" o cos PJtg' B + ts' 0 Bi|noeia: coslB : cos cx' cos F; cos lC = sin2 q, cos ztvi = --_--:. "o" cJtg' c + tg'? B 616. SaYMoBoro' ZMKA: ZMNA, ro,qi MN : MK AK : AN' L,MNK: NK2 = MNz + MK2 -2MN MK'cosP = = 2M N2 (l - cosD) = 2MN'1 Z sin' !; B NK = zMN sin;; LMAN (tMAN : SA"): AN : MN cos cl; M-l{K: costKAN=ra*y*' -zttr'r'":2;!#!'"h'2= -" lr.ln "' 9, cos'a-2sin'f 2sin'! ^ ,,KAN ^ 'i"'!. r - cos tKAli = t - --;"e;- = -*J;t z.'n' -- = "' ;"", o' ^,-B f"*9] .i"!r srn , 1_,KAN = ur".inl 2 Ir IKAN =ZarcsinJ.sinlIKAN =-=3| ;ZKAI =arcsrtrll^"1, I' cosu"" 2---'- cos c 2 cos ct l .Bsln - Bidnoeila: 2 arcsin -l ' 976 i,,ll.:l reouerplg Eees F. rl. ra ix' IlpoQinrnraft pieexu tt{i:ti{ii::fil{tl;i1] .2 srn u cos ct ffi g 16. OproroHarbHe npoeKTyBaHHf, 631. /.B 1 cr, BF, BC - noxra.ni, AF iAC - ix npoerqii; BC > AC, BF > A.F, ro.qi Af : 16 cm. LBAF (LBAF :9O"): BA: 12 cw; LBAC (IBAC - 90"): AC : 9 or. Bi1noaidt': I ctr. 632. MHB (ZA_rrB :90"): ,4-FI : 8 (clt); LCHD (ZCHD: 90'): C.FI : 6 crvr. AC -- A - 6:2 (cwI). Bi1noaida: 2 cll. Hara,qyemo sracrr4socri cepeAuboro nponopqiftroro y HIIKY. h2: a" b". Knalpal Br4corra rrpsMonyrnoro rpl4rcyrurrf Hyar4, * 4o6yron npoexuiii xaterin na rinorenysy. a2 : a".c. Kaa4par rcarera npaMoryrgoro rpr4rcyrul.rxa ro Karera na rinorenyay i riuorenyon (a6o b2 - b"' c). 633. MI{ 1 ct, A-I{ * npoercqis AM, BK - npoerqis, .. orxce; BK > AK, BK - AK: 2 cu. flpone,4elgo MF L AB, roti AF - npoercqin AM ua AB, BF - npoercqis MB uaAB. LAMK (1AKM : 9O'): Ml{z: 102 - x2 (AK : x ctrr; BK : (x + 2) crrt). LKMB (.MKB : eo"): MK2 = (e"!r)' - @ + 2)2; ioo - ,r, : L28 - x2 - 4x -.-,4; 4x:24;x:6;AK-6crvr; MK=J{o'4" =8 (cm). LAKB (ZAKB: 90')::43 : 10 (clr), AKz : AI . AB; AF=4=T=a,o ("*); AB 10 BF : lO - 3,6 : 6,4 (cu). Bi0noeida:3,6 cu i 6,4 cu. 634. ctll 0;,qB: CD:20:lt.,AM tP,DFIs.,AM: MMB: AM2 : 4O0x2 - 256; LCFD DF2 :169x2 - 25; 40Ox2-256:L69x2-25; 231x'z : 231; x2 : 7', x :1 * AoBlrcrrHa 1 vacurnz, na ari nogi.neno ni4pie- *u AB i CD. AB: 20. 1 : 20 (ot), CD:13.1:13(cu). Bi1noai1a:20 crr i 13 cvt. ,'.:*:i:i:i::::J:ii:iii:fi:* FEOMETPI9 Eess F. Il. ra in. flpoQ 32 v"i up. lo 0.. c. I
  • 62. 635. o I F; A.I( L a, BM L a; a n B : a; AI( L a; BK * npoenqir. AB na ct, roli MKB: BK = ,tE6' - 10' = 24 (cv; BM L 9, dJl4 - npoercqia -4-B na B. AM =,Ea'-n'z =,[6.4G = Jr.B.2.n =2l@. Bilnoeida: 24 cv i 2J69 cvt. 636. AM L a, MF - npoexqis -A-F MK LAF, roai AK - npoett"titt AM sa A.Fi AM2 : AK .A-F; 762 4.4.16 4.16 AK=- =-- 20 4.5 5 = ly = 0,8 . 16 = 12,8 (cu). l) Bidnoai1a: 12,8 cm. 637. AB L a, BC - npoenqiaAC na a; BD L AC, -A,D - npoexqis AB sa AC. AA-AB:2ct'r. Hexafi AB - r. cM, AC : x ) 2; MBC: (x * 22 : xz 't 36; x2+4x14-x2-36:0; 4x:32;x:8; l-B : 8 cu; AC: LO cv. AB2 - AD AC; AD=4=g=!j=6,4 (cru). AC 10 10 Bidnoai0o: 6,4 cw. 638. SayMoBoro x L A, U L z, z L xi npoerc4is -/.B na x - b, AB nag - at ABsaz-c, roai AB2 : a2 I b2 I c2i AB=J;b' +e. 639. o r P, B ry, cr'IY. AK L cr, BK : a - npoexqia AB sa a; BF I y, FZ : b - npoexqia AB na Y; BM Lp, AM : c - npoexqia AB :aaP. LBMA (IBMA: 90"): lBz : c2 't BM2i ABFrA: AB2 : b2 * FrB2; LABK: AB2 : AI{2 I a2i ABz : c2 + b2 - .AK2 : c2 + b2 - AB2 + &2; 2f,482-a2tb2 t c2: *=F-':-l: 641 . Cnopucraeuocs Sopmy.noro: Son. : S cos o; Bl}noei}v:6O". 642, SBCD:6'4: 24 (cr,tt2)..o.o = 3 = l,'242' cr, : 60" - nyr uixc npsMonyrHrrrou ra fioro npoer Ilpoexqi,x He MoJfie 6yrra xnagparou, rr;my uo AD : 6, S"n.-6':36+L2: a6o AB :4, S"n. - 42 - 16 + 1'2. BilnoeiOu: 60o; ne uoxe. 644. Bc li o; Br r c. CK I a; BC : FK,.fqMy ,4,r< = 5n[3 c*' s*" =29*= 1006 @v2), AM L FK; MK : FM: 10 crvr; "o"o=E=1: a:60'.702 AM = lb"fu)'- 10' = !6113 - 4) -- ;f,s =15 (cr s^""" = 1 FK . AM = !.ro .1b = 150 (cuz). 22 "."o= 150= =ti9J5=f , o=s0". 100J3 300 2 Bi4noaiOa:3O". 645. MI,p - rpoexqin MBC, BBt I (ABLC). ,T3*i4+1'- " s_,4q = l:--(21 -t3t(2t- 14X21-15) = -JnEls=h-?-2.4.,.'LB=8 2.2 7 =84 (c S*,qq = S*"'cos30o; s ^ = 84=2 - 84 2' h = 28 2,h = 56uE 1crvI'?);- "Nc- l; - D !6 ,*"=#; AB= = rffi = J4.? p*" =t2'174 @*). Bidnoaila: 12JiZ cm. 646. AfiLCrDt - oproroHaJlbHa npoexuin, AtBr: DrCr: 5 cu; ArDr: 16 cm; BrCr: 10 crrn; 16 + 10 So,qq4= Z nV Bhcorarpaneull, LCIKD1 (lCtKDt: 90.: D,K = 16 .10 = 3 (crvI); crK : (cIvr)), '2 Sr,a.r, = 13'4=52 cttt2i s*,,=b2Ji: *"o= u52rr=$: o- +t". Bi1noeidu: 45". i;i:li:'ir:ii:r::r,iii:r fEOMETPlg Seee l-- Il. ra ix. flp 32* 4S*" !€ s7a l{li rEoMETPlg Eees r. Il. ra ix. I'lpoQinrnrfi piBeHb {tlitit::{it{itj{i;
  • 63. 647. BCL ll tx, rogi BclllB,D,ioO,Lo.,; BB1 : DD1 : ooi BBt ll DD, ll oo'. LAOD (ZAOD :9O": /ADO: l2A z 2 = 60'; Ao = AD.sin60' = 20 9= 10!6 (cM); 2 DO : 20 cos 60o: 10 (cu); BD : 2 10 : 20 (cu); BrDr: 2O cna' AC1 - nPoerqia AC na cxi OO, - cepe4na ni:r.ia MC.C (./ACtC :9O"): CCt: 2' 10 : 20 (cu); AC, = JAe -cc? = r/(20.6)' - zo' = =fot G-1) =zoJr. Bilnoai}a:2O cv; 2OJ2 cw. 648. B1D1 ll AO' nA' r (,431C1)' DDl r (ABp)' ool L (AB9J' Brall Dll oo" nn, : oo,. BrD, - npoercqis BD na (ABrCt), BrD, = SIEJ u; oo. =l cc. =5 w.,2 LADP (IADQ : e0': AD=,[AD'Joo: =h44+% =13 rvri DCCTDL - np.sMoryrrra rPaneuia; CE - Cq - DDt : 10 - 5 : 5 (Ivr); ^DEc (zDEc: e0"): DC = Jbt+# = J81+ % = J106- Ba nac.ni,qrou B reopeMrr nocrnycin (g.na napale;rorpaua):' AC2 + BD2 :2(APz + DC2); ACz:2 (169+ 106)-I'2l-2'275-La9' 1 550- 189:361;AC:19m' Bilnoeidv: 19 ttr. 649. ABCD * rerpae,4p, ea yuouoro/-BCD - nparr'urrnufi, toAi DO - rraiora rerpa- e,qpa fipoxoArrrr, qepea qenrp MBC - rouxy O i rci rpaui rerpae4pa - npaBilJILHi pinni tpunytnrarcu. LBOC - npoexqia (DBC) ua (ABC). Hexaft AB : a, ropi BO : OC : ft - pagiyc orrcanoro HaBrcorro M-gCrco.na. oc=+="9-t./3 3 OM L BC, OM : r - pa4iyc Brrr{canoro a LABC ft ""lsKoJra, r=t= 6 | 650, a)MCD - Rpoeftqi.fl LBMD; "o"o = !*- AA,:3a: Mc=?.3a=2a; S*"' 3 ABCD (IBCD :9O'I: He s.--^ = 1-Bc . Dc = !. 2a. a = azi 22 LDCM (tDCM : 90") ro.{i DM = J4a7 = aJ BM = 8. ac = zali; BD = aJ|; LBDM - pinno6egpeiruft , DK - iioro n coTa, DK = JBD' -Er' = Jia" - re' = alii s*o* = f;. a' vG "6cosc=ffi=6,o=5rccos6' ^J6I'idnoatdb: arccos 6 . 6) IlporeAeruo CK L BD, KKtll DDb MN llCK, rogi MN I(BBP, roMy rrlo MN L KKt (CK L BB), MN I BD - oBHaRa fiepnenArrnyJrfipnocri nparuoi i ABND - npoerqiu LBMD rta nrrorl{rzny (BBP. S*ov = a"J6 (tns. a), 11 s*r, = ! BD. NK = ;a.tE' za = a' Ji, OE E T:q'tb tb lo roAl cosc=7G= {at "=arccosv6. r; Bi6noai0a: ur""o..i9. 6 651 , a)ABCDA1B1CTD. * r<y6, BrDr: d. Maeno Bpt Ha npoxo,qnrb qepea np.fiMy, ,Erca napaJleJrbna apy rrpsMa reperilHy naparreJlbua rrepurift npsrvrift, ro6ro s**=!nc BM =;a +=#' tDMo -xyr uinc (ABc) i (DBC. costD*o = * = ## =it lDMo = u.""o* 1. 3 ^z-[q 1 Bidnoaiu: ll9; utcco" : . 980 i{lli rronerele 6eBs F. rl. ra in. rlpo$intnufi piBeHb $jiifir*ll*il{ti:i::$ { t I d2 .[2.] iiilililliit*llitl rEoMErPle seea F. l-1. ra in. flp
  • 64. d o'd (l drE B.F='= =--=:.' d"'13 V3 3 2- Bilnoaito: r t +. 652. ABCDATB.C.D. - riy6, roAi Cp L (ABC)' CD L AD flrc croponr'r KBaApara' 655. .&43C - npaBrrrrbnrzii rerpaeAp, SO L(ABC), O - qeurp AABC. ABOC - npoexqia LSBC na(ABC), roni coszsKo- s** - Q' . " S*u. A HexaitAB : a,'opi S*- =4;uc4 MD LAD 3a reopeMoro nporpr nepnexAranyJrsplr, ro4i IMDC - xyr uixc (ABC) i (MND) IMNAD - nepepie rrJroulr{Horo, sagatr,rii 3a yMoBorol. a ^DcM {tDcM : eoe): tsIMDC=#=i=+t I z MDC = arcts -,-2 ANMD - rpfiMonyrrrraft, s** =f,ac.or = Q, - a"Jg'E *t Q" L2.o'JB- 3' Bidnoai0r: I:3. t aly 26 _o"JB. t2' A1 la2 Bidnoaidu: arct1';; 'lE. 653- 3a yMoBoro SA : SB : SC, roai SO L (ABC, O - qentp MBC; LABM - nepe- pis, aanauait Ba yMoBoro, MB : MA, LAMB - pin' no6egpenraft, MK L AB. LMKC : B - xyr mirrc 1 (AMBIi (ABCI. SwB = tAB M K. MOt 'l SO, roai - oc lJd I't;oo, =s,2 =; =;3= f tco - pa4iyc Kona' A onr4caHono HaBr{oJro rrpaBr?IJrr'Holo Tpr4KyrHr'rft a' oC = B+f, ro4i KO, = oc =t{?, J3 3' LMOTK (ZMOTK:9O): Mx--19, = 16 , s^., -L, l# cosf3 3 cos P ^'" = ,'' 3.o.8 BiAro"iAotl'# 6 cos [3 656. SABC - rpaBr4rrr'Hr{ft rerpae,qp, (AMB) I SC roMy rr{o BM I SC, AM L SC. SM L (AMBJ, rcn BM - npoercqis SB, AM - npoexqia SA, BM -npoexqia CB, AM - npoenqisAC, ro6ro npoexqia rerpaeApa ua (MB) - LAMB. 1 S...,- = :A.B' MK. L,MKB: 2 MK =.tmB' - BKt = - t o,[z o'J2 Swa= oa'-, = n ' Bidnoaidr, o'fi . 4 658. (ua.n. a).HexafiArBrC,D, -npoenqir.ABCDnac.. toABCD.looi l9 - BP u6o 48, =44 ,Zo- " 48, BrC, BrC, BC 25 P.^^^ L orHre, n'q"ur - -i Po,en-2.(25+2Ol:90(crrnh Pn""o S (ua.n. 6). .flxqo (,48C) ll o, roli npoexqia AtBCtD c - pinnrafi np,fiMoxyrHnx. ToAi Pno"o= Po,",",o, = Bi1noai0a: 72 cu; 90 cu. ii{:#ll:+:il}:*::: FEOMETPIS Sees F. fl. ra ix. flp S*.o=AD'MD=a l" a .la" + -v4 Et' a' c ="1 n =t!r. 654. ABCD - rcBaApaa, ro,4i BO f AC, AC ll $l,:itiitt;:;, BF r MD' PF L MD €a reopeMoro npo Sti:::r!i-l rpu nepneHAraxynflPu' frili$it:tli ZPFB -- o - rcyr vilt (MSP) i (ABC). ii;i:!li:i:iii: MSPQN - nepepia, aalanniz ra vMo- ,{ti}it:r.;:;:+: Boro. |fil{if:i MAB1N - oproroHarrbHa npoercqig qso- 'ri:i::Xli:i:i ro nepepiay sa (A-BC), rori Sr*"* =S-"o - S-r" = "' -; ; ;= O t,J3 6cos B MD, -o" = 8 S.- ---., 7a2 c-Jvsrer= """ -S"a""' Bi}noaida: 7a' 8 cos cr ga2 i:::i rEoMETPle 6eBs F. Il. ra ix. l'lpoQinunri piBeHb iiiiiiliii;iii::::iii
  • 65. 659. KO L ct, KA, KB - noxrnai, OA, OB - ix ni4nori4ni npoexqii; OP - npoexqis KP sa ct. LKOB (ZKOB: 90'): KO : 3 ' 4: 12 (cn); LKOA (tKOA: 90"): oA = ^[KP' - oK' = $g' -lr' = "li6 = 5; LAOB: AB2 : OB2't OA2 - 2OA ' OB ' cos l2O" : :256 + 25 + 16.5:256 + 105 :361 : 19'z; AB: 19; sp = 19. rs. 2L OA5PA5 Saynaxcuuo, *o Og =G, pB= L6' ro,qi PO - 6icexrpvrca LAOB, ZBOP : 120" : 2 -* 60"' PBz : PO2 * OB.2 : 2PO' OB cos 60'' (1#)' = x2 + t62 -!6xi xz -r6x +!62-(tu;in)' = o' 212. x2 : 16. 212' x * L62' (21 - 9)' (21 + 9) : 0; 2l: tc2 - !6. 212. x't 162. 2' 4O : O; Dt:82.2L4 - 212.162.40'2:212 ' (82 ' 212 - t6' 80):2L2 882' * _8.2t'Xr[Z_t.88i . 2l' 8.'2t2+21.88 21(8.21+88) 168+88 256 .-1. 212 212 21 21 7 168 - 88 80 ^17x'= 2r =n=ozr' Ba uepiuricno rpi,IxyrHura, MOP: OP < OA + AP; n| < s.# t nf, < s* +]l,,,ro HeMolrcJrr{Bo, or)r(e' 17 Bi}noai)a: 3fr cm. 660. 3a yMoBoro BC ll u, cc,ll nn'll lLr' CC' r o. MrBrCt - npoexqis LABC sa a. AK L BC, ArKL L BLCt, AzK ll A1K L' rogi /-AKA,:30' - xyr mix LABC i MrBtCt. Ocrci.nr,nr BC 11 BPv ros! CC1: BBr BC : BrCr. S*,.q", = S*" 'cos80o; - s-,u., -a'Je'2 -{ ,",".: ol,s*" = cos3,o" =-48 = 2 -r-r w,' 1 sdc =;BC-A]4 MA2K (ZAA2K:9O"): ArK = O'* =+, -- A-K o"!s 'zA](=L=- =Ai cos30o 2'Jg LAKC: ctstKAc = # = + = 2; IKAC : atcc1e 2l ZBAC:LLKAC:2arcclg2. Bidnoeidt: 2 arcclg 2. .984 "t{ii reOrvrrrplg Eees'f. rl. ra ir. rlpoSinunrai piBeHb ll:iifiqil{iit:Xi:il 661. Hexafi ABCD - oproror{aJrr'Ha rpoenqifi .s -- = s-., . s.--^ = lr.Bc + AD.cKt" AtqctDl - cos 45", *8"D KD:(25-7):2:9(cu). CKz:AK.KD; CK =.,@b-rf , =4.8=12 (cu 1 S*,o = i.' ( + 25). 12 =.16. 12 = 192 (cru'z); sn,ac,, = tfr;=Ls2J, @M"). Bilnoeidu: 192'[2 cw2. 662, AB ll CA ll s, rosi AA, L u, BB, L d, CCt L a, D uarorr pieui npoerqii, ro4i ABr: DCt; OO,Lcr; LAOD- LCoCt, 4=42=9, OC BC 28, P =!t LAo:o N LAc,ci oc 2' oot - Ao . ]L =1, sp. =12'J = 2o (cm). cc, AC' cct 5 ' 3 MBB,: AB: 25 cui BF LAn, CD : CrDt, rol.i DDt: CCt. ,a) Poamareuo rpanerin ArBrCrD, (ryra.n. 6) - npoexqil LDrbcr- npauoxyrni piago6eApeHi, ro4i D,o = r t( b,tt , oJr' -r 2,u saqqo, = ;A"c,' D,B' = ;i * i ) = 2' 7to s."-^ = so,q",q - (b + o)t . s^".o = f 1,48 + cD).*MCD cosg 4cosrp' 2' . s,""" (b+a2 2 b+a '= F;= 4""'P(' - b) = 'z"""Q' Bidnoaida: !J b 2cosg 66g. ABCD - saAana rpaneqia (AB : CD). Ocxi.n BBr L s, CC, L a i BB1 - CC1,B1C.: BC' PisHi n Po = 3! ctt. 2T f i I tl{*$i;iii;ii*:*:{: TEOMETPI9 Eeea F Il. ra ix. Il
  • 66. LA.FB: BF = JAB'z - AFz = @1 ")" = ws - +s= Jb?o = 24 (c*t). t F =l.;r*28,t,24=8b.24=840 (crvr)..S*.o =:(AD + BC. B, 2. ' B.r =!.zo $b'z J'z =Sry,q=;(AD+4C11'.. 2 = B'JB.n = ssJz.a.z.n =*+ -aP= rao-Jlr (cu,); L onTL J]T fi1cos<O= *i-= 6 ; q=6rccos 6 lt*t*rpareqr,trMr. BiLnoei0a:840 crvI2: 140JiJ cnn'; u"""o"$' 664- 3a yMoBoro MBC - pieuo6e4pexnfi, CK L AB, ocHi.nrxz LA'BD - pinuo6egpe- nzfi, rosi DK IAB.Ilpone4euo CF L(ABD), FeKD. Maervro: LAIB - npoexqir LABC na (ADB)- MKC (IAKC :90"): cx =,[ac' - ex' - Jls' -tz'= 5 (c.vr). MKC (ZAKD:9O': DK = ,[AD' - AK' - ,li?'z - Lr'? = ,[zs ss = =5.7 = 35(cn). B LcFK (ZCFK :9O': CFz : CK2 - KF2 : 25 - tcz (KF : x); LCFD (ICFD : 9O"): CF2 : CD2 - FD2 : 352 - (35 - r)'9. Maerro pinnsnsa: 25 - x2 : 352 352 'f 7Or - x2; *--?1 --L:coszCKF=KF= 5 =1t^ 70 14' ""--'-' KC 14 5 14' s**, : s*n, . coslcKF; "*,, =i,* ct< =!'24'5 =12 5 = 60 (cru'z); s..-- = 60. 1 = -60 = 99= 4? 1.r'1. 14L477 Bidnoeilt: 4? c*". 7 665. Ocxirrxu DA: DC : DB, ro DO L (ABC), O - cepeszna AB. LAOC - npoexqia (ADC)aa(ABC). OM LAC, roai DM LAC (sa reopeMoro npo rpu rep' ' c.DM=!a.nrr,neH.4l,Ifiy,rffpu), S*. = i,O , s = 1ac -om =Lu 22 LDOM (ZDOM :9O"): DM = ^[Do" * oM' = 666, Ey,qyerto nepepia xy6a; s'egnaeuo A i M, M 1 .ro,ai CrF. ll AM, F - cepe4aua DD', AMCI - nepepia xy6a, za4anufi 3a yMoBoro; AM:MC,_C,F:AF, rogt AMC.F - porvr6. s*,,r -f,.dc,.*r ='UtA "J2 =+. Ocrfursnr,r rpaui xy6a - r(BaAparu, ro AC I B AO L(BB,D),rogiFO - npoexqis.A-F ua(BB'Dt), npoexqis AM na (BB'D'). Ana.norivHo, CrOr L (BBaDr) i MOri FOr - n MC, i FCt ua (BBrDt) ei4noei4Ho. * =lo lEr-b'=uw, costDMOo*,=2' 2 - 4 , - IDMO - ryr uinc (ADC) i {ABC), IDMO = ur Orxe, OMO.F - npoenr4ia/MC.F ua (BBrDr); 11-a2Ji Soro,n.= UOO,.nlf = )a.o.lZ =' it ,o - AoB cx - r(yr wix< (AM,c,) i (oMott,'o'ot "o.o = Y ,iir::;::::i:: {;$:t:.:;l.i rfi91:;:;:1:;;; ii:F-:i:::*: .6 Ri4noaidu: arccosr.t 667. Ilepepia, saganufr Ba yMoBoro, - uecrr4lcyrHr{t< Hrororb noJroBuni 4iaronarri rpaneii, ro6ro ^lo MN = NK = FK = EF = EG = GM ="' . 2 Ocri;rr,rcra AC I BD, FK ll AC, to,qi FK L BD. OO2 L BD, Ba reopeMoro upo rpr{ reptrenAfiKy- rapu OrT L FK. Ilponegeuo DX LTI. Maemo: FK I (DXT); DT I FK i TI I FK - oaHarca trepneu,4rrr{yJrsp- nocri npnmoi i naorqznrz, roai FK L DX. Maeuo: DX L(MNK), ro4i npoexqia OrD na nepepis - O"X. OrX:OrT+TX; OrT * zucorarrpaBuJrbHoro LFO.K, Or, =' 'ff LO"OT a LDXT: g=ob 24 Ao =;J7 +b';,qn = JF *a"; LDOA (ZDOA - 9O"): Do2=ADz-Ao2=r,-l ,t".0' =f -4it -412-a' b2. 124'4' 986 i:i::i TEOMETPI9 6ees F. l'1. ra in. llpoQinrxrr piBeHb i:..:ii;i;iiiiiii::::*iil rx.-rp. rx-or.rp -Loo Loo or - o;i' "'- o,r - 9.&_ 4 oD_;G K tiri{liilf:{iti;it* TEOMETPI9 6ees r. n. ra ix. l
  • 67. BidnoeiOa: 668. 34qanufi nepepis - MNTKEF - npaaa.irrnr,rfi necrrlrcyrrtzfi' Beprfir,rHrr slco- {-c, Ocrci.rrru Bh L (ABC), AArll BB, ro4i,4-1y' - npoexqis MN na (ABC), A?, - npoerqis. ArT ya (ABC, Tr - cepeguna AB, TrK, - npoenqis TK aa (ABC), KrC - npoerc4ia KE na (ABC), FC - npoexqis EF ua (ABC). Orrxe, NAIII(,CF - npoerqis MNTKEF sa (ABC). a" 3a2 , 4 4' "nG ""la sal6 + "J6 +aJ6 "J6 "4L212123 Ocxi.nsnu O" - cepeguna BrD, ro O, * cepeAlIna ii upoexqii, to4i ruyrana npoexqia: ro - cepeArlurz pe6ep ry6a. 1 oJi.flxuo -41| : a. NF = -- AC - -. 22 Bigomo, rqo AJrJ{ rrpaBuJrbHoro [recr],rnyrur,rra pu4iy" A' olncarroro rco;ra 4opinnroe ftono croponi, ro4i S."r"r": 6 ' SAxo. M (O - qeHrp urecturyrnnxa), 1 ^ ^ NF'J' ^ i"'Ji Ba'^1B A Sr*u"o =o.- 4 =o --= 4 . S"ori","" = Sscr - 2S*o, = o' -, t S"rtt."" 3a2 4 I cosc( = - ""o*- s"rr"r" - l, ga,rls- ,[B J5 u = arccosS: - xyt rrixc nepepiaou i ftoro npoerqiero. Bionoatoa: Af, Yt AC:4 (35 : 5):28 (wt); Bc:3.(35:5):21 (u). LACN: AN2: ACz + CNz - zAC.CN cos 45'; 4o0 = 282 + CN2 - za.cN .Jr; CN : x; x" - 28nl2x + 8.48 = 0; o, = iali)' - 8. 48 = L'so.2 -BB4= 892-884 = x=t+JitJs; x" =t4Ji + zJi = 2oJr, rogi BC < cN + NB, rrlo He Bl,rxoHyerhcff rpn IrboMy BHarreHH il x, = 1 Binoaia: l2Ji w. 675. Hexaft SABC - npanra.nrnrafi rerpaeAp, flo6y,4yeuo urorqr,rHy d, ,Erca napaJreJrsna BC, Als qroro uepea rovry A npoBe,qeMo n ll nC, roa KM - ei$clrag::o si,4 BC to a, KM - Bl4cora LKB ABI - ii npoerqia, CCr I ct, roAi SC, - npoexq na o, BrS - npoerqia SB sa tr. Orrre,.43rSCr * npoexqic rerpaeApa SABC na a Ocxi;nnra -A.S L BC, BC ll BrC' ro4i.AS I BrCr, 't-La2 ro4i S6,sc, =;Ot'F.,C, = |a a =;. 677, HexafiABCDA$ptDr - ry6. Ilo6yAyemo fioro npoexqiro Ea nJrorqr?Iny, fiKa rreprrenAur{y- .nspna .uiarona"ri .BrD, Ocxi.nrrrn Bp L AC, rosi PK ll .eC, O e fX, P, K - npoenqiiA i C aiAnosiAno Ha rlro nrro- IIIIIHY. - Bygyeuo uepee A' C1,..Dri B npaui, napa- lerrni BrD, ro4i orpurraeuo na n;rouquni npoercqiro PFKMNQ - mecrr{KyrHrrn, cropo- Hr,r rKoro pinri, rouy qo pe6pa ry6a pirui' PK=AC=aJ2; LPFK- pirno6e4penufi, FD L PK, ZPFD:6O"; P pF = Pp _ o"!i.J _ "Ja . sin60' 2.,18 3 Sru*rrq:6 'Srro (O - qenrp raecruxyrnr,rxa), _^*i c_ ^ pF,Jg _s.y:h=a,.ld.roAr J=o 4 =t I Binoaiu: a2 y!5. 67A. BPI = ACt'KCy HexafrAB: a. 1",*'=Bp:-Kq, ln,x'= AB2 -(Act- KC)zi zal6 3 zal6 3 da.-.---a' 22 t;v6 .i5 arccos -3 670. Hexaft MO L (ABC), to.4i OF L BC, OD L AB. 3a reopeuoro rpo rprr neptreu,qrxyJr.flpu MF L BC i MD I AB. LMBF : LMBD aa rinorenyaoro i xytou, rogi BF : DB. LODB : LOFB sa ABoMa KareraMn, roti IOBF : ZOBD, ro6ro OB - 6icettrpwca IABC. .flnrqo noxrara yrBoproe pisHi xytfl a ABoMa npfiMr4Mr,r, r{o fleperffHalorbcs, ro ii npoerqis Ha rrJrou{rrny qfix np.fiMt4x Jrerfietrb na 6icercrpuci xyra uirx NPfiMITMI,I. 672. MN I (ABC),,4-ly' * npoerqis AB na (ABC), AN : 20 m; BN - npoexrri a M B sa (ABCI, BN : 15 u. 3 yvroeu nraelro: CN - 6icexrpia ca ZACB, tosi # = + = +, toAi -43 : 35 ivu' '' BC l5 3 ', 988 tiiilj reorvrerple Eees r. fl. ra ix. rlpoginuxrafi pieenu ltil:{i*llii{i:L: Fl:iiiii:il$:ii::{: TEOMETPI9 Seso f. Il. ra ix. I
  • 68. ii ,.'t. rr ,lrl t/ [qu'=a'-xcl, la,x' = 2a' - Ba' + 2aJ3. Kc, - KC:i at - rcl = -a' + 2aJB. rc, - xcl; 2oz = 211rEKC,t xC, = fr. [u*"-AB'-AM', lnu'= nci - uci: )a' - am' =za' -(o,ts - d-nt)' . la" - w'=2a2 -3a'+2al3ll4 - AM': * = #, *u = ft i Kct =.ft, to6ro Aiarouam nosineua na 3 pirirr'rx ni,4pianx' v C D 4 B x lllJ a ALctDrBtl b(a - x1: stc' ba-br-c'x:0i -x(b + c): -ba; ba b+c' ba MO=-. b+c ah ab+ac-ab NO=a- b+c b+c b+c Orrxe, MN = Ko-o,= mi W = F _ iu"to'-o, X *i7<o'-w"t' -,lt-S::b+c b+ Bilnoai}a: 6 -1u,-X. B S 17. Bigcrani uixQiryPauu 685. ABCD - trapar'rerrorpau, AC e oc, flpone4euo BK LUiDM Io'. ,{ore4emo, t:qo BK : DM. LBKO: LDMO (sa rinoreuyeoro ra rocrprzM rcyron): (BO : DO), roxi BK : DM ' 686. .4-4, I o, BBt L a, roai OOrll AAr, OO, - ni4craul ni4 cepe4unn nigpiaHa,4S go cr. OO, - cepearn linig rpaneqii. oo. -,13 *7 = 1o (crv). ,2 Bi}noaila: IO cw. 687. AC: CD: DB, AAr L a, CCr L cr, DDt L a, BB1 L a, ADDIA, - rpanerr,ia, CCr - li cepe4ua linin. CC. = a '4.'2 .DD, - cepeAH.fi ;rinia rpaneqii CLCBBD DD- =8+ x . '2 "tL Maeuo: g=+-; 16 4:yl 2 =8*-{, "-6.a : 12; AAt: 12 ctt; 4 ro6ro8e cr. 2 Bi1noei1a: B e u, AAt: 12 cu. 688, -MN : a, MK I cr, NP L a, KP - npoexqin MN sa u. LMKO: MO : x; xo=r[*r_u,, LNP} lzNPo - e0"): OP = IA - ,' - "'I MKMObxAMOKa ANOC: -'PNNOcax 99O :::::j TEOMETPI9 EeB3 f. ll. ra in. IlpoQinsnrai pieexu iiti:::::ii:ii:i:iiiiii 6A9. MA L (ABC), MA - ascraut sis M 4o AB; AB L BC, roti MB L BC, MB - nigcrant ei4 M Ao BCi BD LAC, rosi MO I BD, MO - niAcrass sig M go BD. AMAB: MB: 5 (rnn); ,qc=3rti: Ao-+l t;; Bidnoaida:4 am: 5 a*; f .t*. 690. MO L (ABC, O - iientp Btrrcarroro rora e A rogi M - pinnoni44aneira nifi cropin ALBC' OK LAts,MK LAB, MK ai4crans ei1 M to cropin rprxyrurHa: L.MK: MK = "tFf * r'. Bidnoeida: ,Tn't ,'. 692. MB L(AtsC. 3a yuororo BC I AC, rogi MC L AC, MC - eincraus sil M to AC; LMCA; AC = ITM' - Mc' = Jn. s4 -15' = = Jsoa -zx= rlET = 9 (cnn). BC : Ac : 9 cnr; LMCB: MB = nffic" * BC' = ^lr% - 8l = JL44 = 12 (m 6gb. LACB - np,ffMonytnrrii, roAi qesrp ftorra' orr Horo HaBr{oJro rpr,rr(yrHu}ca, - cepeAl{Ha rinoreuy OF 11AC, ropi KF L BC, KF - si4craus eig K 4o 1A: 60" (nporu 6inrtroro rcyra B rprr{yrnl'rxy i::;:iri:i;i:::::::.;.i:i::;r::. FEOMETPI9 Eees l'. Il. ra in. Bo =+l
  • 69. flpone4euo OP ll BC, ro.qi PO L AC, KP L AC, KP - rigcranr six K ao AC. LKOF: OF : 3 . 2: 6 (cu). AOFB: BO : L2 cm sn r{arer, ulo Jle)i{r{rb flporlt r(yra 30', to4i,43 : 24 cmi lo MCB: BC = A,B.cos30" =rn ;=tzJS: p6 =! ps = 6ja,, 2A LKOPT -1 Kp = ,l Koz + po2 = V64 + (6J2) = J6a + r08 = =,8?2 = ^lr.Bo = ^lr.24z =2J4s. Bidnoailu: 2J43'cwr. 696, a)Ocxi.nrxu PB L(ABC)' roti PB - aigcranr Bia P ao (ABC), flpone4euo PK L AC, ro,qi BIf I AC, rouy qo pinui noxr,r.ni, nponegeni s oAniei rotrrtt, Marorb pirni npo- exqii, ro.qi BA: BC, K - cepeluna AC. A ^.PKC (ZPKC :9O'): PK =.{Pc' - Kc' = Jbo' - 1# = ft2.68 = = JL6.2.2.u = arEZ (cm); LBKC (:BKC:9o'): BK = "[Bc" - Kc'= Jgo' - 1s' = J-tz.aa = .,6 4.16€ = 4.2.3 = 24 (cut), (Pno," : AC + 2AB; AB : BC : (P - AC) : 2 : (96 - 36) : 2 : 30 (clt)); LPBK (IPBK : 9O'): pB=,lFr" -at<' -J6[:s4-14' = CZ{s4 -r) =8JB =40 (or). Bi7noai1o: 4O cttr. 6) Ilpoee4ervro BP L PK,uaerro: AC L(PBK),to.qiAC I FB, otxte'BF rr"prr"",q"- rcyrrrrpna ABoM rrpaMrrM PK i AC, rlo reperr,ruarorsca, rogi BF L (APC), BF - r.i.g- craur ri4 B ao (APC); LPBK (ZPBK :9O'): BPz : PF PK, PF = # = #- = T- ='#= it?f ' LBFP (LBFP :90'): - f " 1od.a4 Bp=Jnp'-PFz=,14o'z--=i -- L72 o-- 3a yuonoro: AN : NC = +, BN ='l lN' NC = J9 16 x - AoBsxtina o4riei vacrnnu, na axi uogineno rino l2x : 24; x : 2. AC : (9 + l-6)r : 50 (cu)' AB2 : ap =',1g.2 25.2 = 3. 5.2= 3o (cu); Bc = JcN. Ac = J16.2.25.2 = 4 5 2= 40 (crvr); AB+ BC - AC 30+40 - 50 =10 (cv):,= z = _ z LM'K (zMoK : 9o"): Mo = JMK' - oK" = Jr6'z Bidnoeida: 24 c:ll. 698. .RO L (ABC), axrqo R pinnoni4gariena ni4 cropin OK I AB, RK L AB; -RK - nigcrant r'iq R ao AB -RO - tsiAcranr nit R to (ABC); oC = L: oX = J=; LROK ZROK :9o"1: Js 213 @"' - "' "Jn "Jtz-t 12 = 2J{ = 6 RO= Bidnoeiha: oJt+t . 6 699, Ocxi.nrrcu r. O pisHosi{Aarena nig cropin rprnxyr nraxa, ro4i ii npoerqin Ha rrJrorqnHy rptnyrHrixa -qeHrp Brrrrcagoro HoJra, or:rce, OK I (ABC), KM LA OM L -AB, OM * aiActalgr si,4 O ,4ci cropin rpr'rxy HITICA, Eo *L7 +21 ^s.."- =.1^" '-' '-^ (24-ro)(24-t7i(25-2t) =-wuz/ ' =J24.t44.s =.,tb.6.211 s =3'7 4=84 (cu'z 1 S*" = ,P*"'KM (KM : r - paniyc rnucanoro ro.na); KM =2='-+=3,5 (cu)r P4a LOKM {LOKM :9O")t oK = JoM' - KM" = '11r,5" - 3,5' = J9.16 = 12 Bilnoai0u 12 cttt. 7O2, PA L a, PK, PF - noxruri; PK : L7 n, PF : 1 AK - AF: 9 wr. AP,41{ (ZPAK : 9O"): PA2 : 172 - LPAF (LPAF :9O"z pA2 : poz _ ({ - g)r: 172 - x2 : 10' - x2 * 18x - 8 I8x:172 - 10'z-I- 81; 270 30 sg = :_:: = ":; 15 (u). PK : 15 u; 182 LPAK: PA= j1?'-ls' =,lr'n = 8 (m). Bi4noai}t': 8 ttt. {liii:lf = W_*F * - llr.ur,o,l. 17 -r5'4 -::i&:#:::: 'l L7' 17 :::i,Si.iil fi?.4ooL8 20.sh4 60.,,6t :[H:ili;: - Li t7 t7 ::?r*lii:! ^ 6oJa4 Bldnoaldb: - CM. t7 A 697, Ocxi.nrrra M pisgosiA,qa.neua niA cropin AABC' ro ii npoercqia na (ABC) - qeurp Brrr4cauoro rcolrla. Orxe, MO L (ABC), OK L AB, MK L AB' MK -ni4cranr, si[ M to AB, OK : r - paAiyc BlrucaHo- no ftoJra. a' t2 gg2 i{:} TEOMETPI9 Eese f. Il. ra in. tlpoQinexr,tft piBeHb l{l';ltii;fl{ill$il ij;i;ii;i:i:::ii:i:iji;iit:i: TEOMETPIC Eees f. fl. ra ix. Ilp
  • 70. 7O4. AB e d, CD llABi CF L a, CF - rigcraur si,4 CD ,4o oc, OM L a, OM - siAcraur ei.q O ,4o c[. ^oMA* tcFA: cF = 49 =? t cF:2d. iianouiao, zi. oM Ao 1' 7O5, AAt I a, CC, L u. ATACC, - npnMonyrHa rpa- neqis, OO, - cepe4na rinis. 66. =L3)J!= 1b (crrn). ,2 Ana.noriqno BPDDT - npflMox)mHa tpaneqia, OO, - cepegna ;riuia, BB. + DD.()().= ' ':,2 DDr: 266, - BBr: 2 . L5 - 14 : 16 cm. Bi1noaidt': 15 crvr; l6 cu. 7O7. SABC - rpaBrrbnufi rerpaeAp, ftoro nucora SO, SO I (ABC). IJexair AB : a. LSOC (ZSOC :9O')l oc = ?;so = ,l4c' -oc" = ./3 t; .^ 4vo |lldnoqtdb: -. 3 7O8. Hexafi M - rouxa treperr{Hy ve4ias LABC, BBt L d., AA, L a, KK, I a. AtABBt - trp.sMoxyrHa rpaneqis, KK, - cepe4nr .ninia, xx, = L=14 + BB,) = (4o + 24):2 = 32 (cvl. ' 2' KTKCC. - trp.fiMoxyrHa rpaneui,n, CM i MK :2: T, ro4i F - cepeguuaCM. MM, =!1Kx,+ rr,l:' 2' ri, FFr: 2MMr - KKr: 2 ' 38 - 32 : 44 (cm); ji;i;:1:3f:;:;ifi: MM, + CC, i:ffr:'!:.i:;l:.i: .Ft, = -'i jliili{i:i:i:::h cc,: 2pp, - MMt: 2 . 44 - 38 : 50 (cru). j:itli:::il:::!:::::: Bionoaioa: so cll.. ::it::::::::::::l::: 7oe. o ll 0;,a e c, I e B;*'r'r:.:.i:nt: AK L g, rogi AK - nigctaar ui:n a i B. 1 BC L0', ro+rAK: BC; LAKB (IAKB :9O"): BK - 30 cm. Bi1noeida: 3O cv. I 994 ii::: rEoMETPtg Eeeg I. rl. ra ix. tlpoginsnri pigeHb iiii;i!i;:iiii:.fi;i:':x 713. (MNK) ll(,43c), oot L (ABC), OO, - ri4cranr uix (MNK) i (ABC). Lpo.K N tpoa' PL = !! =L'* -"' PO PC 2' Po=JPC" -co'= EF "Jo !e * s' t: r^ PO. =OO. = ovo = gjb. 3.2 6 .t: Bidnoaila: alo b 714. KA r @BC), KA L BC, ro4i KB I BC, BC L CD BC - oi4cranr uinc KB i CD, BC : AB : a. flponeqeuo AN L KB, roai AD L (AKB), AD I AN ,4-A/ - ni4cranr ruixc KB i AD. AKAB (ZKAB :9O'): KB = J4a" + a' = a1$; AB2 : BN . BK; BN=M'=!-=a-"G,BK 'Fso J5 5 ' I ., ,qN = J,qB' - BN2 = ,!a' -?u 25 q,l^ Bidnoaida: a' '-::a. 5 715. flporegetro OM L BrD, AC L (BBp), rcsi AC L Oiui, omxe, OM - ni4cranr uix BrD i,4n. O - cepe4una AC, LOMD o ABrBD: OM OD "4= Bp' o* = "";:,.9oo = trf = *-:8 =,6 ("*). Bi}noaila: J6 cu. 716, BD !AC san.racrunicrro giaronaaeft rcBa,4para iCp LAC, rogi OC - ni4cranr nitrt BD i CCr, orxte. oc =i* = * Bionouido, aJ2. , 717. SABC - npaurnrnuft TerpaeAp, MNLBC, MN L AS ffK Br.rcora pinxo6e4penoro LBMC i AANS MN = JMB' - BN' = l ;2 I A _tt Jil :i{iiii:#tl;;iii;i:j: |EOMETPI9 Eees F. Il. ra in. flpo
  • 71. hT-l,.ffii ,ry[fi**.{it ;:l{h1:::.:{{-:;! 'm-&*s v$l;:**d "JzI3idnonidb: -. 2 718. ABCD{BTCTD, - xy6, BC L (DCC), ro4i BC L ClD i BC L (ABB), ro4i BC LA'B. Or*ce, BC - ni4ctanr witx CtD iAtB' 'toni BC : a. Bi}noeiio: a. 72O, AK r(BC), AM LBC, KM LBC,KM - r;i-scna:g;rsiAK pBC;AN LCD,roIi KN L CD, KN - ni4cranr nig K Ao Cll. Maeuo: AM : 'A'ly' (nigcranr uix napa- JreJrbglrMr4 cropoHaMrr pou6a), ron.i KM : KN. AK - nigcranr' sia K lo AB ipAD. I MOD (IAOD: 90'): /D = &5" - rO' = 25 (cn). s^^", =! dc-BD =:.30.40 = 600 (crvr'z); lo22 Sosao: AM ' BC: AXI ' 25; AIvr =6o0 = 24 (crvr). 25 L,KAM (LKAM: sLz / KM = Jlo" +24" = 26 (cm). Bilnoaila: 10 cnir i 26 cu. ' 72"1, Hexafi AB : BC, MO L (ABC). Ocxi.nrrra M pianorig4a;reria ni,q ciopiE rpIIICyr- Hlrxa, ro O - qenrp Brrr{cauoro e MBC rora.' MO: 12 cl., MK LAB, OK LAB. Ba ynronoro BO z DO:5:4, ro4i BO : KO:5:4, ro,qi BK :3x (x cM - AoB)nt4rra 1'racrrnr'r, ra .axi noai.neno BD).,41{: 60 - 3r' 3a n.nacrr'rricrro ni,4pia- xin 4ornvnr,rx, npoBe,qenrrx e o4niei rouiu: AK - ADi A AD:60 - 3x. MDA: AB2 : ADz + BD'; (60 - 3x)2 + (9x)2 : 6O2; 60, - 360* + 9.f2 + 8lx2 - 602 : O; l 9Ox2 - 36ox:0; 90r(* - 4):0; x: 4, OK:4.4:16cu. LM7K (zMoK :9o'): MK = ^tMo ox' = Jtz' *ta' = zo Bi0noai1v: 2O cttr. 722. ABCD - rpaneqia, PO L(ABC). Ocxi;rrxz P pinnoni44aaena oiA cropiu rpaueqii, ro O - qeHrp Bnl,fcaHoro e rpaneqiro Ho.na. PF L BC, OF L BC. 3a n.nacrrasicrro orrucauoro qorrlpr{KyrHr4xa' BC + AD: : AB + CD, to.qi P*cr: 2 (8 + 18) : 52 (cu). CT L AD,'to+i TD : (18 - 8 : 2 : 5; CD:AB: (8+ 18):2:13 (cn); A 996 {!} reouerplg 6eee I. ll. ra in. flpoQinuxui (cr'n). P Ttu . ]:.i.:+:r:+J:.;.:!.:il..;q': plBeHb f:l:ti:Iri:i:i:lf:,':!:l:l:l.i'i cr = Jla, 4 = 12 (crvr). s*- = i ta + 18). cr = 1 s*"o = ip*"o. r (oF : n: r =2jf; ="#= 6 LP1F (lPoF: 90"): Po = JPFz - otz = $o' 4 Sayna*rennr: CT : 2r, rogi .B : 12 : 2: 6 (ctt). Bilnoai1v: 6 cvr. 723. SO L (ABC); S pisuosiAAarena nig cropin rpane g rpaneqirc xonat OK L BC, OK : r; SK L BC' SK : 26 ciw. LKOC : AMOC' rosi OC1 : OC : 3 DO - 6icercrpuca ZCDM, ro4i DO L CCt; LC,OD: C,D : 50 cttt; so",oo = f,cro . oD a6o s*- = |om .crD; otur =22:!p' =w#= q%lq = 24 (crvr) - pa4iyc nnzcanoro rcorra. LSOK (ZSOK:9O'): so = ,[s* - oK' = ,tE6' - 14' = 1o (cm). Bidnoei1a: lO c*t. 3a sracrnsicrro 6icercrpucr': #=#= 3 =:' AK-FBc IANK, KB =9, !==L, -- --- AK AN' 3 6-x' 24 - 4x:3x;7x: 24; tt x ='-]: LMKF (IMKF :9O"): l4(-7.6", _ z&es 77 725. MK L (ABC), MN L AC, MF L BC' MN : MF, cM. 727, MN I (ABC), M - piniroai,44a.nena ni4 -48 i BC . NK L AB, MK LAB, MT I BC, MK : MT : 10 C =!Nr.Bc,S*; = ; NI{. 15: Suru, z 1 Sru" = S*" * Sor"" = ;NI((l5 + 2l) = t8NK' s*" = JrO '15 .o. 9 = 15 ' 2 'sJE = 90Jg (or'); eor6 = 18NK; NK = e9f = rJd (crvr).18 LMNT (ZMNT :90'): MN = JMr' - Nr' = fio'? -5'? € = Jioo-?5 = 5 Bi]noeido: 5 cvt. [iiiti:lt*i4}$l:"ti rEoMETPlg Eeee l-. IL ra in. Ilp
  • 72. 725, MBC - piBgo6e.qpeHvfri CCr L cr, AA.L cr; AK : KC; KKt'L a; KKt niHin rpaneqii AtAcCt, **,=12f=16 (crvr); O ..._ qenrp Brrr,rcaHoro n LABC ro.na. CO - 6icer- rpuca, rogi KO : OB: 3 : 7; FF1 I qi KO : OF : 3x, OO, - cepeArrs ;rinia rpaneqii K.KFFT; OOr=4!L; FFt: 2OO1 - KKr: 2 l9 - 16 : 22 (cu); FF, - cepe4us .nini,a tpaneqii O1OSS1, S"lt.t rqoaaJS r1r.-..--... OF=-e o' fr$:iti:l:i.:.:t oJa - 2.h 6 fa$a^.-.,...-...... a ':{$:::##:.r. aVS ,tf$;i;:;{.;:1:;: Bidno7idb' -.1J.U;:li::::::i:i 6 $$illlliiil lss. KL - ni4craxr uilx AC i PB, sasc : 25o cv2i 'rrli.:.:+li: I SAsc : 300 cv2: S*, = UPK AC: ," =' ?:o = 25 (cM); s*" =f,nr'nc; A uu --''::o = 30 (cM); s**u = L LK BP; TT, = FF, +-BB, : TTr: xi BBt: a,,2 roAi r = 4lJ: ss, - cepe.rlut ilnittTrTBB,:'' 2 ss.=?4*BB,.2s=!)-!,'22 2x:22 + A; A:2x - 22;50: x + AiA:50 rt 2x - 22.: 50 - x;3r: 72; x : 24; TT, : 24 cu; BB, : 50 - 24: 26 (cnt). Bi1noeiOu: 26 c:ll. I 731 . HexafiABCDAtBrCtDt - xy6, AB: a; BrD, L(AAri) (sa osuaxoE:J9 rDr LArCu AAt L BlDt), rogi BrDt L AK. Ilpone4euo OF LAK (K - touxa neperzuy giaroualez ACr i BtD). OF e (AAtCr)' roxi OF I BrD. Orrxe, OF I (AB.D), OF - ei4c'ranr' ni,q O go (ABp)" AoFK- IKI,A:#,=#, oo,=+: ox=f,; AI{ = _"Js _"J6.- Ji- z' 19 + SS. FF- = ': '2 SS, : 2 ' 22 - 19 : 44 - 19 : 25 (cu); BS : S", rogi TT, - cepeAgn niriia -FtFBBt; A1 - cepeAHs 998 ::ii: rEoMETPlg Eees r. Il. ra in. IlpoQinunxft piBeHb iliitiiiiiiiiiii{iii:: LKPB pirno6eApenlrft . s.^..- =L sx - Po = !. Bo.,[r5' - 16' = 22 LK =2' !!o = 2.B. 4 = 24 (c't). 25 | .ao.zu Bi1noeida: 24 cv. 734. a)PABC - r,pasliallrlrair rerpaeAp. PO L {ABC), ro4i PO L OM, LBOC - pinro6e4peN OM L BC. Oqn<e, OM - siactagr mitx PO i BC. Ocri.rrxr.r O - 4erarp LABC, Bionoeioa: dE. 6 oM=r= "==oJe. 2t3 6 6) Ilponegeuo LK ll BC, LPLK - pinno6e4penzft, PL : PK - Brrcoru 6iualrx rpaHefi; PT L LK, LT ITK; T C AM; LK IAM' ros LK L (APM), LK L MS. flpone4eniro M S I PT, ro4i MS L (LP K), BC ll (LPK) i BC. LPSM (IPSM: 90"): MS'z - PM2 - PS2. He LMST (IMST - 90'): MS' - MT2 - TS'?: ivlT =!* S? PT-PS-PT-x. 2 LPTK (zPTK :9O"): Prz = PK2 -rKz =(+)' -lXl =+ #=8 3a2 . 3a2 ( o,ll1 ' } MacmopinHauxx: 4--* = fO -l. + -tJ t 2Oa2 ""lll 5a'.2 5a 16 2 ^' ^ + aJtL 2,1f7' . Ba2 25az 33a2 - 25a2 8a' 2a2 MS" =-"'" 4 44 44 44 LL' MS=4=9E' Jrt 11 Bionoaiou: St Oana.noriuno 6). 737. (ABC',) r @BD), CF L AB, DF L AB, .DEC : 90 flpone4euo FK L DC. Maeuo: AB L (DFC), rcai AB L FK, orrxe, FK - ri4cranr ttitx DC iAB. ^ABC: cF = tr€ = tVU, MDB: DF = JIO'z 8' = 6 (cu). LDFC (IDFC :9O"): DC = ^lcF' * nr" = {6t€ + 36 = Ji92.36 =.EB :i:i:i:i:i:i:i:iii::C:i:::i:ira FEOMETPIS Sees f. Il. ra ix. Ilpo
  • 73. s** =f,nr .FC a6o s*," =f;rr nc. rx=25 = DC Bidnoai1a: DF .FC 6.8J5 24",tE ."t5.JLs 2J57 57 24.}JLs 24^lls 3 .19 19DC 24JLg 19 cM. 738, a)Hexaft ABCD - npanNlrnrafi TerpaeAp, BC:A, K? - niAcrans ral.r" BC i AD, KT= o"li A 2 Bihnoeio: a& 2 6) rlposeAelro ME llec, twr, =,8,#Jy!! F$ L{Anc), F,s - nigcra:$ mirxAC i Er, .qs = lr,r: - pS= Jt+)' -[rfu)' = 87,, = E = E = + Bidnoeioa: a&. 6 s) .or ll Bc, oM L Bc, rogi OrMl - ri4cranr virx DO i BC, OlM1llOM. or=+=#r=*, o,,=*. Bionoaida: aG, 'A t2 r) MN - oi4crar* rvrirx.A-D i BC, EF ll BC, M$llMN,ro.ui MN =5, *,*=+ -nigcrans Bionoaiha: a& . 4 739, Hexaft.ASCDAtBrCtDt - xy6, ArB , AD | - rvrrnno6irrcni 4iarona;ri rpauefi AAtB ,B i AA.D 1D . (ABC2,) ll (AD'C), Al orxe, ni4cranr Nrirx Erilvrz - ni4craur wlj.w AtB i ADt. Bp L(ADp), DrO I BrD; DO.: r, rogi AD'OD: DrO2 : DrD2 - x2i LBTODT: DLO2 = 4Dl - BLO2: a" - x" =(o{z)' -(oJs - r)" ; A a' - x" = zaz -Ba" +2anlvx - x"t 2a' = zarl|x; * = 2a"- = o:t - za"ls- Jg' LD:OD : LBOtBt, B:OL = DO -- *, OOr= oJd - 2a - a t^ J3'--' J5-"5' g;6n66i6o; !!9. 3 looo liltj rEoMETPlg 6ees F. l'1. ra ix. IlpoQinsnni piaenr $i:,.ili:'i,t:i.*.1:i:$,rt C witx AD i EF. 740. Toqxa, pianouiggaaena ni4 cropiu rpfiltyl rlul(tr, qeuTp Brrr,rcaEoro B Tpr,rryrrrrrx xo.na. ffxu4o flpoBecrI XO L (ABC), roAi 6yAr'-.grca rorrxa qboro reprreg,ql4 KyJrfipa pionorigga.nena ri4 cropin rpr4nyrur'rra' S 18. Kyru e crepeonerpiT 762, LHAB : LHAC (sa AsoMa xateraura), ZHBA: IHCA, orxce, noxrani HB i HC yrBopuorb pinni xyru s (A ,flrrqo noxzri Ao nJrou1lrHlr, axi npone4eni e o4niei TOrrRI,, MarOTrt pieUi npoexqii, To Borrtt yrBopKlrorB pinni xyrr,r s qiero n.noulranoro. 765. ASOM (tSOM:90"): OM :8 cw; LSOKT OK :8; LMOK: KM2 : OW + OI( - 2OM 'OK' cos 120o : 82 + 82 KM = d,t5. Bilnoeila: A16 axa. frcrqo 'lovxa piruonig4a.rena ni4 nepurun lrnoro- r(yrrruxa, ro ii npoenqin Ha nJrorqrruy Mnoroftyr- HI4Ka - "qelrTp KOJra' oflIlcaHoro HaBr{oJro qBoro MHOTOKYTIIIIKA. 766. flrqo rovxa pinnoli4ga;rena ri4 Beprrruu KBa,4p KBa,4paTa - rIeHTp onl4caHoto rcoJla HaBKoJIO fiBaAp MO I (ABC')i IMAC * rcyt vrix MA | (ABCI' AC 6rE ^EA()=-=-=iaJz. 22 LMOA (tMOA: SA'): coszoAM =#=+=+' Bidnoai}a:45. 767. MA: MB: MC, roni MO L (ABC), O - qenrp onrrcaHoro xora ranxo;ro MBC, OB * ftoro paAiyc' IMBO * rlegrp Haxllrry MB ao (ABC). 'Ioai s LMOB (ZMOB : 9O")i costMBo=oB =-Y3--A'MB JB .2a 2 ZMBO:30'. Bi1noai0t:3O". ii;tj*:i:l|{tiitl{;i rEoMErPlc Eees r. Il. ra ix' Ilp toAM - 45".
  • 74. 77O. SAtsC - npaBnrrbnrzft rerpaeAp' SO L(ABC), O - qesrp M-BC' ASoB: cosZSBO=#=-{*= t; ISBO = u"c"o"*. Jt. 3' BiAnoaiAb: 771. Hexait SABCD - trpaBrlJrbHa qorlapunyrHa [rpaMrAa' rctiABCD - IoBagpar. SO L(ABC)' O - qenrp xnagpara; ISCO - xyt rrint 6i-rnuru pe6poru i u.norqunoro ocHoBr'{' oc oJz ,12 ^soc (tsoc : 90'): cosZS"o = fr=; = 7 t""- xaftAB:4, aa yMoBoro SB: SC: SD: SA - BD - a); tsco:45". Bi0noai6v:45". 772. PABC - rerpaeAp, PA: PC: PB aa yMoBoro' ro;ri P0 I (ABC, O - qestp r{orra' onrlcaHoro HaBfio- to LABC, OB : R - Pa,qiYc qboro Korra. A LPOB (IPOB: 90"): PO : PB sin s : c sin ct' Bi}noai}a: c sin n. 774. MA: MB : MC, ro1i MO L (ABC), O - qenrp KoJra, orficaHoro nasroro AABC. MBC - npaara.nrnufi, roAi OK LAC' MK IAC, MK - nigcraut' sa M lo AC. LMOA (IMOA - 9O"l: 1 ft clEc AO : h cLg a: OK =:ZAO - -j:; LMOK (tMOK : SO": MK =,[MooK' = ," h'ctg2a ,* 4 Bionoaida: h ,l+ ,.tnm. 2' 776, ABCDB.CrD, - napa.ne.nenine4, y -- S."". ' h : Ss"o A' LAAC (IAAC: 90"): AA,: 4 sin P; AC : d cos B; Aa d.cos} J2 AD=- Jz2A, ,. - ( a co" FJi l' ., "r, u - za' :o"' j:! onl - d3 cos'? Bsin B ,-l 2 /'.",'.r,= 4 = 2 Bionoeioa: d3 sin E cos'? I . 777, V: S."".'lr: S*"r' DDr: an AD'DD'; A ^B$D (LBBD - 90": BBr : I sin a; BD : I cos cx; LBtClD (ZBPP: 90'): B'C' : I sin B; J5 arccos - i,.r.{i:t'!l.ld{ lOO2 tl::i ;EOMETPI9 Eees F. Il. ra in. IlpoQinuxrai piBeHb i::i:i:':+:i:i:i:i:i:i;::i:':i AD : BlCf LABD (ZABD :9O"): AB = JBD" - AD" = J'z cos'? cx +rr'? F - l.f*'o y = ,G;%:;;'p'tsinB.tsinoc = = 1'"ino"i.rBG*'o *irf F = = 13 sin o. sin Bviilsln' CI - sln' p = = l' .in,, "in pl6-d 0- "in' o. Bilnoai}a: lt sin cr sin B.ftffi - .t"' ". 779, ct L [], lX L a, BF L a; a o F : a; BF L a; IBAI :30'; -4r r 0; /ABK : 45". LAFB (ZAFB : 9O'): BF=l an=a: 22' MKB (IAKB :9O"): oJi BK = AB. "o"45" = -Z i ABFK (ZBFK :9O"): ru eK =Jnx'- sF' -.l!--! = 2 4 Bilnoai}a: 9. 2 78O. Ocxilsxa MB, MA, MC, MD oAHaHoBo saxn, MK L (AtsC), K --- rlenrp oilucaHoro HaBKoJIo rpa neqii ltora. 3a yuonorc IACD:90', roti K - c peAuEa AD, ornte, KA : KB : KC - KD : R - pagiyc riscro xota. ZMDK : 45" xyr aaxuny MD to {ABC). Hexaia CD:'15 cu, CH L AD, CH : 12 cv, ro4i HD: 9 cnn; CD2 '=. I{D . AD; AD=g: - t!+ - 25 (cM); HD9 KD =!.25 = 12,5 (cw); AMKD (zMKD : 9a: 2 Bidnoaido:-12,5 cv. 78'l , PO f @BC, O - qeHrp rtora, onr,rcaHoro sasn rpaneqis, roti lD: ZA:60". 3a sacliAHorr s reopeMr{ ctnycin gna LACD: Ac = 2R (R: oBl: sin ./ D R=:c _=4+=2 (crvr). 2sin lD 2.,Js 2 LPOB (IPOB :9O")t costPBO=Bo =?=L, zPao:00".BP42' Bi1noei}a:60". l;i:::ii;ii:l:i;i{ii:il.:.:. FEOMETPI9 Eeee f. fl. ra in. l-
  • 75. 782, MB L (ABC). IlponeAeIrno,4l{ L BC,'ropi MK npoerqig KM ua (MBC)' MKB (ZAKB :9O"): AK= A13"i,,60'= "j3 ; 3g =r- aP = 91 22 ttwax (ztwnr :90"): MK = ^lme" * nx' = LMKA (IMKA: 9O'): rs ZKMA = # = *# = $; zeux : zo'. Bi1noai1t:30'. 783. KC I (ABC), IKAC - rcvr uim AK i (ABC' Ilpone4ervro A/ L BC, rop.i KF L BC' waewo: Ar L 6BC), /Alff - rvr uirx AK i (KCB)' Hexaft AC: a, toAi MCK (ZACK - SO''): ,q.x= AC =$="Js; onr*i'Xox:utno'l' - =a sinao" =df ; sinIAKF = # = *=f,; z'+xr : 'tr.'. 7A4. AO : 4,5 te, a; AO : 1,5 tg 2a;4,5'tg cx - 1'5 tg sin c sin 2ct 3._--=U;cos rI cos Zo[ 3 sin cx cos 2cr - cos o sin 2o - O. cos ct cos 2ct 2 sin cr cos 2cr * sin(o - 2a):0; 2 sin cr cos 2n - sin cr: 0; sin cr(2 cos 2e - l):0; 2 cos 2cx: l; 1 cos 2c, = j; 2a : 60; s: 30". AB:1,5: cos 60' : 3; Bilnoeila:3 i 3J5. L AK i AK L (MBC), toti MK - M .E*7 =to,v42 ri:t":--..-.. '1qli:::*Si:?l 4.52 9 gJt -,aAC = 4.5:cos30" = ::=G = s =.r!6. V'O 785. 3a yMoBoro tr - 0 : 60' MOC: /'O : 3 sin cr; MOBI AO: 8 sin B; 3 sin cr: 8 sin 0; 3 sin(60' + B) - 8 sin B- 0; I lo r tlf *"B + isinFJ - 8sinB = s; SJicosB + SsinB - 16sinP = 6; toO4 .*ili rEOMETplg Eegs r. fl. ra in. rlpoQmuxraft piEeHb *li{itl:ti:l*ll:i* 3JdcosP = l3sinF; ^I; 13teg=e$: tg0=H. LAIB: tgp= #t #= f ' uo : a: Ao= *" 64 = rlE"l13 .) , 64.169 a" = -' 196 Ao2 =64 (9" =v4;q, "o= F!-iw + a'i 27az + 169a2: 64 . 169; 8.13 4.1.3 52 -- tl, - z 7' SaAaqi ga roroBr4Mlt MarKtHKaMlt Bapiaxr A 1. 1) Kyr mint npanzwrr.r ABri CD "*yt .ix* DCi D 2) Ityr uixc npsmuuu AB, i BD e Hyr nnilrc BD i C 3) Ityr uix,4.B, i BC, e rcyr uix .AJ9, i ADr ZD/'B IAo2 =244- Bo2. 2. l--- - -- -' 244-BO2:40O-BCz;BO:O IAO'=4OO-OC"i nopqifinocri; BO : 5x; OC :8tc;244 - 25x2: 400 BO : LO; OC : 16; BCz:BO2+OCz-2.BO.OC cos 60'; BC' =10 Moc: co = JAc' - Ao'z -- Bionoaior,:l"*i € "*.77 BC2 : 196; BC : 14; [dl,t' = 244 - BM2, 244 lltr't" = 400 - (14 - BM)': 2a BM:AO; BM =19, 4114''7 lG.z' !497 - BM2 :- 400 - 19 = 244 -ff; dm 3. Po=*o=I+=+' PM'z:2 Poz: PM2 =2.Lf;, ,, =lJut ,",: PM2 + MC2' ,r' =u :u +76: pc2 =ff, ,"= + 4, AII : CH : 5; DH2 : AD2 - O11z; DH2 : 64; AII : DCz : AH2 - CHz - 2 . AH' CH . cos a; 49 - 50 - 11cos o( = -- : O( = arccos - . 50' 50 !:i:ill!i;ii:i:i;ij:i:i:::it TEOMETPI9 Eees t-. tl. ra ir. Ilp ,l I i ;l 4.108 -49
  • 76. 5. AB2 : ACz I BCz; AB2 : 625; AB : 25. lcK'=.tc -.aI<2- I--^ : ACz -AI(: BC2 - BK2;225-AKz:4OO - (25 -AK)'; lcK'=BC'-BK"i 225 - AK2: 400 ' 625 '1 50 .A-I( -.AK2; 50 AK : 45O; AK : L CKz : 228 - 81; CK2 : 144; CK : 12. ' co 6 =-lr cr:80".slnd=AC; slncl=i2 2, - 6.AB:BC:AC:a; BM=aGz 2' sin60o = BO , Bo =!o. "lnaoAo = BO . BM. 4 AB' sin.1AOB = 3-: IAOB = IOC B = arcsin q . 4'4 7. MK : OM : NK : ON : a. OK2 : 2a2: te6}" = g . J5=4, aN=aJl; oN' sin6o.= #, +="f ; oa-zo; cosIAOK = ff, "o":nOK = {: , ./AOK:45". 8. ABCD - npanrz;rnufi rerpaeAp. AB : a. EOtll DOi AE : ED; "o=? +=+, Do2: Ar2 -AO2. Do2 = a2 -";, oo = rf1, "o, = 4, ,,o,=rr;,o=f; ruo,=+ +=+t ME'=EOi+MOi; ME'={+{: mn--L.6 3 """ J'' 10 r -or: -t,! : ?:.. ?:.' 9I = 4, oN= 1^0 ij MB BE 9 12' -" 3 5. AB : 25; A,ll, . 24. A e rr, Be p; AA, - E;BB, Kyr ruix rr i fi ./(:A,11. 111. ll A,Br. AC" : AB2 - AC! : Z'tt 241!i A(t 7. AC'| - C4 + A4 -2CA 49 : 89 Il0 <xts/(tA,A: I coslcAtA -' i t (:A,/ 69o. 6. sAABc : 48 cu:r; sn,,,,, ., Ji}.2.I2.4 (s = .[1p soo" = 24J3 r:m:,; t:,srr - ?!!1 =f ; " : ao". 7. BK: KC; Il{: A(:i AKtt . AB2 + BK2 - 2AB.BK BC : AC . di Att uJz; IiCt : 2KC2; az : 2KC2 r,^2 a' ,, Jlxc- = -: ,((' : AIi2 : BKz - AKzi 2:tt"- 2a' 3u' Fza" = _+ AK-,i nn. _ -; ; eX = al|; !o'=2o' ,u' 2,,J2 ',',.cos1ABK: !o, =5ro2222*2- 2a2 cosZABK c"; r.r,s . ABK = !: ZABK - GO" 2' 8. Bigcranruixr,4/l i lltl) - ei4cransrvrintBrD iDC B,D =8J3; OI) 4J:1. oM,: oD, DM2; oM2 OM2 :32; OM 4J2 ,'.. Tecroei sae.qaxxn 1. a) nepnengrxylrprri. 2, 6) nepueH4uxyrurprri. 3. a) (BCC.) rre llel)norr,rttrKyJrspEa Ao rrrou{rrrn (AAp 4.6) LSOD - lr;uuoxvtuurT. 5. 6) npoxo4urb ,repe3 r-eHrp onucauoro r{oJra. 6. a) npoxo4rrb trepea rIeHTp BtrEcaHoro KoJra. 7, a) 2a. 8. a) 30". 9.6) tDMA. 1O. 6) 90.. fililililiftiitl rEoMETpte 6eeg F. n. ra iu. np 3. r=S-ut-- p BE2 '- R(:2 f,p(p - 1sxp -'lsxp - 10) 13+13+10 2 [iC'/: IJE2 - 169 . 25; BE : 72 crt. 10 3' no, = f,oo; Bapiaxr 6 1. AC I BM iAC L NM =+ AC L(BNM) = AC L BN. 2. LBAD: 30'; BE ll OX: np =7. AB = 12 9 (cu); 2 OK : 6 cvr: OM : 8 cm; OK : 6 cm + MK : 10 ctvt. S*"o: AD . BE : 40' 12 : 480 (crvr'z). S*"o: CD 2'OL;2'OL:2O; OL: lO cvt;.ML2: MO2'f OL2; ML2 - 764t ML = J164. 1006 ii{i rEoMETPle Eeeg F. Il. ra in. rlpooinuxrfi piBeHb f$ll{tilliti::iii:l