2. I
II
<l>op3al. 1 . ·
UEHTPA.nbHllBn~CAHIKYT~
c
LABC = ~ uAC
LAOC=uAC
TEOPEMA <l>AJIECA
HKIIJ;O OA 1 = A~2 = A 2A3 i A lB l II A~2 II A3B3,
TO OBI = B 1B2 = B2B3
3. - -
I
i _· _____ __ _ <2>op3a'-' 2
nEPWA 03HAKA nO.Q16HOCTI TP"1KYTH"1KIB
.QPYrA 03HAKA no.Q16HOCTI TP"1KYTH"1KIB
~ B
1
ACA~CI I
TPETSI 03HAKA no.Q16HOCTI TP"1KYTH"1KIB
B
A Bl
A~C~A1 C1
4. A. r. Mep3n~K
8. 5. nonOHCbKll1C1
M. c. 5=1Kip
rEOMETPIR
niAPY'-IHVlK AJlfl 8 Knacy
38f8JlbHOOCBITHIX H8Bl.f8JlbHll1X 38KJ18)::IB
PeKoMeHp,oeaHo
MiHicTepcraoM oceirn i H8YKM YKpaiHi--r
XapKis
«riMH83i~»
2009
9. 1. 4oTMpMKyrHMK Ta Moro e11eMeHTM
Bi,n;pi3KM AB i BC, ao6pameHi Ha pncyHKy 1, MaIOTb TiJibKli
o,n;Hy crriJibHY ToqKy B, .HKa e KiH~eM KomHoro a HMX. TaKi
Bi,npi3KM HaanBaIOTb cyci.D;HiMH. HarrpnKJia)J;, Ha pncyHKY 2
KO:>KHi ,D;Ba Bi,n;pi3Kli e cyci,n;HiMM.
B
c
A
Puc. 1 Puc. 2
BiApiaKn AB i CD Ha pncyHKY 3, a), 6) He e cyci,n;HiMn.
TaKo.m Hee cyci,n;HiMn BiApi3KM AB i AC Ha pncyHKy 3, B).
'
A C B
• A B c
B
·7D
a) 6) 6)
Puc. 3
PoarJI.HHeMo cpirypy, .HKa CKJia.n;aeThC.H 3 't!OTMpnox TO't!OK A,
B, C, D i 't!OTMpbox Bi,n;piaKiB AB, BC, CD, DA TaKnx, ~o
Hi.HKi ,n;Ba cyci,n;Hi Bi,n;pi3Kli He JiemaTb Ha o,n;HiM rrp.HMiM i Hi-
.HKi .z:i;Ba Hecyci,nHi Bi.n;pi3Kli He MaIOTb crriJILHliX TO'C!OK (pliC. 4).
<l>irypa, YTBOpeHa ~liMli Bi,n;piaKaMM, o6Memye 'C!aCTliHY
IIJIO~liHH, BM,D;iJieHy Ha PHCYHKY 5 3eJieHHM KOJibOpOM. I.1;10
'tlacTHHY rrJio~IIHII paaoM a Bi,npiaKaMn AB, BC, CD i DA Ha-
B B
A
Puc. 4 Puc. 5
6
10. 1. LJon'1plt1KYTHlt1K ra tlloro eneMeHTltl
A B
D c
a) 6)
P:nc. 6
3MBaIOTL qoTHPHKYTHHKOM. ToqKn A, B, C, D HaanBaIOTL
BepwHHaMH "t!OTHpMKYTHMKa, a Bill;pi3KM AB, BC, CD, DA -
CTOpoHaMH "t!OTHPHKYTHMKa.
Ha pncyHKY 6, a), 6) 306pameHo cpirypM, .HKi CKJia,n;aIOTLC.H
3 qoTnpLox BilJ;pi3KiB AB, BC, CD, DA i -qacTMHH IIJIO~MHH,
.HKY BOHM 06Memy10TL. IIpoTe u;i cpirypn Hee 'tJOTMpMKYTHMKa-
MM. IlO.HCHiTL "t!OMy.
CTopoHM qoTMpMKYTHMKa, JIKi e cyc~HiMM Bi,n;pi3KaMn:, Ha-
3MBaIOTL cyci.nHi.MH CTopoHaMH l.!OTMpMKYTHMKa. BepIIIMHH, .HKi
HaJie:>KaTL O,D;Hili CTOpOHi, Ha3MBaIOTh cyci;J;HiMH BepwHHaMH
"t!OTMpMKyTHMKa. CTopoHM, .HKi He e cyciAHiMM, HaanBaIOTL npo-
THJie:>KHHMH CTOpoHaMH "t!OTlipHKYTHMKa. Hecyci.n;Hi BepIIIliHli
Ha3HBaIOTb npOTHJie:>KHRMH BepwHHaMH "t!OTlipHKYTHMKa.
Ha pncyHKy 7 306pameHo qoTnpnKyTHHK, y .Hi<.oro, Harrpn-
KJia.n;, cTopoHH MQ i MN e cyci~HiMM, a CTOpOHM NP i MQ -
rrpoTMJiemHMMM. BepIIIMHM Q i P - cyci,n;Hi, a BepIIIHHH M
i p - npOTH.Jie:>KHi.
qOTliPHKYTHliK Ha3HBaIOTb i Il03Ha"tlal0Tb 3a lioro BepIIIli-
HaMM. HanpnKJia,n;, Ha pncyHKY 4 so6pameHo "t!OTMpHKYT-
HMK ABCD, a Ha pHcyHKY 7 - "tIOTMpnKyTHMK MNPQ. IlpH
Il03Ha"tJeHHi "t!OTHPMKYTHMKa 6yKBli, w;o CTO.HTb nopyq, Bi,n;rro-
Bi,UaIOTb cyci~HiM BepIIIMHaM "t!OTlipliKYT-
HHKa. HanpHKJia.n;, "tJOTMpMKyTHMR, so6pa- P
meHMH Ha PMCYHKY 7, MO:>KHa rro3Ha"tJMTM
w;e TaK: PQMN, a6o MQPN, a6o NPQM
TOIIJ;O.
Bi.n;pisoK, .HKMH crrony-qae rrpOTMJiemHi
BepmHHli qQTlipHKyTHMKa, Ha3MBaIOTb ~ia
rouamno. Ha pMCYHKY 8 Bi.n;pi3Kn AC i BD -
,n;iaroaaJii qoTMPMKYTHMKa ABCD.
7
Q
Pnc. 7
11. B A
D
c
a) 6)
PHc. 8
KyTH ABC, BCD~ CDA, DAB (pHc. 9) Ha3HBalOTh KyTaMH
'IOTHpHKYTHHKa ABCD. Y ~hoMy 'tIOTHPHKYTHHKY Bci BOHH
MeHrni Bi,n; poaropHYToro KyTa. TaKHH 'tIOTHPHKYTHHK Ha3HBa-
IOTh onyKJIHM. 0,n;HaK y 'tIOTHPHKYTHHKY Of);HH 3 KYTiB MO.me
6YTH 6iJihIIIHM aa poaropHYTHH. HarrpHKJia,n;, Ha pHcyHKY 10
KYT B 'tIOTHPHKYTHHKa ABCD 6iJihIIIHH aa 180°. TaKHH 'tIOTH-
PHKYTHHK HaaHBaIDTh ueonyKJIHM
1
•
B
c
A
D
D
PHc. 9 PHc. 10
Teo p e M a 1.1. CyMa icymi6 'tomupuicymnuica iJopi6n10e
360°.
B
A
D
a)
c
PHc. 11
D
6)
1
,l1;0KJI8JJ;Hillle 3 IlOH.HTT.HM «OnyKJiiCTbl> BH n03HRHOMHTecb y nyHKTi 19.
8
12. 1. Lfor111p111KyTHll1K ra iltoro eneMeHTVI
/{OB eiJ eHHJl. 0 Y qoTRpRKYTHRKY rrpoBe.n;eMO .n;iaroHaJib,
SIKa poa6RBa€ iforo Ha .n;Ba TPRKyTHRKR. HarrpRKJia.n;, Ha pR-
cyHKY 11 :u;e .n;iaroHarrh BD. To.n;i CYMa KYTiB qoTRPRKYTHRKa
ABCD .n;opiBHIO€ cyMi KyTiB TPRKYTHRKiB ABD i CBD. OcKiJih-
KR cyMa KyTiB TPRKYTHRKa .n;opiBHIO€ 180°' TO cyMa KyTiB
qoTRpnKyTHRKa ~opiBHIO€ 360°. .&
CyMy .n;oBmRH ycix cTopiH qoTRPRKyTHRKa HaaRBaIOTb ne-
pHMeTpOM qoTRPRKYTHRKa.
0-. 3 a~ a 'I a. !{oeeiJimb, zito iJoBJ1Cuna 6yiJb-.HKoi" cmoponu
1lomupuKymnuKa Menwa BiiJ cyMu OOBJICUH mpboX inwux uozo
cmopin. B
Po a e' a a a n n a. PoarJISIHeMo .n;o-
BiJihHRH qQTRPRKYTHRK ABCD (pRC. 12).
IloKameMo, HarrpRKJia,n;, ~o AB <AD +
+DC + CB. A'---~I
IlpoBe~eMo ,n;iaroHaJih AC. 3 fl ABC C
aa HepiBHiCTIO TPRKYTHRKa oTpRMY€Mo:
AB < AC + CB. AHarroriqHo a fl ADC: D
AC <AD + DC. P11c. 12
0Tme, AB < AC + CB < AD + DC + CB.
l;>
• 1. noflCHiTb, flKi BiApi3Klt1 Ha31t1Bal0Tb cyciAHiMlt1.
2. noflCHiTb, flKY cpirypy Ha3111Bal0Tb 40T111p111KYTH111KOM.
3. flKi CTopOHlt1 LfOT1t1p1t1KyrH1t1Ka Ha31t1Bal0Tb cyciAHiM1t1? npon'1ne:>KHll1Mll17
. 4. flKi BepW111Hlt1 LfOT1t1p1t1KyrH1t1Ka Ha31t1Bal0Tu cyciAHiM1t1? npoTlt111€:>KHlt1Mlt1?
5. flK Ha31t1Bal0Tb i no3Hal.fal0Tb YOT1t1p1t1KyrH1t1K?
6. Ll..lo Ha31t1Ba10Tb Aiar0Ha111110 Yor111p111KyTH1t1Ka?
7. flK1t1i11 YOT1t1p1t1KYTHlt1K Ha3111Bal0Tb onyKJ11t1M? HeonyKJ11t1M?
8. CcpopMy1110illre reopeMy npo cyMy KyriB YOTt11p111KyrH111Ka .
9. Ll..lo Ha3111Bal0Tb nep111MerpoM l../OT1t1p1t1KyrH1t1Ka?
0 nPAKTI1l!HI 3ABAAHHSI
I
~ ~ ~
- - - - - _L..;.:_-· ...
l.
0
HaKpecJiiTh orryKJIRH qoTRPRKYTHRK, y SIKoro:
1) TpR KyTH Tyrri;
· 2) ABa cyci,n;Hix KYTR - rrpSIMi, a ABa iHIIIRX He€ rrpSIMRMR;
9
13. ~ § 1. Yornp111KyrH111K111 _
3) O,l(Ha ,z:i;iarOHB.Jlh TO'lKOIO nepeTHHY ,z:i;iaroHarreii ,l(iJIHThCSI
HaBniJI, a ,z:i;pyra He ,n;iJIHThCSI HaBniJI;
4) ,z:i;iaroHarri nepneH,l(HKYJISipHi.
2.
0
HaKpecJiiTh .noBiJihHHH 'lOTHPHKYTHHK, noaHa'lTe iioro
BeprnHHH 6yKBaMH M, K, E, F. YKamiTh nap11 iioro npoTH-
Jie:>RHHX cTopiH, npoTHJiemHHX Beprµ11H, cyci,n;Hix cTopiH. Ha-
Be,n;iTh SIKi-He6y,z:i;h TPH Il03Ha'leHHSI Uhoro 'lOTHPHKYTHHKa.
3.° HaKpecJiiTh onyKJIHH 'lOTHPHKYTHHK, y SIKoro:
1) TPH KYTH rocTpi;
2) ,l(Ba npOTHJie:>RHi KYTH npSIMi, a pernTa He€ np.HMHMH;
3) ,z:i;iaroHarri TO'lKOIO nepeTHHY ,l(iJISITbCSI HaBniJI.
~nPABlr1 _ .
4.° Cepe,z:i; <}>iryp, ao6pameHHX Ha pncyHKY 13, yKamiTh 110-
THPHKYTHHRH.
oca) 6) B) z)
r) iJ) e) €)
PHc. 13
5: Cepe,z:i; 'lOTHPHKYTHHKiB, ao6pameHHX Ha pncyHKY 14,
yKamiTh: 1) onyKJii; 2) HeonyKJii.
K E
DM F
PHc. 14
6.
0
HaBe.n;iTh SIKi-He6y,n;h 4 noaHa11eHHSI 110THPHKYTHHKa,
ao6pameHoro Ha pncyHKY 15. YKamiTh:
10
14. 1) BepIIIHHH qQTHpHKYTHHKa; K
2) lioro CTOpOHH;
3) yci nap11 cyci.n;Hix sepIIIHH;
4) nap11 rrpoTHJieJKHHX BepllIHH; M
c
5) yci rrap11 cyci,ZJ;Hix CTopiH; A
6) nap11 npOTHJie.mHHX CTOpiH;
7) .n;iaroHani TIOTllpHKYTHHKa.
PHc. 15
7.., "LloMy .zi;opiBHIO€ qeTBepTHH KYT tfoT11p11KyTH11Ka, .HKr.qo
TPH lioro KYTH si.n;rroBi,lJ,HO .n;opiBHIOIOTh 78°, 89° i ·93°?
8.
0
3Hali.n;iTb KYTll qQTllPllKYTHHKa, .HKI.qO BOHll piBHi Mi.m
co6oro.
9.· y qQTHPllKYTHHKY ABCD LB= 150°, LA== L c= L D. .
3Hali.n;iTb HeBi,n;oMi KYTH qoT11p11KyTHHKa.
10.
0
0,lJ,HH 3 KyTiB qQTllPllKYTHHKa y 2 pa3H MeHlllllH Bi,lJ,
.n;pyroro KyTa, Ha 20° MeHllIHH si.n; TpeTboro i Ha 40° 6iJibllIHH
aa qeTBepT11li. 3Hali.n;iTb KYTH qoTHPHKYTHHKa.
ll.
0
3Hali.n;iTb KYTH qQTHPHKYTHHKa, .HKI.qO BOHll rrporrop-
uiliHi .n;o q11ce.JI 2, 3, 10 i 21. OrryKJIHH q11 HeorryKJIHH ueli
qQTllpllKYTHHK?
12.
0
3Hail,lJ,iTb KYTH qoT11p11KyTH11Ka, .HKI.qO TPH lioro KYTH
rrporropu:iliHi .no q11ceJI 4, 5 i 7, a qeTsepTHH KYT ,nopiBHIO€ i'.x
rriscyMi. OrryKJIHH q11 HeonyKJIHH ueli qoT11p11KyTHHK?
13.., qll Mo.me qQTllpHKYTHllK MaTH:
1) Tpll rrp.HMJIX KYTH, a qeTBepTllH - rocTp11li;
2) Tpll np.HMJIX KYTH, a qeTBepTllH - Tyn11if;
3) qoTHPll rrp.HMJIX KYTH;
4) qoT11p11 rocTp11x KYTH;
5) ,lJ,Ba rrp.HMJIX i .n;sa TYIIHX KYTH;
6) ,n;Ba rrp.HMHX, o.n;11H rocTp11li i O,lJ,HH Tyn11li KyT11?
y paai II03llTHBHOI Bi~IIOBi.n;i Hap11cyfiTe TaKHH qQTllpHKYT-
HUK.
14.
0
Ilep11MeTp qoT11p11KyTHJ1Ka .nopiBHIO€ 63 CM. 3Hali,niTb
., 2 ..
11oro CTOpOHll, .HKI.qO .zi;pyra CTOpoHa CTaHOBllTb 3 nepIIIOl,
TpeT.H - 50 o/o .zi;pyroi, a qeTBepTa - 150 % rrepllloi.
15.
0
3HaH,lJ,iTb CTOpOHH qQTllpllKYTHHKa, .HK~O 0,lJ,Ha 3 HllX
Ha 2 CM 6iJibWa aa JJ,pyry, Ha 6 CM MeHllla Bi)l TpeThOl, y 3 pa3H
MeHllla Bi,lJ, qeTBepTOl, a rrep11MeTp ~opiBHIO€ 64 CM.
11
15. § 1. Yon1p111KyTH111K111
16.
0
Y qOTHpHKYTHHKY ABCD cTopoHR AB i BC piBHi, a ,nia-
roHa.rrh BD YTBOproe a URMH CTopoHaMR piBHi KyTH. ,ll;oBe.niTh,
IQO CTopoHR CD i AD Te.m piBHi.
17.
0
.D;iaroHaJii qOTHpHKYTHHKa TOqKOIO nepeTHHY ,n;iJI.HThC.H
HaBniJI, o,n;Ha 3 iforo CTOpiH ,nopiBHIOC 6 CM. "'lloMy ,n;opiBHIOC
npoTH.Jie.mHa lH CTOpOHa qoTHPHKYTHHKa?
18.
0
y qQTHPHKYTHHKY MNKP MN=NK, MP=PK, L M=
=100°. 3Hali~iTh KYT K.
19.
0
Y qOTRPRKYTHRKY ABCD ,n;iaroHa.Jih AC yTBoproe ai
cTopoHaMR AB i AD piBHi KYTH Ta ai cTopoHaMR CB i CD Te.m
piBHi KyTR, AB= 8 CM, BC= 10 CM. 3Hali,n;iTh nepRMeTp qOTH-
pnKyTHRKa ABCD.
20.• Y TPRKYTHRKY ABC L A = 44°, L B = 56°. BiceKTPRCH
AK i BM TPRKYTHRKa nepeTRHaIDThC.H B Toqui 0. 3Haif,n;iTh
KYTH qoTRPRKYTHHKa: 1) MOKC; 2) AOBC.
21: y TPHKYTHHKY ABC L A= 36°, L B =72°. BHCOTH AE
i BF TpRKYTHRKa nepeTRHfilOThC.H B TOII~i H. 3Han,n;iTh KyTR
qoTRPRKYTHRKa: 1) CFHE; 2) ACBH.
22: 3Hali,n;iTh ,niaroHa.Jih qOTHPHKYTHHKa, .HKIQO iforo nepR-
MeTp ,n;opiBHIOC 80 CM, a nepRMeTpH TPRKYTHHKiB, Ha .HKi IJ;.H
,n;iaroHa.Jih po36RBae ,n;aHHH qQTHPHKYTHHK, ,n;opiBHIOIOTh 36 CM
i 64 CM.
2a.· "'llH MO.iKYTh CTOpOHH qQTHPHKYTHRKa ,n;opiBHIOBaTR:
1) 2 ,ZJ;M, 3 ,ZJ;M, 4 ,ZJ;M, 9 p;M; 2) 2 ,ZJ;M, 3 ,ZJ;M, 4 ,ZJ;M, 10 lJ.M?
24.
00
Y qoTRPRKYTHRKY ABCD LA= L C = 90°. .D;oBe,n;iTb,
~o 6iceKTPHCH ,ZJ;BOX iHIIIHX KyTiB qQTHPHKYTHHKa a6o napa-
.JieJihHi, a6o JiemaTh Ha o,n;Hiif np.HMiif.
25:· .D;oBe,D;iTb, ~o KO.JIM 6iceKTPHCH ,D;BOX npOTRJiemHHX
KyTiB onyKJIOro IIOTHPHKYTHHKa napa.rreJibHi a6o JiemaTb Ha
OlJ.HiH np.HMiH, TO )lBa iHIIIHX KYTH qOTHPHKYTHHKa piBHi.
26:· Ilo6y,n;yliTe qQTHPHKYTHHK aa iforo CTOpOHaMH Ta O,ll-
HHM 3 KyTiB. .
27.•• IIo6y,nyifTe qOTRPRKYTHRK aa TpboMa cropoHaMR i ,nBo-
Ma ,n;iaroHaJIHMR.
28.·· Ilo6y,n;yifre qOTHpHKYTHHK aa iforo CTOpOHaMH i O,D;Hiero
s ,niaroHa.rreiL
29.* IIo6y,n;yifTe qoTRPRKYTHHK ABCD aa KyTaMR A i B,
cTopoHaMR AB i BC Ta cyMoro cTopiH AD i CD.
12
16. 1. YoT111p111KyTHll1K Ta tl!oro eneMeHTll1
" "roTY€MOOI ,QO BMB4EHHSI HOBO!TEMM ------
30. Ilp.HMa c nepeTMHa€ KO.mHy 3 np.HMHX
a i b (pHc.16). YKa.miTh napM piaHocTopoHHix
i napH Of(HOCTOpOHHiX KyTiB, .f.IKi rrpM UhOMY
YTBOpHJIMC.f.I. 5IKe B3a€MHe p03Mil.I(eHH.H npsr-
MHX a i b, .HKII(o: 1) L 1 =L 4; 2) L 1===20°,
L 3 == 170°?
31. y lIOTHPHKyrHHKY ABCD (pHC. 17) L c =
=110°, L D= 70°. ,IJ;oaef(iTh, II(O BC Ii AD.
B ____
A------n A
P1w. 17 PHc. 18
P11:c. 16
a
b
32. y llOTHPHKYTHHKY ABCD (pHC. 17) LA== L B == 90°,
L C =100°. Y cTaHoBiTh B3a€MHe po3MirqeHHH npsrMHX BC i AD
Ta npsrMHX AB i CD.
33. Ha p11cyHKY 18 AD= BC, L ADB = L CBD. )1;0BetJ:iT1>,
II(O AB= CD i AB II CD.
34. BitJ:pi3oK BK - 6iceKTp11ca TPHKYTHHKa ABC. Ilpsr-
Ma DK napaJieJihHa CTopoHi AB i nepeTHHa€ cTopoHy BC
y TOliu;i D, L BDK= 116°. 3Ha:Hf(iTh L BKD.
IloHOBiTb y naM'srri 3Mit'1' llYHRTiB 12, 13, 14 Ha c.. 197-
198,.
i;:Ji.tcnoCTEPlrA_lilTE, P&.1CY"1TE,-
~ KOHCTPYIOVITE, <t>AHTA3YVITE
35. BiJiy IIJIOII(HHY tJ:OBiJI1>Ho aa6pH3KaHO lIOpHoro cpap6oro.
)J;oBef(iTh, II(O Ha IIJIOII(HHi 3Ha:Hf(eThC.H Bi~piaoK aaB,nOB.'.>KKH
1 M, KiHu;i .HKOro 3acpap6oBaHo Of(HMM KOJibOpOM.
13
17. 2.napanenorpaM.BnacT~eocTinapanenorpaMa
0 a Ha "<I e H H .H. IT a pa JI e JI or pa-
M 0 M Ha3HBalOTL 'IOTllpHKyrHHK, y JIKO-
I'O KO:llCHi ~Bi npOTH.ae:llCHi CTOpoHH na-
paJieJibHi.
Ha p11cyHKY 19 ao6pa.meHo napane-
.rrorpaM ABCD: AB II CD, BC II AD.
PHc. 19
Teo p e Ma 2.1. .V napaaeaozpaMa npomuae:HCni cmoponu
pi6ni.
!{ 0 6 e ae H H Jl. 0 Ha p11cyHKY 20 ao6pa.meHO rrapa.rreJiorpaM
ABCD. IlpoBe.n;eMo .n;iaroHa.rrb AC. .noBe.n;eMo, ~o TPHKYTHHKll
ABC i CDA piaHi.
Y :a;11x TpllKYTHHKax cTopoHa AC - cniJihHa, KYTH 1 i 2
piBHi HK piaHOCTopoHHi npH napa.rreJibHllX rrp.JIMHX BC i AD Ta
ci"<IHiii AC, KYTll 3 i 4 piBHi HK piaHOCTopoHRi npH napa.JieJib-
HllX np.HMllX AB i CD Ta ci"<IHiii AC. OT.me, ~ABC=~ CDA aa
.n;pyroro 03HaKoro piBHOCTi TpHKYTHHKiB. 3ai.n;c11 AB = CD
i BC =AD. A
Teo p e Ma 2.2. .V napaaeaozpaMa npomuae:HCni Kymu
piBni.
!{ o 6 e iJ e n n si. 0 Ilp11 .n;oBe.n;eHHi nonepe.n;HhOI TeopeMH 6y.rro
BCTaHOB.rreHo, ~o ~ABC=~ CDA (p11c. 20). 3ai.n;c11 L B =L D.
TaKo.m a piBHOCTeii L 1 = L 2 i L 3 = L 4 BHIIJIHBa€, ~o L 1 +
+ L 3 =L 2 + L 4. OT.me, L BAD= L BCD. A
Teo p e Ma 2.3. .V napaaeaozpaMa iJiazonaai mo"l£KOJO
nepemuny iJiaambca na6nia.
!{ o B e iJ e n H si. 0 Ha pncyHKY 21 ao6pa.meRo napa.rre.rrorpaM
ABCD, .n;iaroHB..Jii .HKoro nepeT11Ha10TbC.H B To"<I:a;i 0 . .noae.n;eMo,
~o TPHKYTHHKM AOD i COB piBHi.
B C
t:??JA DA
PHc. 20 PHc. 21
14
18. 2. napanenorpaM. Bnacr11mocri napanenorpaMa
)l;ilicHo, L 1 i L 2 Ta L 3 i L 4 piBHi HK piaHocTopoHHi npR
napane.JibHRX npsrMRX AD i BC Ta citIHHX AC i BD Bi;i;noBi,n;Ho.
3a TeopeMOIO 2.1 AD= BC. OT.me,~ AOD =~COB aa ;i;pyroro
03HaKOIO piBHOCTi TpRKYTHRKiB. 3Bi.n;cR AO= OC, BO= OD. .A.
0 a Ha lI e H H sr. B H co Toro n a pa JI eJio rp a Ma Ha3HBa-
IOTL nepneH,IJ;HKYJUip, onym;eHHH 3 6y,ZJ;L-JIKOi TO'IKH npHMOi,
HKa MiCTHTL CTOpOHY napanenorpaMa, Ha np.RMY' w;o MiCTHTL
npOTHJie:>KHY CTopoHy.
Ha pRcyHKY 22 KO.manii
3 Bi,D;pi3KiBAF, QE, BM, PN,
CK € BRCOTOIO napaneJiorpaMa
ABCD.
3 Kypcy reoMeTPil 7-ro KJia-
.cy BR 3Ha€Te, mo BCl TOlIKR
0,D;Hi€i: 3 ,ZJ;BOX napaJieJibHRX
npSIMRX piBHOBi,D;,D;aJieHi Bi,D;
;i;pyroi npsrMoL ToMy AF =QE
i BM=PN=CK.
p
N
PHc. 22
Ka.myTb, mo BHCOTH BM, .CK, PN Bi.zi;noBi.zi;aroTL CTOpOHaM
BC iAD, a BRCOTRAF, _QE - CTopoHaMAB i CD.
0-- 3 a Jl a 'I a. f{06eOimb, ll.{-0 np.RMi, JlKi Micm.Rmb 6UCOmU
mpuKymnuKa, nepemuna10mbc.R 6 ooniu mo1lqi.
p 0 3 6' Jl 3 a H HJl. 'l.J:epea KO.mHy BepIIIRHy ;i;aHoro TPRKyT-
HRKa ABC npoBe;i;eMo npsrMy, napane.JibHY npoTRJiemHili CTO-
poHi. 0TpRMa€MO TPRKYTHRK A1
B1
C1
(pRc. 23).
3 no6y~OBR BRilJIRBa€,
ru;o lIOTRPRKYTHRKR AC1BC B
1~---tt--------.A_____--.c
1
iABCB1
- napaJieJiorpaMR.
3Bi,ZJ;CR AC1
= BC = AB1
•
OT.me, TOlIKa A € cepe;i;RHoro
Bi.n;piaKa B1
C1
•
IlpoBe,D;eMo BRCOTY AH
TPRKYTHRKa .ABC. OcKi.Jib-
KH BlCl II BC, TOAH .1. BlCl.
3Bi.,n;cR npsrMa AH - cepe-
,D;~HHHH nepneH,D;RKYJIHP
15
PHc. 23
19. cTopoHH B1C1 TPHKYTHHKa A1
B1
C1
• AHa.rrori"llHO Mo.mHa ,n;oBec-
TH, II.O rrpHMi, HKi MiCTHTb ~Bi iHllli BHCOTH TpHKYTHHKa ABC,
€ cepe,n;HHHHMH rreprreH,n;HKYJIHpaMH cTopiH C1
A1
i A1
B1
TpH-
KYTHHKa A1
B1
C1
•
HK B~OMO, cepe,n;HHHi rreprreH,n;HKYJIHPH CTopiH TpHKYTHHKa
rrepeTHHaIOTbC.H B o,n;Hiii TO"ll:a;i.
IT pH K JI a11.. BiceKTpHca Tynoro KYTa napa.rreJiorpaMa ,n;i-
JIHTb rrpOTHJie.mHy CTOpOHY y Bi,n;HOllleHHi 2 : 1, paxyIO"llll Bi.n;
BepllIHHH rocTporo KyTa. 3Haii,n;iTb .cTopoHH napa.rreJiorpaMa,
HKII.O iioro rrepHMeTp ,n;opiBHIO€ 60 CM.
Po 3 e' .R 3 a H H .R. BiceKTpHca TY-
-------C rroro KyTa B rrapa.JieJiorpaMa ABCD
(pHc. 24) rrepeTHHa€ cTopoHy AD.
y TO"ll:a;i M, AM : MD::= 2 : 1.
KyTH ABM i CBM piBHi aa yMo-
A BOIO.
KyTH CBM i AMB piBHi HK pia-
PHc. 24 HOCTopoHHi rrpH BC IIAD Ta ci"ll-
Hiii BM.
To,n;i L ABM= LAMB. OT.me, tJ. BAM - piBHo6e,n;peHHii,
AB=AM.
Hexaii MD =x CM, To,n;i AB =AM= 2x CM, AD =3x CM. Ile-
PHMeTp napa.rreJiorpaMa ,n;opiBHIO€ 2 (AB +AD). YpaxoByIO"llH
YMOBY, MaeMo:
2 (2x + 3x) = 60;
X= 6.
OT.me, AB= 12 CM, AD= 18 cM.
B i ,n; II 0 B i ~ b: 12 CM, 18 CM.
~ ~ .;A ':_t , ) - - - - -
• 1. flK111i71 LfOTlllPlllKYTHlllK Ha3111Bal0Tb napanenorpaMoM?
2. flKy snacr111sicTb MalOTb npor11111e>KHi cropoH111 napanenorpaMa?
3. flKy snacr111sicTb MalOTb npor11111e>KHi Kyr111 napanenorpaMa?
4. flKy snacr111sicTb MalOTb AiaroHani napanenorpaMa?
5. Ll.lo Ha3111Ba10Tb s111coro10 napanenorpaMa?
6. 4oMy AopiBHIOE cyMa 6yAb-flKll1X ABOX cyciAHiX KyTiB napanenorpaMa?
BiAnoBiAb o6rpyHryl71re.
16
20. 2. napanenorpaM. BnacTl-1BOCTi napanenorpaMa
~l'IPAKTM~HE~B,QAHH . • •. ~~36.
0
Ha pncyHKy 25 ao6pameHo rrapaJieJiorpaM ABCD.
3po6iTb TaKHif pncyHoK y 30lllHTi. Ilpoae,n;iTb a TO'llOK Bi M
BHCOTH, .HKi Bi,n;noBi,n;aIOTb CTOpOHi AD napa.rreJiorpaMa, a 3 TO'll-
KH K - BHcorry, .RKa Bi,n;noai,na€ cTopoHi AB.
B
B M c BM
A c
A D
D
a) 6) 6)
PHc. 25
-
' ----- - ----- -- - --- -
37.
0
.D:Bi napa.rreJibHi np.HMi nepeTHHaIOTb TPH iHilli napa-
JieJibHi np.HMi. CKiJibKH npn IJ;bOMY yTBOPHJIOC.H napa.rreJiorpa-
MiB? .
:~.
0
Ha pncyHKY 26 ao6pameHo napa.rreJiorpaMH. 3Haifp;iTb,
He BHKOHYIOtIH BHMiproaaHb, Ha .HKHx pncyHKax BeJIH'llHHH
KyTiB a6o ,D;OB.>KHHH Bi~pia1dB Il03Ha"<IeHO HenpaBHJibHO (,D;OB-
.>KHHH Bip;piaKiB HaBep;eHo B caHTHMeTpax).
6
a)
6
z)
6) 6)
6 2
Puc. 26
17
21. § 1. YOT111p111KYTHl/1Kl/1
39.
0
"'llH BHCTatIHTb 40 CM ,n;poTy' :rn;o6 BHrOTOBHTH 3 HbOI'O
napaJieJiorpaM ai cTopoHaMH:
1) 14 CM i 8 CM;
2) 16 CM i 4 CM;
3) 12 CM i 6 CM?
40.
0
IIepHMeTp napaJieJiorpaMa ,n;opiBHIOe 112 cM. 3Hali,n;iTb
:H:oro CTOPOHH, .HK:rn;o: 1) o,n;Ha 3 HHX Ha 12 CM MeHrna Bi,n; ,n;py-
I'Ol; 2) ,n;Bi lioro CTOpOHH Bi,ZJ;HOC.HTbC.H .HK 5 : 9.
41.° 3Hali,n;iTb CTOpOHH napaJieJiorpaMa, .HK:rn;o O,n;Ha 3 HHX
y 5 paaiB 6iJibrna 3a ,npyry, a nepHMeTp napaJieJiorpaMa ,nopiB-
HIOe 96 CM.
42.
0
Y napaJieJiorpaMi ABCDAB= 6 CM, AC= 10 CM, BD =8 CM,
0 - To-ciKa nepeTHHY lioro ,n;iaroHaJieli. 3Hali,n;iTb nepHMeTp
TPHKYTHHKa COD.
43.
0
0,n;HH 3 KyTiB napaJieJiorpaMa ,n;opiBHIOE: 70°. 3Hali,n;iTb
pernTy :Moro KyTiB.
-44.
0
Y TPH~yTHHKy ABC LA= 35°. "'C.Jepea ,n;oniJibHY To-ciKy,
.HKa HaJieJkHTb cTopoHi BC, npone,n;eHo ,n;ni np.HMi, rrapaJieJibHi ·
cTopoHaM AB i AC TpHKyTHHKa. BHaHa-ciTe BH,n; -cioTHpHKyTHH-
..... • • u
Ka, :rn;o yTBOpHBC.H, Ta 3HaH,n;lTb yc1 HOI'O KyTH.
45.
0
3Ha~iTb KYTH rraparreJiorpaMa
ABCD (pHc. 27), .HK:rn;o L ABD = 68°,
LADE= 47°.
_ 46.
0
Y napaJieJiorpaMi ABCD ,nia-
roHaJib· AC yTBoproe ai cTopoHOIO AB
B C
A;:s;zKYT, .HKHH ,n;opiBHIOe 32°, L BCD= 56°. PHc. 27
3Ha.li,n;iTb L CAD i L D. .
0-. 47.
0
BiceKTPHCH KyTiB A i B napaJieJiorpaMa ABCD rre-
peTHHaIOTbC.H B To-ciui M. BH3Ha-ciTe BH,n; TpHKYTHHKa ABM.
48.
0
3Hali,n;iTb KYTH napaJieJiorpaMa, .HK:rn;o:
1) cyMa ,n;BOX :Horo KyTiB ,n;opiBHIOE: 100°;
2) pi3HHU.H ,ZJ;BOX lioro KyTiB ,n;opiBHIOE: 2.0°;
3) ,n;Ba lioro KyTH Bi,n;HOC.HTbC.H .HK 3 : 7.
49.
0
3Hali,niTb KyTH napaJieJiorpaMa, .HK:rn;o 0,ZJ;HH 3 HHX:
1) y 2 pa3H 6iJiblIIHH 3a ,n;pyrHH;
2) Ha 24° M€HIIIHH Bi,n; ,n;pyroro.
50.° CTopOHH rrapaJieJiorpaMa ,n;opiBHIOIOTb 6 CM i 10 CM. "'llH
Mome o,n;Ha 3 lioro ,n;iaroHaJieli ,n;opiBHIOBaTH 16 cM?
18
22. 2. napanenorpaM. BnaCTll1BOCTi napanenorpaMa
51: BHcoTa BK napaJieJiorpaMa ABCD ,niJIHTh lioro cTopo-
Hy AD Ha ai.n;pi3KH AK i KD TaKi, m;o AK= 4 CM, KD = 6 CM.
3Hali,niTh KYTH i nepHMeTp napaJ1eJiorpaMa, SIKIQO L ABK= 30°.
52: 0,n;HH 3 KyTiB napaJieJiorpaMa ,nopiBHIOe 45°. B11:c0Ta
napaJieJiorpa:Ma, npoae,neHa 3 aeprnHHH Tynoro KyTa, .n;opiBHIOe
3 CM i ,n;iJIHTh CTOpOHy napaJieJiorpaMa HaBniJI. 3Hali,n;iTh I.UO
CTOpOHy napaJieJiorpaMa Ta KyTH, .HKi yTBOpIOe ,n;iaroHaJib, IQO
CilOJiyqae BepIIIHHH TYilHX KyTiB' 3i CTOpOHaMH napaJieJio-
rpaMa.
53.• Y napaJieJiorpaMi ABCD L C =30°, BHCOTa BH, npoae-
,n;eHa ,no cTopon0 CD, ,n;opiBHroe 7 CM, a nepHMeTp napaJieJio-
rpaMa - 46 CM. 3Ha:il,n;iTh CTOpOHH rrapa.,"IeJiorpaMa.
54.· ,IlaHo napaJieJiorpaM ABCD i TPIIKYTHHK MKN. qH
MO.iKYTh o,nHoqacHo BHKOHYBaTHCSI piBHOCTi LA= L M, L B =L K,
L C=L N?
55.• ,IIoae,n;iTh, ~o aepIIIHHH B i D napaJieJiorpaMa ABCD
piBHOBi,n;,n;aJieHi ai,n np.HMo1 AC.
56.• ,D;oae,niTh, .mo 6y,n;h-HKHH ai,npi30K, SIKHH rrpoxo,nHTh
qepe3 TO'tfKy nepeTHHy ,niaroHaJieii napaJieJiorpaMa i KiH:ui
.HKOro HaJie.maTh npOTHJie:>KHHM CTOpOHaM napaJieJiorpaMa,
,n;iJIHTbCSI u;iero TOqKOIO HaBniJI.
57.• ITepHMeTp napaJieJiorpaMa ABCD ,n;opiBHIOe 24 CM,
L ABC = 160°, ,n;iaroHaJih AC yTBoproe 3i CTOpOHOIO AD KyT 10°.
3Haii,niTh CTOPOHH napaJieJiorpaMa.
58.• ,D;iaroHaJih BD napaJieJiorpaMa ABCD yTaoproe .3i cTo-
poHoro AB KyT 65°, LC= 50°, AB= 8 CM. 3Hali,n;iTh nepHMeTp
napaJieJiorpaMa.
59.• 3Hali,n;iTh KyTH rrapaJieJiorpaMa ABCD, SIKIQO BD 1- AB
i BD=AB.
60.· ,D;iaroHaJib napaJieJiorpaMa yTaoproe 3 lioro cTopoHaMH
KyTH 30° i 90°. 3Hali,n;iTh CTOpOHH napaJieJiorpaMa, SIKIQO lioro
rrepHMeTp .n;opiBHIOe 36 CM.
61: Ilo3a napaJieJiorpaMOM ABCD npoae,n;eHo npsiMy, napa-
JieJihHY lioro ,n;iaroHaJii BD, HKa nepeTHHae npsiMi AB, BC, CD
i AD y ToqKax E, M, F i K ai,nnoai,n;Ho. ,D;oae,n;iTh, IQO
MK=EF.
62.• IIapaJieJibHO ,n;iaroHaJii AC napaJieJiorpaMa ABCD npo-
Bep;eHo npsiMy, siKa nepeTHHae ai,n;pi3KH AB i BC Ta np.HMi AD
19
23. § 1. LJon1p111KyTHl/1Kl/1
i CD y TOqKax M, N, P i K Bi,zu10Bi,n:Ho. n:oBe,n:iTb, m;o
PM=-NK.
63: 0,n;MH 3 KyTiB, YTBOpeHMX IlpM nepeTMHi 6iceKTpMCM
Kyra napaJieJiorpaMa 3 iioro CTOpoHoro, ,n:opiBHIOe 24°. 3Haii,n;iTb
KyTM rrapaJieJiorpaMa.
64.• BiceKTpMca KyTa A napaJieJiorpaMa ABCD nepeTMHae
cTopoHy BC y Toqu;i M. 3Haii,n;iTb rrepMMeTp ,n;aHoro napaJie-
JiorpaMa, SIKID;O AB= 12 CM, MC= 16 cM.
65.• EiceKTpMca rocTporo KyTa rrapaJieJiorpaMa ,n;iJIMTb lioro
CTOpOHy y Bi,l(HOIIIeHHi 3 : 5, paxyIOl:.IM Bi,n; BepIIIMHM Tynoro
KyTa. 3Hali,n;iTb cTopOHM napaJieJiorpaMa, HKm;o iioro nepMMeTp
,n;opiBHIOe 66 CM.
66.• EiceKTpMca KyTa B napaJieJiorpaMa ABCD rrepeTMHae
cTopoHy CD y TO"t.ID;i K TaK, m;o Bi,n;piaoK CK y 5 paaiB 6iJibIIIMH
aa Bi,n;pi30K KD. 3Haii,n;iTb cTopoHM rrapaJieJiorpaMa, SIKID;O lioro
rrepMMeTp ,n;opiBHIOe 88 CM.
67; Y napaJieJiorpaMi ABCD AD= 12 cM, AB= 3 cM, 6iceK-
TpMcM KyTiB B i C rrepeTMHaIOTb cTopoHy AD y TO"CJKax E i F,
rrpM"CJOMy TO"t.IKa E JiemMTb Mim TO"CJKaMM A i F. 3Haii,n;iTb ,n;oB-
mMHY Bi,n;piaKa EF.
68.• KyT Mim BMCOTOIO BH rrapaJieJiorpaMa ABCD i 6iceK-
TpMcoro BM KyTa ABC ,n;opiBHIOe 24°. 3Haii,n;iTb KYTM napaJie-
JiorpaMa.
0-. 69.• ,D;oBe,n;iTb, m;o KYT Mim BMCOTaMH rrapaJieJiorpaMa,
rrpoBe,n;eHMMM 3 BepIIIMHM Tyrroro KyTa, ,n:opiBHIOE rocTpoMy
KyTy napaJieJiorpaMa.
0-. 70.· ,D;oBe.n;iTb, m;o KyT Mim BMCOTaMM napaJieJiorpaMa,
rrpOBe,n;eHMMM 3 BepIIIMHM rocTporo KyTa, ,n;opiBHIOe TyrroMy
KYTY rrapaJieJiorpaMa.
71.• KyT Mim BMCOTaMM napaJieJiorpaMa, rrpoBe,n;eHMMM 3 Bep-
IIIH~M Tynoro KyTa, ,n;opiBHIOe 30°. 3Haii,n;iTb rrepMMeTp rrapa-
JieJiorpaMa, SIKID;O iioro BHCOTM ,n;opiBHIOIOTb 4 CM i 6 CM.
72.· BMcOTM rrapaJieJiorpaMa, rrpoBe,n;eHi a BepIIIMHM rocTpo-
ro KyTa, yTBOplOIOTb KYT 150°' CTOpOHM napaJieJiorpaMa ,n;opiB-
HIOIOTb 10 CM i 18 CM. 3Haii,n;iTb BMCOTM napaJieJiorpaMa.
73•• 1Iepea ,n;oBiJibHY TOl:.IKy OCHOBM piBHo6e,n;peHoro TpnKyT-
HMKa npoBe,n;eHO rrps.1Mi, rrapaJieJibHi iioro 6i"CJHHM CTOPOHaM.
20
24. 2. napanenorpaM. BnacnrnoCTi napanenorpaMa
.D;oae,n;iTh, m;o nepHMeTp yTBOpeHoro "l!OTHpHKYTHHKa ,n;opiBHIO€
CyMi 6i"l!HHX CTOpiH ,n;aHOI'O TPHKyTHHKa.
74.• qepea KO.iKHy aeprnHHy TpHKyTHHKa ABC rrpoae,n;eHo
np.HMy, napaJieJihHY npoTHJie.mHili cTopoHi. CyMa nepMMeTpia
ycix YTBOpeHHX napaJieJiorpaMiB ,n;opiBHIO€ 100 CM. 3Hali,n;iTh
rrepHMeTp TpHKyTHHKa ABC.
75: 1Io6y,n;yii:Te rrapaJieJiorpaM:
1) 3a ,l(BOMa CTOpOHaMH i KyTOM Mi.iK HHMH;
2) 3a ,ll;BOMa ,n;iarOHaJI.HMH i CTOPOHOIO;
3) 3a CTOpOHOIO, ,n;iaroHaJIJIIO i KyTOM Mi.iK HHMH.
76: IIo6y,n;yliTe rrapaJieJiorpaM:
1) 3a ,ll;BOMa CTOPOHaMH i ,n;iaroHaJIJIIO;
2) 3a ,ll;BOMa ,n;iaroHaJI.HMH i KyTOM Mi.m HMMM.
77... ;r:t;aHO TpH TO"l!KH, .HKi He Jie.maTh Ha O,n;Hill np.HMlli.
IIo6y,n;yliTe napaJieJiorpaM, aeprnHHaMH .HKoro e: ,n;aHi TO"l!KH.
CKiJihKH poaa'.H3KiB Mae aa,n:aqa?
78:· ToqKa nepeTHHY 6iceKTPHC ,n;aox cyc~Hix KyTiB napa-
JieJiorpaMa HaJie.iKHTh HOI'O CTOpOHi. 3Hali,n;iTh Bi,n;HOIII€HH.H
. . .cyc1,n;a1x CTOplH napaJieJiorpaMa.
79." Ha cTopoHi BC napaJieJiorpaMa ABCD icHye: TaKa Toq-
Ka M, m;o BM =MD =CD. 3aaii.n;iTh KyTH napaJieJiorpaMa,
.HKlIJ;O AD = BD.
80:• Ilo6y,l(yHTe rrapaJieJIOrpaM:
1) 3a CTOpOHOIO, npoae,n;eHOIO ,n;o H€l BHCOTOIO i ,n;iarOHaJIJIIO;
2) 3a ,ZJ;BOMa ,n;iarOHRJI.HMH i BHCOTOIO; .
3) 3a rocTpHM KyTOM i ,n;BOMa BHCOTaMH, .HKi Bi.zi;noBi,n;aIOTh
,ll;BOM cyCl,ZJ;HlM CTOpoHaM.
s1:· IIo6y,n;yliTe napaJieJiorpaM:
1) aa ,ZJ;BOMa CTOpOHaMH i BllCOTOIO;
2) 3a ,zi;iarOHaJIJIIO i ,ZJ;BOMa BliCOTaMH, .HKi Bi,n;rroai,n;aIOTh ,ZJ;BOM
cyCI,l(HlM CTOpOHaM.
82.* 3 aeprnHHM B rrapaJieJiorpaMaABCD orrycTHJIM rrepneH-
,n;HKYJI.Hp BE Ha ,n;iaroHaJih AC. 1Iepe3 TO"l!KY A rrpoae,n;eHo
np.HMy m, nepneH,n;HKyJIHpHy ,n;o np.HMoI AD, a qepea TO"l!Ky C -
rrp.HMy n, rreprreH,n;HKyJI.HpHy ,n;o npHMoi: CD. ,D;oae,n;iTh, i.u;o
TO"l!Ka rrepeTMHy rrp.HMHX m i n HaJie.iKMTh rrp.HMiii BE.
83.* IIo6y,n;yliTe rrapaJieJiorpaM aa cTopoHoIO, cyM~ro ,n;iaro-
HaJieii i KyTOM Mi.m ,n;iaroHRJI.HMH.
21
25. • i § 1. '-lornp11KyrHl1Kl1
84.* Ha cTopoHax AB i BC napaJienorpaMa ABCD no3a HMM
no6y,n;oaaHo piBHOCTOpOHHi TpHKYTHHKH ABM i BCK. )lo-
ae,n;iTb, :rn;o TpHKYTHHK MKD - piaHocTopoHuili.
85.* qepe3 TO"CJKy, SIKa HaJieJKHTb KyTy, npoae,n;iTb np.HMY
TaK, ~06 Bi,n;pi30K, .HKHM cnonyqae TO"CJKH nepeTHHY u;ie! npsi-
MOI 3i CTOpOHaMH KyTa·, ,n;aHOIO TOqKOIO ,n;iJUIBCSI 6 HaBniJI•
..t)ffBnPABlll ,[VlSl nOBTOPEHHSl "
86. ,[J;oamHHa ai,n;pisKa AB ,llOpiBHlOE: 24 CM. To"CJKa C HaJie-
mHTb npsiMili AB, npH"CJOMY BC= 5AC. Ha ai,n;pis:Ky AB nosHa-
"t.IeHo TO"t.IKY D TaI<, mo ai,n;pisoK AB= 4BD. 3Hali,n;iTb ,n;oamHHY
ai,n;pisKa CD.
87. CKiJibKH icHye HepiBHHX Mim co6oro:
1) np.HMOKyrHHX TpHKYTHHKiB ai CTOpoH-OIO 5 CM i KYTOM 45°;
2) piBH06e,n;peHHX TpMKYTHHKiB 3i CTOpOHOIO 6 CM i KYTOM 30°;
3) npSIMOKYTHHX TpHKYTHHKiB si CTOpOHOIO 7 CM i KYTOM 60°?
88. ,[(iaronani AC i BD "CJOTHPHKYTHHKa ABCD e ,n;iaMeTpaMH
1<ona s u;eHTpoM 0. ,[J;oBe}liTh, :rn;o AB II CD.
~~ cnoCTEPlrAlilTE. PlllCYlilTE.
~ KOHCTPYIOfATE, <bAHTA3Y£'iTE
89. qH MOmHa KBa,n.paT p03MipoM 10 x 10 KJiiTHHOK po3pi-
3aTH Ha 25 cpiryp, mo CKJia,n;aIOTbCSI 3 "CJOTHpbOX KJiiTHHOK i Ma-
IOTb TaKHli BHrJIH,ll ~ ?
3. 03HaKM napa11e11orpaMa
03Ha"CJeHHSI rrapaJieJiorpaMa ,ZJ;03BOJISIE: cepe,n; "CJOTHpHKYTHHKiB
p03Ili3HaBaTH napaJieJiorpaMH. IJ;ili caMiH MeTi CJiyryIOTh TaKi
TpH TeopeMH-03HaKH.
T e o p e M a 3.1 (o 6 e p H e H a ,n; o T e o p e M H 2.1). HKUfO
B "tomupuKymnuKy Komni iJBi npomune:HCni cmoponu piBni,
mo q.eii "tomupuKymnuK - napanenozpaM.
22
26. /{oBeiJeHHR. 0 Ha pHcyH-
Ky 28 ao6pameHO qQTHpHKYTHHK
ABCD, y.HKOMyAB=CDiBC=AD.
,lJ;oBe,n;eMo, mo AB II CD i BC II AD.
IlpoBe,n;eMo ,n;iaroHa.Tib AC. TpH-
KyTHHKH ABC i CDA piBHi aa Tpe-
TbOIO 03HaKOIO piBHOCTi TpHKYTHH-
3. 03HaK111 napanenorpaMa
PHc. 28
KiB. 3Bi~cH L 1 = L 3 i L 2 =L 4. KyTH 1 i 3 e pi3HOCTopoHHiMH
rrpH np.HMHX BC i AD Ta ciqHili AC. 0Tme, BC II AD. AHa.Tio-
riqHo a piBHOCTi L 2 = L 4 BHITJIHBae, m;o AB II CD.
0Tme, y qoTHpHKyTHHKy ABCD KOmHi ,n;Bi rrpoTHJiemHi
CTOpOHH rrapaJieJII>Hi, a TOMY .a;eii qQTHPHKyTHHK - rrapa.Tie-
JIOrpaM• .A.
Teo p e Ma 3.2. HKUlO 6 11,omupuKymnuKy iJBi npomuae:HC-
ni cmoponu piBni i napaneabni, mo ifeu 11,omupuKymnuK -
napanenozpaM.
JI, o 6 e iJ e H Ha. 0 Ha pHcyH-
Ky 29 ao6pameHO qQTHpHKYTHHK
ABCD, y .HKOMy BC=AD iBC II AD.
AoBe,n;eMo crroqaTKY, m;o AB= CD. A_.___._......,____
IlpoBe,n;eMo ,n;iaroHaJib AC. Y TpH-
KyTHHKax ABC i CDA MaeMo: BC=
PHC. 29
=AD, KyTH 1 i 2 piBHi .HK· pi3HOCTOpOHHi npH rrapaJieJibHHX
rrp.HMHX BC i AD Ta ciqHiii AC, a cTopoHa AC - crriJihHa.
0Tme, /),.ABC=/),. CDA aa rrepmoro 03HaKoro piBHOCTi TPHKyT-
HHKiB. 3Bi,n;cH AB= CD. 0Tme, y qOTHpHKYTHHKy ABCD Kom-
Hi ,n;Bi rrpoTHJiemHi cTopoHH piBHi. ToMy aa TeopeMoro 3.1 qo-
THpHKYTHHK ABCD - rrapaJieJiorpaM• .A.
T e o p e M a 3.3 (o 6 e p H e H a ,n; o T e o p e M H 2.3). H KUlO
6 11,omupuKymnuKy iJiazonani mo11,Koro nepemuny iJinambCR
nasnin, mo lfeU 11,omupuKymnuK -
napanenozpaM.
/{ o 6 e iJ e It Ha. 0 Ha pHcyH-
KY 30 ao6pameHO qQTHPHKYTHHK
ABCD, y HKOMY ,n;iaroHaJii AC
i BD rrepeTHHaIOTbC.H B Toq.a;i 0,
23
PHC. 30
27. . z§ 1. liOT111p111KYTHV1Klt1
B _____.,.C
A D
Pnc. 31
rrpH1lOMy AO= OC i BO= OD. ,JJ;oae,n;eMo, ~o BC= AD
i BC II AD.
Oc1d.JihRH RYTH BOC i DOA piBHi HK aepTHKaJihHi, TO ~ BOC=
=~ DOA aa rrepworo 03HaKoro piBHOCTi TPHKyTHHKiB. 3ai,n;cH
BC =AD i L 1 = L 2. KyTH 1 i 2 e pi3HocTopoHHiMH rrpH npsi-
MHX BC i AD Ta cillHiii AC. OT.me, BC II AD.
TaKHM llHHOM, y llOTHPHKYTHHKY ABCD ,n;Bi rrpoTHJie.mHi
CTOPOHH piBHi ii rrapaJieJihHi. 3a TeopeMOIO 3.2 l:J:OTHpHKYTHHK
ABCD - rrapaneJiorpaM• .&
BH 3HaeTe, ~o TpHKYTHHK o,n;H03Ha1lHO aa,n;aeThC.H iioro CTO-
poHaMH, T06TO aa,n;a-qa rro6y,n;OBH TpHKYTHHKa 3a TpbOMa CTO-
poHaMH Mae e,n;HHHM poaa'si3oK. lHma cnpaaa 3 rrapaJieJiorpa-
MOM. Ha pHcyHKY 31 ao6pa.meHo rrapaJieJiorpaMH ABCD,
AIB1C1Dl, A2B2C2D2, CTOpOHH HKHX piBHi, T06TO AB =A1B1 =
=A2
B2
i BC= B1
C1
= B2
C2
• IlpoTe o-qeaH,n;Ho, ~o caMi napane-
JiorpaMH He e piBHHMH. ToMy rrapaneJiorpaM, Ha Bi,n;MiHy Bi,n;
TpHKYTHHKa, Hee .mopcTKOlO <}>iryporo:
ll;.H BJiaCTHBiCTb napa.JieJiorpaMa IIIHPOKO BHKOpHCTOByeThC.H
Ha npaKTHrti. 38.B,ZJ;HKH iioro pyxoMocTi JiaMny MOmHa BCTfillOBJIIO-
BaTH B 3p~He JJ;JIH po6oTH IIOJIOmeHHH, a p03CYBHY pemiTKY -
Bi.n;CYBaTH Ha IIOTpi6Hy B.i,n;CTaHb y ,n;BipHOMY npopiai (pHc. 32).
a) 6)
Pnc. 32
24
HepyxoMa '
BepnrHHa
Bicb o6epTaHH.H
Pnc. 33
28. 3. 03HaKLI! napanenorpaMa
Ha p11cyHKY 33 ao6pameHo cxeMy MexaHiaMy, .flKHli Haa11-
BaIOTh napanenorpaMOM YaT'M Ha YiecTh BHHaxi,n;HHKa rrepmoi
YHiBepcaJibHOI rrapOBOl MaIIIHHH. IlpH a6iJihIIIeHHi IIIBH,ZJ;KOCTi
o6epTaHHH oci KyJii Bi,n;,n;aJI.HIOTbC.fl Bi,n; Hel rri.n; ,n;iero Bi,n;:u;eHT-
poBOl CHJIH, THM caMHM ni,n;HiMaIOYIH aac.niHKy, .flKa pery.nroe
KiJihKiCTb rrap11.
0-. 3 a ,n; a q a• .HKU{O B ttomupu1eymnu1ey KOJICHi OBa npomu-
JleJ1Cni 1eymu pieni, mo qeu ttomupu1eymnu1e - napa.neJLozpaM.
P o 3 e~R 3 a n n R. Ha p11cyHKY 34
ao6pameHO qoTHPHKYTHHK ABCD,
y .HKOMY LA=L C, L B=L D.
3a TeopeMOIO rrpo cyMy KyTiB "tIOTH-
pMKYTHHKa LA + L B + L C + L D =
::: 360°. To.n;i LA + L B = L C + L D =
=180°.
PHC. 34
OcKiJibKH LA i L B - o,nHOCTopoHHi KYTH np11 npSIMHX AD
i BC Ta ciqHili AB, a 1x cyMa .n;opiBHIOe 180°, TO BC II AD.
AHanoriYIHO Jl,OBO.l(HThC.H, m;o AB II CD.
0Tme, qoTHpHKYTHHK ABCD - rrapa.ne.norpaM.
- ~£~ --~f'!t.. ·-4 ;. :' .....-......... ;., ...... -~~ - . '.
• 1. flKi 0111 3HaEre o3HaK111 napa11e11orpaMa? C¢opMy11toilire i·x.
2. CepeA BJlaCT111BOCTeil1 i 03HaK napa11e11orpaMa BKa>KiTb B3aEMHO
o6epHeHi reopeM111.
3. flKa s11acr111sicTb napa11e11orpaMa W1!1pOKO Bll1KOp111CTOBYETbCfl Ha
npaKr111~i?
- - - - - - - - ---- .. - -
90.
0
~oBe,n;iTh, II(O KOJIH cyMa KyTiB, rrpHJI€I'JIHX .n;o 6y,n;h-
.flK0l i3 cyci,n;Hix CTOpiH qQTHPHKYTHHKa, .n;opiBHIOe 180°' TO
:u;eli llOTHpllKyTHHK - rrapa.neJIO-
rpaM.
91.
0
qOTHPHKYTHHKH ABCD i
AMKD - rrapaneJiorpaM1:1 (p11c. 35).
~OBe,n;iTh, m;o YIOTHPHKYTHHK
BMKC - rrapaJieJiorpaM.
25 .
B C
A<] DZM K
PHC. 35
29. . 92.· Bi,n;pi30K AO - Me,n;iaHa TpH-
KYTHHKa ABD (pHc. 36), ai,n;piaoK BO -
Me,n;iaHa TpHRYTHHKa ABC. .D;oae,n;iTh,
mo "CJOTHpHKYTHHK ABCD - napaJie-
JiorpaM.
93.
0
Ha ,n;iaroHaJii AC rrapaJieJiorpa-
t?<lcA D
P11c. 36
Ma ABCD rroaHa"CJHJIH TO"CJKH M i K TaK, II(O AM =CK. )l;o-
ae,n;iTh, m;o "CJOTHPHKYTHHK MBKD - rrapaJieJiorpaM.
94.
0
,Uaa KOJia MaIOTh crriJihHHM neHTp 0 (pHc. 37). B o,n;Ho-
MY 3 KiJI npoae,n;eHo ,n;iaMeTp AB, y ,n;pyroMy - ,n;iaMeTp CD.
;D;oae,n;iTh, m;o qoTHPHKY'fHHK ACBD - rrapaJieJiorpaM.
95.
0
To"CJKH E i F - cepe,n;HHH cTopiH BC i AD rrapaJieJio-
rpaMa ABCD ai,n;rroai,n;Ho. ,ZJ;oae,n;iTh, m;o "CJOTHpHKyTHHK
AECF - rrapaJieJiorpaM.
96.
0
Ha cTopoHax AB i CD rrapaJieJiorpaMa ABCD Bi,n;KJia-
,n;eHo piBHi ai~pi3KH ·AM i CK. ,IJ;oae,n;iTh, m;o "CJOTHpHKyTHHK
MBKD - rrapaJieJiorpaM.
97:Ha cTopoHax rraparreJiorpaMa ABCD (pHc. 38) Bi.n;KJiaJIH
piBHi ai,n;pi3KH AM, BK, CE i DF. ,Uoae,n;iTh, m;o "CJOTHPHKYTHHK
MKEF - rrapaJieJiorpaM.
0-- 98.• Y TpHKyTHHKY ABC Ha rrpo,n;oBmeHHi M~.n;ifilIH AM
3a TO"CJKY M ai,n;KJiaJIH ai.n;piaoK MK, .flKHM .n;opiBHIOe ai,n;pia..:
KY AM. BH3Ha"CJTe BH,n; 110THpHKyTHHKa ABKC:
99: Y qOTHpHKYTHHKy ABCD AB 11 CD, L A= LC. BH3,Ha"CJ-
Te BH,n; "CJOTHPHKYTHHKa ABCD.
100.· BiceKTpHca KYTa A rrapaJieJiorpaMa ABCD rrepeTllHae
cTopoHy BC y TO"CJUi M, a 6iceKTpHca Kyra C - cTopoHy AD
y Toqu;i K. ,[(oae.n;iTh, m;o "CJOTHpIIKyTHHK AMCK - rrapaJieJio-
rpaM.
PHC. 37 PHC. 38
26
30. 3. 03HaK111 napanenorpaMa -. ..
A
P11c. 39 PHc. 40
101.· Ha p11cyHKY 39 qoTHpHKYTHlfK ABCD - rrapaJieJio-
rpaM, L BCP =L DAE. ~oBe,n;iTb, m;o qoTHpHKyTHHK APCE -
rrapaJieJiorpaM.
102: Ha pHCYHKY 40 qoTHPHKYTHHK ABCD - napaJieJio-
rpaM, L BEC =L DFA. ~oBe,n;iTb, ~o qoTHpHKyTHHK AECF -
napaJieJiorpaM.
103.· 3 BeprnHH B i D rrapaJieJiorpaMa ABCD rrpoBeJIH rrep-
rreH.n;HKyJIHPH BM i DK .n;o .n;iaroHaJii AC. ,D;oBe.n;iTb, ~o 'tlOTH-
p11KyTHHK BKDM - napaJieJiorpaM.
104.• BiceKTpHCH KyTiB A i C rrapaJieJiorpaMa ABCD rrepe-
THHaIOTb lioro .n;iaroHaJib BD y TO't!Kax E i F Bi.n;rroBi,n;Ho. Ao-
Be,niTb, ru;o 'lIOTHpHKyTHHK AECF - napaJieJiorpaM.
105.·· 11epe3 cepe,n;HHY 0 .n;iaroHaJii NP rrapaJieJiorpaMa
MNKP rrpoBe.n;eHo npHMy, aKa rrepeTHHae cTopoHH MN i KP
y TOqKax A i B Bi.n;rroBi.n;Ho. ~oBe,niT:&, m;o qoTHpHKYTHHK
ANBP - napaJieJiorpaM.
106.•• 11epe3 ToqKy rrepeTHHy .n;iaroHaJieli rrapaJieJiorpaMa
CDEF rrpoBe.n;eHO lJ.,Bi rrp.HMi, OlJ.,Ha 3 .HKHX rrepeTHHae CTOpOHH
CD i EF y ToqKax Ai B Bi.n;noBi.n;Ho, a ,n;pyra - CTOpOHH DE
i CF y TO'l!Kax M i K Bi.n;rroBi,n;Ho. ~oBe.n;iTb, ~o qoTHpHKyTHHK
AMBK - rrapaJieJiorpaM.
101:· ToqKH M, N, Ki P - cepe.n;HHH cTopiH AB, BC, CD
i AD rrapaJieJiorpaMa ABCD Bi.n;rroBi.n;Ho. ~oBe.n;iTb, ~o qoTH-
pMKyTHHK, BeprnHHaMH .HKoro e TOqKH rrepeTHHy rrp.HMHX AN,
BK, CP i DM, - rrapaJieJiorpaM •
.IJfBnPABM .AllSl nOBTOPEHHSl
108. IlpaMi, Ha HKHx JiemaTb 6iceKTpHcH AK i BM TPHKYT-
HMKa ABC, rrepeTHHaIOTbCH rri.n; KyToM 74°. 3Hali.n;iTb L C.
27
31. § 1. LIOTV1p111KYTHV1KV1
109. KyT, npoTuJiemHHH ocHoBi piBHo6e~peHoro TpHKYTHH-
Ka, ,n;opiBHI0€ 120°' a BllCOTa, npoBe,n;eHa ,n;o 6i-ciHOI CTOpOHll, -
8 CM. 3Han,n;iTb OCHOBY TpllKYTHllKa.
t;.fft cnOCTEPlrAf'TE. PlllCYHTE.~
~ KOHCTPYI0"1TE, <l>AHTA3Y"1TE
110. Y-ciuTeJib aanponoHyBaB yl'.!HeBi BupiaaTu a KapTOHHoro
JlllCTa p03MipoM 8 x 8 KJiiTllH;OK BiciM KBa;::t;paTiB p03MipoM 2 x 2
KJiiTHHKll aa yMOBll He ncyBaTH KJiiTHHKH, ~o 3a.JIHIIIH..1IHC.H.
IloTiM Bll.HBllJIOC.H, ~o IlOTpi6eH ~e O,ZJ;llH TaKllH KBa,n;paT. qll
3aBm,ZJ;ll MOmHa n;e 3p06llTH ia 3aJIHIIIKiB JIHCTa?
KOnM JPO&nEHO YPOKM
. Jieo6xi;xno i JlOCT8THbO
3 Kypcy reoMeTpii: 7-ro KJiacy BH ,n;i3Ha.JIHC.H, ~o 6iJibIIIiCTb
TeOpeM CKJia;::t;aeTbC.H 3 ,Zl;BOX l'.faCTHH: yMOBH (Te, ~o ,n;aHo) i BHC-
HOBKY (Te, ~o Tpe6a ,Zl;OB€CTH)•
.HK~o TBep,nmeHHSI, .HKe BHpamae yMoBy, noaHal'.fHTH 6yKBOIO
A, a TBep,n;meHH.H, .HKe BHpamae BHCHOBOK, - 6yKBOIO B, TO
cpopMyJIIOBaHHSI TeopeMH MOmHa ao6pa3HTH TaKOIO cxeMoro:
.HK~O A' TO B.
HanpHKJia,n;, TeopeMy 2.3 MOmHa ccpopMyJiroBaTn TaK:
A
"CJOTHpHKyTHHK
e napaJieJiorpaMOM
B
' TO ,n;iarOHaJii l'.fOTHpHKYTHHKa
TO"CJKOIO nepeTHHY
. .,ZJ;lJl.HTbC.H HaBIIlJI
To,n;i TeopeMy 3.3 MOmHa ccpopMyJiroBaTH TaK:
B A
.HK~O ,n;iarOHaJii l'.fOTHpHKYTHHKa ' TO --"CJOTHpHKYTHHK
TOl'.fKOIO nepeTHHY e napaJieJiorpaMOM
.n;iJI.HTbC.H HaBniJI
qacTO y IlOBC.HK,ZJ;€HHOMY lliHTTi y CBOIX BHCJIOBJIIOBaHHSIX
MH KOpHcTyeMoc.H cJioBaMH «Heo6xi,n;Ho», «.n;ocTaTHbO». HaBe-
,ll,eMo KiJibKa rrpHKJia,n;iB.
28
32. Kon1t1 3po6neHo ypoK1t1
• .IJ.;u1 Toro mo6 yMiTM poaB'.HayBaTM aa,n;aYii,
Heo6xiihto aHaTM TeopeMM.
• .HKl.QO BM Ha MaTeMaTMYJHill OJiiMnia,n;i p03B'.H-
3a.JIM npaBMJibHO BCi aanponOHOBaHi 3a)laYJi, TO
UbOrO iJocmamllbO ,Zl;JI.H TOI'O, ~06 3aBOIOBaTM
neprne M1cu.e.
• .Il.JI.H Toro ~o6 cTpiJieUh BJiyt.IHB y MirneHh B
B
0
PHc. 41
(pMc. 41), noMy iJocmamHbO BJIYt.IMTM B MirneHh A. ,D:JI.H Toro
~06 BJIYlIMTM B MirneHb A, 1teo6xiiJHo BJIYlIMTM B MirneHb B.
B:i:KMBaHH.H cJiiB «Heo6xi,n;Ho» i «AOCTaTHhO» TicHo noB'.HaaHe
a TeopeMaMM.
PoarJI.HHeMo oYieBMAHY TeopeMy:
A B
.HK~o .___H_a__T_y_p.._a_JI_h_H_e_ YJ._M_c_JI_o__ , TO In;e 'IHCJIO KpaTHe ~- KpaTHe 10 . - _J
3poayMiJio, ~o YMOBaA «'CJMCJIO KpaTHe 10» e iJocmamHbOKJ
,n;JI.H BMCHOBKY B. PaaoM a TMM no,n;iJihH:icTb YIMCJia HaIJ,iJio Ha
5 (TBep,n;:t.KeHH.H B) Heo6xioHa ,n;JIH no,n;iJihHOcTi YIMCJia HaIJ,iJio
Ha 10 (TBep,n;:t.KeHH.H A).
HaBe,n;eMo ~e o,n;HH npMKJia,n;.
Y TeopeMi
A
,n;Ba KYTH e
B€pTHKa.JlbHHMM
, TO
B
. . .lll KYTH plBHl
TBep,n;:t.KeHH.H A e ,nocTaTHhOIO yMoBoro ,ZJ;JI.H TBep,n;:t.KeHH.H B,
T06To ,n;JI.H ·Toro, ~06 ,n;Ba KY™ 6yJIM piBHMMM, iJocmamHbo,
~06 BOHM 6yJIM B€pTMKa.JlbHMMM. Y Uill CaMill T€OpeMi TBep,n;-
.meHH.H B e Heo6xi.,I(uoro yMOBoro ,D;JI.H TBep,n;:i:KeHH.H A, T06To ,n;JI.H
TOI'O, ~06 ,ZJ;Ba KYTH 6yJIM BepTHKaJibHMMM, neo6xiiJHO, ~06
BOHM 6yJIM piBHMMM. 3aaHaYJMMO, ~o TBep,n;:t.KeHH.H B He e ,n;o-
cTaTHhOIO YMOBOIO ,ZJ;JI.H TBep,n;:t.KeHH.H A. CnpaB,n;i, .HK~O ,n;Ba
KYTM piBHi, TO IJ,€ 30BCiM He 03HaYJaC, LUO BOHM B€pTMKa.JlbHi.
.HK~O cnpaBe,ZJ;JIMBa' He TiJibKM TeopeMa
.HK~O A, TO B, .
aJie .ii o6epHeHa TeopeMa
.HK~O B, TO A,
29
33. TO A 'e ueo6xi,n;uoro i ~ocTaTHLOIO yMOBOIO ,z:pu1 B, a B - Heo6-
xi,n;Horo i ,z:i;ocTaTHhOIO yMOBOIO .D;JUI A.
HanpHKJia.n;, TeopeMH 3.3 i 2.3 e BaaeMHO o6epHeHHMH.
MoBOIO «Heo6xi,n;Ho - ,n;ocTaTHhO» n;eli Q:>aKT MO.mHa cQ:>opMy-
JIIOBaTH TaK:
iJasi mozo Uf06 '1-0mupuKymnuK 6ye napa,n,e,n,ozpaMoM,
neo6xiiJno i iJocmamnbo, Uf06 uozo iJiazona,n,i mo'l-KOJO
nepemuny iJiauaucsi naenia.
IIi,n;KpecJIHMo, ~o KOJIH B TeopeMi e cJioBa «Heo6xi,z:i;HO» i «.D;O-
CTaTHhO», TO BOHa o6'e,n;Hye ,D;Bi TeopeMH: IIp.HMY i o6epHeHy
(rrp.HMOIO TeopeMOIO MO.me 6YTH 6y.n;h-.HKa 3 .D;BOX TeopeM, To.n;i
,n;pyra 6y,n;e 06epHeH010). OT.me, ,n;oBe,n;eHH.H TaKoi'. TeopeMH Mae
CKJia,n;aTHC.H 3 ,D;BOX YJaCTHH: ,D;OBe,n;eHb IIp.HMOl Ta o6epHeHOl
reop~M. Ta1d TeopeMH B MaTeMaTHn;i Ha3HBaIOTh KpHTepi.HMH.
IHo,n;i 38.MiCTh «Heo6xi,z:i;Ho i ,n;ocTaTHhO» roBop.HTh «To,n;i ii TiJih-
KH TO,lll».
HarrpHKJia,n;, BaaeMHO o6epHeHi TeopeMH 2.1 i 3.1 o6'e,n;Hye
Q:>opMyJIIOBaHH.H:
'£omupuKymnuK e napa,n,e,n,ozpaMOM. moiJi u miabKU moiJi,
Koau KO:HCni iJei uozo npomuae:HCni cmoponu pieni.
Crrpo6ylire caMocTiliHo TeopeMy 2.2 i KJIIOYIOBY aa,n;aYiy a rryHK-
TY 3 cQ:>OpMyJIIOBaTH y BHrJI.JI,D;i 0,D;Hie1 TeopeMH.
4. npSIMOKyrHMK
IIapaJieJiorpaM - ~e YIOTHpHKYTHHK, rrpoTe OYieBH,n;Ho, ~o
He KO»CHHH YIOTHpHKYTHlIK e rrapaJieJiorpaMOM. "'Y ~bOMY BHIIa,n;-
KY I'OBOp.HTb, ~o rrapa.JieJiorpaM - n;e OKpeMHH BH,ll YJOTHpHKYT-
HHKa. CxeMa, ao6pa.meHa Ha pHcyHKY 42, iJirocTpye n;eli Q:>aKT.
€ Ta:Kom OKpeMi BH.D;H rrapaJieJiorpaMiB.
PHc. 42
0 3 H a YI e H H .H. II p .HM 0 Ky T H H-
K 0 M H83HB8IOTL napaJieJiorpaM,
y .HKOro BCi KYTH np.HMi.
Ha pHcyHKY 43 ao6pameHO rrp.HMO-
KYTHHK ABCD.
30
34. B_....----~~C
A......,____...-_.D
PHc. 43 PHc. 44 PHc. 45·
3 HaBeAeHoro 03HaqeHH.H BMilJUIBa€, mo np.HMOKYTHMK Mae
BCi BJiaCTlIBOCTi napaJieJiorpaMa: y np.HMOKYTHMKY npOTMJie.mHi
CTOpOHM piBHi, ,n;iaroHaJii TOYIKOIO nepeTMHY ,n;iJUIThC.H HaBiliJI.
IIpqTe np.HMOKYTHMK Mae CBOIO oco6JIMBY BJiacTMBiCTh.
Teo p e Ma 4.1. ,l(iazonaai npstMoKymnuKa pieni.
/I, oB e iJ e n H .R. 0 Ha pMcyHKY 44 ao6pameHo np.HMOKYTHMK
ABCD. .IJ;oBe,n;eMo, ~o lioro ,n;iaroHa.rri AC i BD piBHi.
Y np.HMOKYTHMX TpMKYTHMKax ABD i DCA KaTeTM AB i DC
piBHi, a KaTeT AD - cniJihHMM. ToMy tJ. ABD = tJ. DCA aa ,n;Bo-
Ma KaTeTaMM. 3Bi,n;cM BD =AC. .&
OaHaYieHH.H np.HMOKyrHMKa ,n;ae 3Mory cepe.n; napa.rreJiorpaMiB
poaniaHaBaTM np.HMOKyTHMKM. U:i:M: caMiH MeTi CJiyryIOTb TaKi
,ZJ;Bi TeopeMM-03HaKM.
· Teo p e Ma 4.2. HKiq,o oiJun a Kymie napaaeaozpaMa npsi-
Muii, mo Zfeii napaaeaozpaM - npsiMoKymnuK.
,D;oBe,n;iTh :a;ro TeopeM:y caMOCTiHHO.
Teo p e Ma 4.3. HKiq,o e napaaeaozpaMi iJiazonaai pieni,
mo Zfeu napaaeaozpaM - npstMoKymnuK.
/I, o Be iJ e H H .Jl. 0 Ha pMcyHKY 45 ao6pameHo napaneJiorpaM
ABCD, ,n;iaroHaJii AC i BD .HKoro piBHi.
PoarJI.HHeMo TPMKYTHMKM ABD i DCA. Y HMX AB =CD,
BD =AC, AD - cniJihHa cTopoHa. OT.me, u;i TpMKYTHMKM piB-
Hi aa TpeThOIO oaHaKoro piBHOCTi TPMKYTHMKiB. 3Bi,n;cn L BAD=
= L CDA. IlpoTe rd KYTM e o,n;HocTopoHHiMM npM napa.rreJihHMX
np.HMMxAB i DC Ta ciYIHiliAD. OT.me, L BAD+L CDA= 180°.
To.n;i L BAD= L CDA = 90°. ToMy aa TeopeMoro 4.2 napa.rreno-
rpaM ABCD - np.HMOKYTHMK• .&
31
35. ~
• 1. s:lKy cpirypy Ha31t1Bal0Tb npHMOKYTHVIKOM?
2. s:lKy oco6n111ay anaCTVIBiCTb MatoTb AiaroHani npHMOKYTHVIKa?
3. s:lKi anaCTVIBOCTi Ma€ npS1MOKYTHVIK?
4. 3a HKVIMll1 03HaKaMV1 MO)l(Ha BCTaHOBVITVI, lJ.tO napanenorpaM € npH-
MOKyTHVIKOM?
~OPAKTlll~HElABAAHH111.
0
HaKpecJiiTb np.HMOKYTHIIK. 3a ,n;onoMororo JIIIme niHili-
KII 3Haii.n;iTb TOqKy, .HRa piBHOBip;,n;aJieHa Bi,n; fioro BepIIIIIH.
---- - ------- - - --:-- - -:.~:J'.
112.
0
,ll;oee.n;iTb, m;o qoTIIPHKYTHHK, yci KYTII .HKoro np.HMi,
€ Ilp.HMORyTHIIKOM.
l t ~t~ ;::I;iaroHani np.HMOKYTHHKa ABCD B~----"""""::ll'IC
(pHC. 46) nepeTHHaIOThC.H y TO~II~i 0. ,ll;o-
ee.n;iTL, ~o TPHKYTHHRH AOB i AOD -
pieHo6e.n;peHi. A~----_,;;v D
l :t·4..' ,ll;iaroHaJii np.HMQKYTHIIKa ABCD
(pHC. 46) nepeTHHaIOThC.H y TOqli 0,
L ABD = 64°. 3Hafi.n;iT:& L COD i L AOD.
PHc. 46
t ·~ ·:ti ,ll;iarOHaJii np.HMOKYTHIIRaABCD (pHC. 46) nepeTIIHa-
IOTbC.H y TO'tlli 0, L ADB == 30°, BD = 10 CM. 3Hali.n;iT:& nepIIMeTp
TPHKYTHHKa AOB.
116.° Kyr Mim .n;iaroHaJI.HMII np.HMOKYTHIIKa ,n;opiBHIOe 60°,
a MeHma CTOpOHa np.HMORYTHIIKa ,n;opiBHIO€ 8 CM. 3Hafi,n;iTh
.,n;1arOHaJlh np.HMOKYTHIIKa.
117.· Ha .n;iaroHani AC np.HMOKYTHHKa ABCD Bi,n;Kna.n;eHo
pieHi Bi,n;piaKH AM i CK (ToqKa M JiemIITh Mim ToqKaMII A i K).
,ll;oee,n;iT:&, ~o qoTIIpHKYTHIIK BKDM - napaJieJiorpaM.
118.· Ha npo.n;oemeHHi ,n;iaroHani BD np.HMOKYTHIIKa ABCD
aa TOqKy B B3.HTO ToqKy E, a Ha npo,n;oemeHHi aa TO'tlKY D -
TOlIKY F True, m;o BE=DF. ,ll;oee.n;iTh, m;o lIOTHPHKYTHIIK AECF -
napaJienorpaM.
32
36. 119: To~Ka M - cepe,zi;HHa CTOpOHH BC rrpHMOKYTHHKa
ABCD, MAJ_ MD, rrepHMeTp rrpHMOKYTHHKa .zxopiBHIOe 36 CM.
3ttali~iTh CTOpOHH rrp.HMOKYTHHKa.
120.· IIepHMeTp npHMOKYTHHKa ABCD ,zi;opiBHIOe 30 CM,
a 6iceKTpHca KyTa A rrepeTHHae cTopoHy BC y T0~1.J;i M TaK,
mo CM - BM= 3 cM.. 3ttali,zi;iTh cTopoHH rrpHMOKYTHHKa.
121: I'irroTeHyaa piBH06e,zi;peHOI'O IIpHMOKYTHOI'O TpHKYTHH-
Ka ,zi;opiBHroe 55 cM. Ilp.HMOKYTHHK ABCD rro6y,zi;oBaHo TaK, mo
ABi noro BepIIIHHH Ai D tta.n:emaTh rirroTeHy3i, a ,zi;Bi iHIIIi -
KaTeTaM. 3Hali,zi;iTh cTopOHH rrp.HMOKYTHHKa, .HKm;o AB : BC=
= 3: 5.
122.. y TpHKYTHHKY ABC L c =90°' AC= BC= 6 CM. Ilp.H-
MOKYTHHK CMKN rro6y.n;oBatto TaK, ~o TO~Ka M HaJiemHTb
KaTeTy AC, TO~Ka N - KaTeTy BC, a TO~Ka K - rinoTeHyai
AB. 3ttali~iTh rrepHMeTp rrpHMOKYTHHKa CMKN.
123.· ,Il;oBe.n;iTh, mo KOJIH ,zi;iaroHa.n:i rrapaneJiorpaMa yTBO-
proroTb piBHi KYTH 3 O,ll;HiE:IO 3 llOI'O CTOpiH, TO n:eif napa.rreJIO-
rpaM e rrp.HMOKYTHHKOM.
0-. 124: .ll:oBe,zi;iTb, ~o Me.n;iaHa rrp.HMOKYTHOI'O TPHKYTHH-
Ka, rrpoBe,zi;eHa ,ll;O rirroTeHy3H, ,zi;opiBHIO€ iI IIOJIOBHHi.
125: IIo6y,zi;yliTe rrpHMOKYTHHK: 1) aa ,zi;BoMa cTopoHaMH;
2) aa ,zi;iarOHaJIJIIO i KYTOM Mim ,zi;iarOHaJIJIIO i CTOpOHOIO.
126: Ilo6y,zi;yliTe rrp.HMOKYTHHK: 1) aa CTOpOHOIO i ,zi;iarOHaJl-
JIIO; 2) aa .n;iaI'OHaJIJIIO i KYTOM Mim ~iaroHaJI.HMH.
127:· Cepe,zi;HHHHH rreprreH~HKYJI.Hp ~iaroHa.n:i AC rrpHMO-
KYTHHKa ABCD rrepeTHHae cTopoHy BC y To~u;i M TaK, mo
BM : MC= 1 : 2. 3HaiiJiiTh KYTH, Ha HKi ,zi;iaroHaJib rrp.HMOKYT-
HHKa ,zi;iJIHTb lioro KYT.
128... Y rrp.HMOKYTHHKy ABCD L BCA : L DCA = 1 : 5,
AC= 18 cM. 3Haii.n;iTh Bi~cTaHh Bi~ TO~KH C ,n;o ,zi;iaroHa.n:i BD.
129:· ,Il;oBe,zi;iTh, m;o 6iceKTpHcH KYTiB rrapa.n:eJiorpaMa, rre-
peTHHaro~Hcb, yTBOproroTb rrp.HMOKYTHHK.
130... Ilo6y.n;yliTe rrpHMOKYTHHK aa CTOPOHOIO i KYTOM Mim
.n;iaroHaJI.HMH, .HKHll rrpOTHJI€JKHHll ,zi;aHili CTOpOHi.
131.* IIo6y,zi;yliTe rrp.HMOKYTHHK:
1) aa ,zi;iaroHaJIJIIO i pi3HHu;ero ,ll;BOX CTOpiH;
2) aa rrepHMeTpOM i )J.iaroHaJinro;
3) aa rrepHMeTpOM i KYTOM Mim ,zi;iaI'OHaJIHMH.
33
37. . . ~ § 1. YornpVIKYTHVIKVI
..t)ffenPABlll AJ1H nOBTOPEHHH
132. 'Y TpHKYTHHKy ABC L c = 48°, Bi,npi3Kll AK i BM -
ii:oro BHCOTM. 3Ha:ii,n;iTb KYT Mim npHMHMH AK i BM.
133. Ha cTopoHi AC TpHKYTHMKa ABC noaHaYieHo TOYIKY D
TaK, I.QO L A= L CBD. 3a HKOI yMOBH TpMKYTHHKH ABD i BCD
I.Qe MaIOTb piBHi KyTn?
134. Bi,zi;piaoK AD - 6iceKTpnca TpMKYTHMKa ABC. "lfepea
TOYIKY C npoBe,n;eHo rrp.aMy, HKa rrapa.TieJihHa rrp.aMi.M AD i ne-
peTMHae np.aMy AB y TOYIIi E. BnaHaYITe en,z:i; TpMKYTHMKa
ACE.
4"cnOCTEPlrA._MTE, PlllCYt.'iiTE,~
"VKOHCTPYt0111TE, <l>AHTA3Yll1TE
135. Ha IIJIOI.QHHi noaHaYJeHo 1000 TOYIOK• .D;oBe,zi;iTb, mo
icHye rrp.aMa, y KOJKHiii niBIIJIOIUHHi Bi,D;HOCHO .flKOl JiemaTb no
500 TO"l!OR.
5. PoM6
0 3 H a YI e H H .fl. p 0 M 60 M Ha3HBalOTb napaJieJIOI'paM, y JIKO~
1'0 BCi CTOpOHH piBHi.
Ha pHcyHKY 47 ao6pameHo poM6 ABCD.
3 HaBe,n;eHoro 03HaYieHHH BllIIJIHBae, I.QO poM6 Mae BCi BJiaC-
TllBOCTi napaJieJiorpaMa.
ITpoTe poM6 Mae ceo1 oco6JIMBi BJiaCTMBOcTi.
Teo p e Ma 5.1. l{iazona.ni poM6a nepneniJuKy.napni i € 6i-
ceKmpuca;n,u iiozo KymiB.
A
PHC. 47 Puc. 48
34
38. 5. PoM6
Po Be iJ en n a. 0 Ha pHcyHKY 48 ao6pameHo poM6 ABCD,
,n;iarOHaJii .HKOI'O nepeTMHalOTbCSI B TO"tIUi ·o. ,D;oBe,D;eMO, ~o
BO .l AC i L ABO:::: L CBO.
OcKiJihKM 3a oaHa"CieHHHM yci cTopoHM poM6a piBHi, TO TpH-
KYTHMK ABC - piBHo6e,n;peHn.M (AB =BC). 3a BJiacTMBicTro
,n;iaroHaJie.M napaJieJiorpaMa AO =OC. To,n;i BO e Me,n;iaHOIO
TpHKYTHMKa ABC i, KpiM Toro, 6iceKTpHco10 i BMCOTOIO :o;horo
TPMKYTHMKa•. 0Tme, LABO= L CBO i BO .l AC. A
Po3niaHaBaTM poM6H cepe,n; napaneJiorpaMiB ,n;o3BOJISie He
JIMIIIe 03Ha"tieHH.H pOM6a, a .M TaKi ,D;Bi TeopeMM-03HaKM.
Teo p e Ma 5.2. HKUlO iJiazona,n,i napa,n,e,n,ozpaMa nepnen-
iJuJCy,n,apni, mo lfeu napa,n,e,n,ozpaM - poM6.
Teo p e Ma 5.3. HKUlO iJiazona,n,& napa,n,e,n,ozpaMa c 6iceK-
mpuco10 uozo Kyma, mo lfeu napa,n,eiiozpaM - poM6.
,Il;oBe,n;iTb :u;i TeopeMH caMOCTiHHO•
.,• 1. SlKy cpirypy Ha3111Bal0Tb poM6oM?
2. SlKi oco6n111si snacr111socri MalOTb .o.iaroHani poM6a?
3. SiKi BJlaCTll1BOCTi MaE: poM6?
4. 3a flK111Mll1 03HaKaM111 MO)l(Ha BCTaHOBll1H1, w,o napanenorpaM EpoM-
6oM?
~nPAKTM~HEJAIAAHH. _,
- . ~ _!.
...rrl._~-
136.
0
HaKpecJiiTh pOM6 3i CTOPOHOIO 5 CM i KYTOM 40°. Ilpo-
Be,n;iTh ,D;Bi BllCOTll 3 BepIIIHHll iioro I'OCTpOI'O KYT8 i ,D;Bi BllCOTll
3 BepIIIMHM Tynoro KyTa.
- - - - -
.._
- - --- - - -
l:J7~(I ,D;oBe~iTb, ~o KOJIH ,n;Bi cyci,n;Hi CTOPOHH napaJieJiorpa-
Ma piBHi, TO BiH e poM60M.
138.
0
.Il,oBe,n;iTh, m;o "CIOTMPMKYTHMK, yci CTOpOHll SIKOI'O piB-
Hi, e pOM60M.
35
39. 139.".Il;iaroHaJih AC poM6a ABCD (pHc. 49)
YTBoproe ai cTopoHOID AD KYT 42°. 3Haii,n;iTh yci
KYTM poM6a.
140.
0
"Y poM6i ABCD LC= 140°, a )'.:(iaroHa.Jii
B
rrepeTHHalOThC.H B TOqli 0. 3Haii,n;iTh KYTM TpH-
KYTHHKa AOB.
141.
0
0,aHa a ,aiaroHaJieii poM6a .uopiBHIDe
iioro CTOpoHi. 3Haii,n;iTh KYTH poM6a.
142.
0
3Haii,n;iTh KY™ poM6a, .HKmo iioro rre-
pHMeTp ,n;opiBHIDe 24 CM, a BHCOTa - 3 CM.
A
D
c
PHc. 49
143.
0
3ttaii,n;iTh rrepHMeTp poM6a ABCD, .HKl.I~O L A =60°,
BD=9 CM.
144.° KYT D poM6a ABCD y 8 paaiB 6iJihlllHH aa RYT CAD.
3Hali,n;iTh L BAD.
145.° KyTH, .HKi cTopoHa poM6a YTBOproe a iioro ,n;iaroHaJI.H~
MH, Bi,D;HOC.HThC.H .JlK 2 : 7. 3ttaii,n;iTh KYTM poM6a.
146.
0
ToqKH M i K - cepe,n;HHH CTopitt AB i BC poM6a
ABCD Bi,n;rroBi,n;Ho. ,ll;oBe,n;iTh, ~o MD= KD.
147.
0
ToqKH E i F- cepe,n;HHH cTopiH BC i CD poM6aABCD
Bi,n;rroBi,n;Ho. ,Il;oBe,n;iTh, mo L EAC :::: L FAC.
148.
0
,Il;oBe,n;iTh, 1.QO BHCOTH poM6a piBHi.
149.· BHcoTa poM6a, rrpoBe,n;eHa a BepIIIHHH iioro Tyrroro
KyTa, ,n;iJIHTh cTopoHy poM6a HaBrriJI. MeHIIIa ,n;iaroHaJih poM6a
,n;opiBHl0€ 4 CM. 3Hali,n;iTh KYTM Ta rrepHMeTp poM6a.
150: ,ll;oBe,n;iTh, m;o ,n;iarOHaJih poM6a ,ZJ;IBHTh H8BIIiJI KYT Mi.iK
BHCOTaMH poM6a, rrpoBe,n;eHHMH 3 Tiei caMOl iioro BepIIIHHH,
mo li ,n;iaroHaJih.
151: Ha cTopoHax AB i AD poM6a ABCD Bi,n;KJia,n;etto piBHi
Bi,n;pi3KH AE i AF Bi,n;rroBi,n;Ho. ,ll;oBe,n;iTh, I.QO L CEF = L CFE.
152.· Bitij)i3oK AM - 6iceKTpHca TpHKYTHHKa ABC. "lfepea
TOYIKY M rrpoBe,n;eHo rrp.HMY, napaJieJihHy cTopotti AC, .HKa nepe-
THHae cTopony AB y To~i K, i IIp.HMY, napaJieJihHY cTopoHiAB,
.HKa nepeTHHae cTopotty AC y ro~i D. ,Il;oBe,n;iTh, I.QO AM .l DK.
153.· BiceKTpHCH KYTiB A i B rrapaJieJiorpaMa ABCD rrepe-
THHaIDTh iioro cTopOHH BC i AD y TOqKax Fi E Bip;rroBi,n;tto.
BH3HaYJTe BH,n; YIOTHPHKYTHHKa ABFE.
154: "Y TPHKYTHHKY ABC rrpoBe,n;etto cepe,n;HHHHii neprren-
,n;HKYJI.Hp iioro 6iceKrpHcH BD, .HKHii rrepeTHHae cTopoHH AB
36
40. 5. PoM6
i BC y TOqKax Ki P BinrroBi,uno. BII3HaqTe BII.D; qoTIIpIIKYTHII-
Ka BKDP.
155: Ilo6y,n;yi1:Te poM6:
1) 3a CTOpOHOID i KYTOM;
2) aa ,Zl;BOMa ,n;iarOHaJI.HMII;
3) sa BIICOTOID i KYTOM.
156: Ilo6y,n;yi1:Te poM6:
1) 3a CTOpOHOID i ,n;iarOHaJIJIID;
2) aa BIICOTOID i .n;iarOHaJIJIID.
157.•• "'Y np.HMOKYTHIIKY ABCD AD= 9 CM, L BDA =30°. Ha
cTopoHax BC i AD noaHaqeHo Bi,n;noBi,n;Ho TOqKII M i K Ta:R,
~o YTBOpIIBc.H poM6 AMCK. 3Han,n;iTh cTopoHy n;horo poM6a.
158.•• Ilo6y,n;yliTe poM6 aa ~iarOHaJIJIID i KYTOM, 3 Bepllll!HII
.HKoro BIIXO,ZJ;l!Th II;.H ,n;iarOHaJih.
159.·· Ilo6y,n;yliTe poM6 3a ,n;iaroHaJIJIID i KYTOM poM6a, npo-
TIIJie»eHHM u;iii ,n;iarOHaJii.
160.* IIo6y,n;yliTe poM6:
1) aa cyMOID ,n;iarOHaJieH i KYTOM Mim ,n;iaroHaJIJIID Ta CTOpO-
HOID;
2) aa rocTpIIM KYTOM i pi3HIID;eID ,n;iarOHaJien;
3) aa rocTpHM KYTOM i cyMOID CTOpOHII H BIICOTII;
4) aa CTOpOHOID i cyMOID ,n;iaroHaJien;
5) aa TYITIIM KYTOM i cyMOID .n;iaroHa.Jien;
6) aa CTOpOHOID i pi3HII:a;ero ,n;iarOHaJien.
161.* )];aHo TO"tJKII M, Ni K. Ilo6y,n;yi1:Te poM6 ABCD TaK,
~06 TOqKa M 6yna cepe,n;IIHOID cTopoHII AB, a TOqKII Ni K -
ocHOBaMII BIICOT, npoBe,n;eHIIX a BeplllHHII B ,n;o cTopoHII AD
i a BepIIIIIHII D ,n;o cTopoHII BC Bi,n;noBi,n;Ho.
~BnPABlllAJl$1 nOBTOPEHHSI.
162. Ha cTopoHax KYTa a BeplllIIHOID B Toq:u;i A Bi,n;Kna,n;eHo
piBHi Bi,n;piaKII AB i AC. qepea ToqKII B i C npoBe,n;eHo np.HMi,
.HKi nepneH,n;IIKYJI.HpHi .n;o cTopiH AB i AC Bi,nnoBi,n;Ho i nepe-
TIIHaIDThc.H B Toq:u;i D. ,UoBe,n;iTh, ~o npoMiHh AD e 6iceKTpn-
coro KyTa BAC.
163. Ha npo,n;oBmeHHi cTopOHII AC TPIIKYTHIIKa ABC aa
ToqKy A noaHaqIIJIII TOqKy D TaKy, ~oAD=AB, a H:a npo,n;oB-
37
41. § 1. YOTV!pVIKYTHV1KV1
IBeHHi u,ie1 cTopoHH 3a ToqKy C - ToqKy E TaKy, ~o CE =BC.
3Hali.n;iTL KYTH i nepRMeTp TPRKYTHHKa ABC, .HK~o DE= 18 CM,
L BDA =15°, L BEC =36°.
c~ cnOCTEPlrAlilTE, PlllCYli1TE,
~ KOHCTPYto~TE, tl>AHTA3Yli1TE
164. Ha nanepi B KJiiTRHKY Bn6pa.rrn ,noBiJibHO 100 KJiiTH-
HOK. ,ll;oBe.n;iTb, r.u;o cepe.n; HHX MOJKHa 3Ha:HTli He MeHille HiIB
25 KJiiTMHOK, .HKi He MaIOTb CIIiJILHHX ToqoK.
0 3 H a q e H H .H. K B a .zi; p a T o M naauBaJOTL np.sMOKYTHH:K,
y .H:Koro BCi CTOpOHH piBHi.
Ha pncyHKY 50 3o6pa:meHo KBa,npaT ABCD.
3 HaBep;eHOI'O 03HaqeHHH BHIIJIHBae, ~o KBa.n;paT - ue poM6,
y .HKOI'O BCi KYTH piBHi. 0TIBe, KBa,npaT e OKpeMHM BliJJ;OM
i IIP.HMOKYTHHKa, i poM6a. 3Bi,ZJ;Cli BHIIJIRBae, ~o .uiaroHaJii
KBa.n;paTa piBHi, nepneHJlHKYJIHpHi i e 6iceKTplicaMM lioro
KyTiB.
B-----C
Ilp.HMO-
A D
PRC. 50 PHC. 51
Te, ~o RBa,npaT e o.n;HoqacHo i np.HMOKYTHRKOM, i poM6oM,
iJIIOCTpye cxeMa, 3o6pa:meHa Ha pMCYHKY 51.
I»• 1. ~KY cpirypy Ha3111Ba.OTb KBa.o,paroM7
2. ~Klt1til poM6 EKBa.o,paroM7
3. ~Ki BJlaCTlt1BOCTi MaE KBaApar?
38
42. 6. Ksa.apar
..
l65.D ,Uoae,n;iTh, ~o KOJIH o,nHH a KyTiB poM6a np.HMHH, To
n:eii poM6 € KBa.n:paTOM.
166.
0
,II;oae,niTh, ~o KOJIH ,nai cyci,ZIHi CTOpOHH np.HMOKYTHH-
Ka piBHi, TO :u;eii np.HMOKYTHHK € KBa,npaTOM.
167.
0
,Il.ia:roHaJih BD KBa.ll;paTa ABCD ,nopiaHIO€ 5 CM. HKa
,noamHHa ,n;ia:roHaJii AC? qoMy ,nopiBHIOIOTh KYTH TPHKYTHHKa
AOB, II,e 0 - ToqKa nepeTHHY ,nia:roHaJieii KBa,n;paTa?
168.
0
Ha CTopoai BC KBa,ZJ;paTa ABCD (pHc. 52) noaHaqnJu1
ToqKy K TaK, ~o LA.KB= 74°. 3Haii,n;iTh L CAK.
169.
0
Ha cTopoHi BC KBa,npaTa ABCD K cB,_..._,,.__
(plic. 52) noaHaqHJIH ToqKy K TaK, ~o AK;::
= 2 BK. 3Haii,niTh L KAD.
170.
0
qli € npaBHJihHHM TBep,nmeHH.H:
1) 6y.n:b-.HKHH KBBJ(paT e napaJieJio:rpa-
MoM;
2) 6y,nh-.HKHH poM6 € KB3ApaToM;
3) 6y.n:h-.HKHH np.HMOKYTHHK € KBa,n;pa-
TOM;
4) 6y,nh-.HKHH KBa,ZJ;paT € Ilp.HMOKYTHHKOM;
5) 6y,nh-.HKHM KBa,npaT .€ poM6oM;
P11c. 52
6) .HKID;O ,n;iaroHa.ni qoTHPIIKYTHHKa piBHi, TO BiH € np.H-
MOKYTHHKOM;
7) .HKll~O .n:iarOH3Jii qoTHPHKYTHHKa nepneH,nHKYJI.HpHi, TO
BiH € poM6oM;
8) iCHY€ poM6, .HKIIH € np.HMOKYTHHKOM;
9) icHy€ KB~paT' .HKHM He € poM6oM;
10) .HKmo ,nia:rOHaJii qoTHPHKYTHHKa He nepneH,nHKyJI.HpHi,
TO BiH He € poM6oM;
11) .HKID;O ,nia:roHaJii napaJieJiorpaMa He piBHi, TO aiH He
Mome 6yTH np.HMOKYTHHKOM;
12) .HK~o ,nia:roHaJih np.HMOKYTHHKa no,niJI.H€ iio:ro KYT Haa-
nin, TO neii np.HMOKYTHHK € KBa,n;paToM?
171.· qepea aepll.IHHH Kaa,n;paTa npoae,n;eHo npHMi, napa-
JieJihHi iio:ro ,n;ia:roHaJI.HM. BnanaqTe BH.ZJ: qoTHPIIKYTHHKa, yTBO-
peHo:ro QHMH np.HMHMH.
39
43. § 1. YOTV1pV1KYTHV1KV1
172.· Y npHMOKYTHoMy TPIIKYTHHKY Yiepe3 ToqKy nepeTHHY
6iceKTpIICII np.HMOI'O KyTa i riIIOTeHy3II npoaeneHo np.HMi, na-
pa.neJihHi KaTeTBM. ,Il;oae,niTh, mo YIOTIIPIIKYTHIIK, HKHH YTBO-
PHBCH, € KB~paTOM.
173.· ToqKH M, K, N, P € B~noai,nHo cepe,nHHaMH cTopiH
AB, BC, CD i AD KBa,n;paTa ABCD. ,Il;oae,niTh, ru;o qoTIIPHKYT-
HHK MKNP - KBa,npaT.
174.· y TPHKYTHHKY ABC L c = 90°, AC= BC= 14 CM. ,Il;Bi
cTopoHII KBa,D;paTa CDEF JiemaTh Ha KaTeTax TpHKYTHHKaABC,
a aepIIIIIHa E HaJie:rKIITh rinoTeHy3i AB. 3Hali,n;iTh nepIIMeTp
KBa,ZJ;paTa CDEF.
175: -Y KBa,ZJ;paTi ABCD no3HaYieHo TOYIKY M TaK, mo TpH-
KyTHHK AMB - piBHOCTopoHHiii. ,IJ;oae,niTb, mo ~ CMD - piB-
Ho6e,n;peHHH.
176: ,Il;oae,niTh, ru;o KOJIH ,niaroHaJii napaJieJiorpaMa piaHi Ta
nepneH,UIIKYJIHpHi, TO Qeii napa.rreJiorpaM € KBa,upaTOM.
177: l:JoTHpHKyTHHKH ABCD, DEFM, MNKL, LPOS,
SQTV - KBa,npaTH (pHc. 53). 3HaiI,niTh cyMy ,noBJKHH THX
CTopiH KBa.n;paTiB, HKi He Jie:rKaTh Ha np.HMiH AV, HKll~O ,noamH-
Ha ai,npiaKa AV ,nopiaHroe 16 CM.
178.· IIo6y.n;yiiTe KBa,npaT aa iioro CTOPOHOIO.
179.·· LI;oae,niTh, ru;o 6iceKTPIICII KyTiB npHMOKYTHIIKa, HKHH
He € KBa,npaTOM, nepeTMHaIOYIHCh, YTBOplOIOTh KBa,D;PaT.
180.... BepmIIHII M i K piaHocTopoHHbOI'O TPIIKYTHHKa AMK
HaJieJKaTh cTopoHaM BC i CD KBa.n;paTa ABCD. .LI:oae,niTh, ru;o
MK II BD.
181:· ,Il;aHo TOYIKII M i K. IIo6y,n;yiiTe KBa,ZJ;paT ABCD TaK,
ru;o6 ToqKa M 6yJia cepe,nHHOIO cTopoHH AB, a TOYIKa K - cTo-
poHH BC.
B
A
c
N K
D 1.11 L
E-'F
p
PRC. 53
40
Q
Is
T
v
0
44. 6. Ksa.Qpar
182.* qepe3 ,ZJ;OBiJihHY ToqKy' .HKa HaJiemHTh KBa,n;paTy' npo-
Be,ZJ;eHO ,ZJ;Bi B3a€MHO nepneH,ZJ;HKYJI.HpHi np.HMi, KO:H<Ha 3 .HKHX
nepeTHHa€ ,n:Bi IIPOTHJiemHi CTOPOHH KBa,n;paTa. ,IJ;oBe,ZJ;iTb, II(O
Bi,ZJ;pi3KH UlIX np.HMHX, .HKi HaJiemaTb KBa,n;paTy' piBHi.
183.* Ilo6y.n;yiiTe KBa,n;paT:
1) 3a cyMOIO .n;iaroHaJii Ta CTOpOHH;
2) 3a pi3HHD;eIO .n;iaroHaJii Ta CTOpOHH.
184.*-Y KBa,n;paTiABCD noaHaqeHo ToqKy 0 TaK, ~o LOAD=
= L ODA= 15°. ,II;oBe,ZJ;iTh, ~o ~ BOC - piBHOCTOPOHHili.
185.* Ha cTopoHax BC i CD KBa,n;paTa ABCD noaHaqeHo
ToqKH M i E TaK, ~o KYTH BAM i MAE piBHi. ,IJ;oBe.n;iTb, II(O
AE=BM +DE.
~BnPABLll,QJUI nOBTOPEHHSI '"· -
186. Ha pHcyHKY 54 AB II CD, AB =AE, CD= CE. ,IJ;oBe,ZJ;iTb,
~o BE 1- DE.
B
A A D K
PHC. 54 PHC. 55
187. Ha pHcyHKY 55 EF 11 AD, BF=KF, CF=DF. ,ll;oBe,n;iTb,
~o EF II BC.
k:."'c n oCTEPlf ~TE,PlllCYil!TE,-
~ KOHCTPYt01'1TE, <PAHTA3Yll1TE
188. ,Il;oBe,ZJ;iTh, ~o B onyKJIOMY ,ZJ;eB'.HTHKYTHHKY 3Haii,n:yTb-
C.H ,ZJ;Bi .n;ia:roHa.rri, KYT Mi:H< SJKHMH MeHIIIe, Hi:H< 7°.
41
45. ~2 § 1. Y ornp'1KYTH'1K'1
7. CepeAHS111iHiS1 Tp111KyrH111Ka
0 3 Ha qe H H.H. Cep e.n;Hi.010 JiiHicro TPHKYTHHKa Wl3H-
eaJOTL empi30.K, .HKHH CilOJiyqa€ cepe.n;HHH 1lBOX HOrO CTOpiH.
Ha pMcyHKY 56 Bi,n;pi3KH MN, NE, EM - cepe,n;Hi JiiHil
TpHKYTHHKa ABC.
Te ope Ma 7.1. CepeiJna .ninia mpuKymnuKa, aKa cno.ny-
'ta€ cepeiJunu d60X uozo cmopin, napa.ne.nbna mpemiii cmo-
poni i iJopiBnroe i""i no.noBuni.
p 0 6 e i) e H H:. Jl. e HexaiI MN - cepe,n;H.H JiiHiSI TPlIKYTHHKa
ABC (pnc. 57). lloBe,n;eMo, ~o MN II AC i MN = ~AC.
B B
c
Pnc. 56 P:ac. 57
Ha npSIMiiI MN no3HaqnMo TOYIKY E TaK, ~o MN= NE
(pnc. 57). CnonyqnMo ToqKH EiC. OcKiJihKM ToqKa N € cepe-
,n;HHOIO Bi,n;pi3Ka BC, TO BN =NC. KpiM Toro, KYTH 1 i 2 piBHi
HK BepTMKMhHi. 0T:H:<e, ~ MBN = ~ ECN 3a nepworo 03HaKoIO
piBHOCTi TPMKYTHHKiB. 3Bi,n;cM MB =EC. 0T:H:<e, EC =AM. KpiM
Toro, L 3 = L 4. Qi KY™€ piaHocTopoHHiMH npn np.HMHX AB
i EC Ta ciqHili BC. To.n;i AB Ii EC.
TaKMM qMHOM, y qoTHpMKYTHMKY AMEC cTopoHM AM i EC
napa.neJihHi iI piaHi. 0T:H:<e, aa TeopeMoro 3.2 qoTMpHKYTHHK
AMEC e napa.nenorpaMoM. 3Bi,n;cM ME II AC, To6To MN II AC.
TaKomME=AC.OcKiJihKH MN=.!ME, To MN=.!Ac. .&
2 2
0-- 3 a JJ; a q a. !(oBeiJimb, ~o cepeiJunu cmopin 1lomupu-
-,.cymnu-,.ca £ Bepwuna.Mu napaJle.nozpa.Ma.
Po 3 B' .R 3 a H H .fl. -Y "llOTMPMKYTHHKY ABCD TOYIKM M, N,
K i P - cepe,n:MHM cTopiH AB, BC, CD i AD Bi,n;noBi,n;Ho
(pHC. 58).
42
46. 7. Cepe.[l,HRniHiR rp1t1KYTHV1Ka
Bi,n;piaoK MN - cepe,n;H.H niHi.H TPMKYT-
HMKa ABC. To.n;i MN II AC i MN = !. AC.
2
AHaJioriqHo ai,n;piaoK PK - cepel{H.H A
JiiHi .H TPMKYTHMKa ADC. To.n;i PK 11 AC,
1
PK= -AC.
2
OcKiJibKM MN 11 AC i PK 11 AC, To
MN II PK. TaKOJK MN= PK= !.Ac.
2
B
c
D
Pnc. 58
0Tme, y 110TMPMKYTHHKY MNKP cTopoHM MN i PK piaHi
i rrapaneJinHi, To6To qoTnpMKYTHMK MNKP - rrapaJieJio-
rpaM.
)I ~·~-:--- - -
---- -~- ~ -- - ~
• 1. LWo Ha3111Batorb cepe,14HbOIO n iHiE:IO rp111KyTH111Ka7
2. HKi BnaCTl/lBOCTi MaE: cepeAHs:I niHis:i Tp111KYTHll1Ka7
3. CKiJlb Kl/1 cepeAHix niHi~ MO)l(Ha ~pOBeCTll1 B Tp111KYTHll1Ky7
- .._.-:;:....._~nPABM189.
11
qM e ai,n;piaoK MK cepe,n;HnOIO JiiHiero TPHKYTHMKa ABC
(pnc. 59)?
190."' qn e a~piaoK EF cepel{HbOIO JiiHiero TPMKYTHMKa MKP
(pnc. 60)?
JUJ." Bil{pi3KM DE i DF - cepe,n;Hi JiiHil TpMKYTHMKa ABC
(pMc. 61). lln e ai.n;piaoK EF cepe,n;HbOIO JiiHiero n;noro TPMKYT··
HMKa?
192~" CTOPOHM TPMKYTHMKa ,n;opiBHIOIOTb 6 CM, 8 CM i 12 CM.
3Haii,n;iTb cepe,n;Hi JiiHil D;bOI'O TpMKyTHMKa.
A 3 CM K 3 C!1: c
Pnc. 59
K
B
~~CM
M 5cM E 5cM P A-----c
F
P11c. 60 P11c. 61
43
47. 4.2 § 1. YOTV1pV1KYTHV1KV1
193.
0
To~n~H M i K - cepeAHHH cTopiH AB i AC TPHKYTHH-
Ka ABC Bi.n;rroBiAHO. 3Haif,niTb rrepHMeTp TPHKYTHHKa ABC,
.HKIQO nepnMeTp TpHKYTHHKa MAK ~opiBHIO€ 17 CM.
194.
0
,ll;oBe,niTb, m;o nepHM€Tp TpHKYTHHKa, CTOpOHH .HKOI'O
€ cepe,nHiMH JiiHi.HMH TPHKYTHHKa ABC, ,nopiBHIO€ noJIOBHHi
rrepHMeTpa TPHKYTHHKa ABC.
1H5.
0
BHaHalITe BHP: TPHKYTHHKa, yci cepe,nHi JiiHil .HKoro
PlBHl.
196.
0
JJ;oBe,niTb, m;o cepe,nHi JiiHil TPHKyTHHKa poa6HBaIOTb
iforo Ha 4 piBHHX TPHKYTHHKH.
197.
0
TolIKH E i F - Bi,nnoBi~Ho cepe~HHH cTopiH AB i BC
TPHKYTHHKa ABC. 3Haif,niTL cTopoHy AC, HKllO BOHa na 7 CM
6iJibIIIa aa B~piaoK EF.
198.
0
.I.J;oBe,niTL, m;o cepe,nHsr JiiHi.H DE TPHKYTHHKa ABC
(To11KH D i E Ha.rremaTb CTopoHaM AB i BC Bi,nnoBi,nHo) Ta iforo
Me,niaHa BM T011KOIO nepeTHHY ~iJIHThC.H HaBrriJI.
199.
0
,I(oBe,niTb, m;o BHCoTa AM" TPHKYTHHKa ABC neprreH-
,z:t;HKYJI.HpHa ,no iioro cepe,nHLOl JiiHil, .HKa cnoJiyqa€ cepe,nHHH
cTopiH AB i AC.
200: 3naif,n;iTb KYTH TPHKYTHHKa, ,ZJ;Bi cepe,nHi JiiHil .HKOI'O
piBHi i rrepneH,nHKYJI.HpHi.
201." Cepe,nH.H JiiHi.H piBHo6e,npeHoro TpHKYTHHKa, rrapa-
JieJibHa OCHOBi, ,nopiBHIO€ 6 CM. 3Haif~iTb CTOPOHH ,naHOI'O
TPHKYTHHKa, HK~O MOI'O rrepHMeTp ~opiBHIO€ 46 CM.
202.· CyMa .n;iaroHaJieif 'llOTHPHKYTHHKa ,nopiBHIO€ 28 CM.
3Hali,niTb rrepHMeTp 110THpHKYTHHKa, BepIIIHHH .HKOI'O € cepe-
~HHaMH CTOpiH ~aHoro 110THPHKYTHHKa.
203.• BH3Ha11Te BH,n i 3HaM,niTb CTOPOHH 110THPHKYTHHKa,
BepIIIHHaMH .HKOI'O € cepe,nHHH CTopiH poM6a 3 ,niaroHaJI.HMH
8 CM i 14 CM.
204.· BHaHa11Te BH,n i 3HaM.n;iTb CTOPOHH qoTHPHKYTHHKa,
. .BepIIIHHaMH .HKOI'O € cepe~HHH CTOplH npHMOKYTHHKa 3 ~1aro-
HaJIJIIO 12 CM.
205.· ,I(oBe,zdTb, ~o BepIIIHHH TpHKYTHHKa piBHOBiMaJieHi
BiA np.HMOl, Ha HKili Jie:tKHTb ftoro cepeAHH JiiHi.H.
206:· Ha cTopoHax AB i BC TPHKYTHHKa noaHalleHo Bi,n-
rroBi.n;Ho TOlIKH Mi K TaK, ~o AM"= 3 BM, CK= 3 BK. .n;o-
BeAiTh, ~o MK AC, i 3HattAiTb MK, HKm;o AC= 16 CM.
44
48. 7. CepeAH~ niHi~ rpviKyTH111Ka
207:· KyTH BAD i BCE - aoaHiIIIHi KY™ TPMKYTHMKaABC.
3 aepIIIHHH B npoae,neHo nepneH.D;HKYJI.HPH BM i BK ,no 6iceK-
TpHc KyTiB BAD i BCE Bi,nnoai,n;Ho. 3Haif)l.iTh ai,n;piaoK MK,
HKI.QO rrepnMeTp TPHKYTHHKa ABC .n;opiBHIO€ 18 CM.
208.•• Ilo6y.n;ynTe TPHKYTHHK aa cepe.ZJ;HHaMH TphOX no:ro
CTOpiH.
209.•• Ilo6y,ZJ;yifTe napaJieJiorpaM 3a cepe,n;HHaMn Tphox noro
CTopiH.
210.* ,IJJaroHaJii onyKJio:ro 'tlOTHpHKYTHHKa ABCD nepneH-
.D;HKYJI.HpHi. qepe3 cepe.n;nHH cTopiH AB i AD npoae,n;eHo np.HMi,
nepneH,n;HKYJI.HpHi ai.n;noai,nHo .n;o cTopiH DC i BC. ,ll;oae.n;iTb,
~o TO"Cil<a nepeTHHY npoae,neHHX np.HMllX HaJieIBHTh np.H-
Min AC.
211.* CToponn AB i CD onyKJioro qoTnpnKyTHHKa ABCD
piaHi. qepe3 cepe,nHHH .n;ia:roHa.rren AC i BD npoae,neno np.HMY,
.HKa nepeTnHae cTopoHH AB i CD y ToqKax M i N ai,nnoBi)l.HO.
,Uoae,n;iTb, ~o L BMN = L CNM.
~BnPABlllAfl~noBTOPE.HH~
212. ,Uo KoJia a ueHTpoM 0 qepea ToqKy C npoae,neHo .n;oTHq-
Hi CA i CB (A i B - TO'tlKH )l,OTRKy). Bi)l.pi3oK AD - )l,iaMeTp
KoJia. ,ll;oae,niTh, I.QO BD II CO.
213. Y TPHKYTHHKY ABC AB =BC, L B =32°, AK - 6iceK-
TpHca TpHKYTHHKa. qepea ToqKy K npoae.n;eno np.HMY, .HKa
napaJieJihHa AB i nepeTnHae cTopoHy AC y Toqui M. 3HaiI.n;iTb
KyTAKM.
214. ,Uia:roH&'lb BD napaJieJiorpaMa ABCD € no:ro BHCOTOIO
i .n;opiaHIO€ cTopoHi BC. 3nan.n;iTb cTopoHy CD napai'IeJio:rpaMa,
.HK~o TO't!Ka B ain.n;ai'leHa ai,n npHMoi CD Ha 4 CM.
c~ cnoCTEPlr~VITE.P"1CYVITE,-
~ KOHCTPYI0"1TE, <PAH1.A3Yfl1TE.
215. II'.HTb TO"CIOK HaJiemaTh piaHocTopoHHbOMY TPHKYTHM-
Ky' CTopoHa .HKOI'O ,nopiBHIO€ 1 CM. ,ll;oae,niTh, II.1;0 3 .a;nx ToqoK
MOJI{Ha an6paTH .n;ai, ai.n;cTaHh MiJI{ .HKHMll He 6iJihIIIa 3a
0,5 CM.
45
49. r2
8. Tpaneu,iS1
0 a Haq e H H .H. T pane u i £:JO Ha3HBaJOTL 'lOTHpHKYTHHK,
y JIKOro JJ;Bi CTOpOHH napaJieJILHi, a JJ;Bi iHmi He napaJieJILHi.
KomHnli a YIOTHPHKYTHHKiB, ao6pameHnx Ha pncyHKY 62,
e Tpane:a;i€IO.
OCH.OBU
OCJtOBa
P1rc. 62 Pnc. 63
IlapaJieJihHi CTOPOHR Tpaneuil Ha3RBaIOTh OCHOBaMH, a He-
napaJieJibHi - 6i'IHHMH CTOpOHaMH (pnc. 63).
Y Tpane:a;il ABCD (BC II AD) KYTH A i D Ha3RBaIOTh KyTaMH
npH ocHoBi AD, a KY™ B i C KyTaMn npR ocHoBi BC
(pnc. 62).
0 a Ha qeH H.H. BHc OTO JO TP aneu11 uaaHBaJOTL nep-
neH,ZJ;HKYJIJip, ony:m;eHHH 3 6y,n;L-JIKOJ TO'IKH npJIMOi, JIKa Mic-
THTL O)lHY 3 OCHOB, Ha np11My, JIKa MiCTHTL ,n;pyry OCHOBy•
.Ha pRCYHKY 64 KOJKHHM 3 B~
pi3KiB BM, EF, DK, PQ € BH-
COTOIO Tpane:a;il ABCD. 3poayMi-
JIO, ~o BM=EF=DK=PQ.
B EC
Ha pncyHKY 65 ao6pa.meHo A 1tf F
Tpane:a;iro ABCD, y .HKOI 6iqHi
cTopoHH AB i CD piaHi. TaKy
Pnc. 64
K
D
Tpaneuiro Ha3HBaIOTh piBH06i'lHOIO a6o piBH06e,n;peHOJO•
Q
p
.HKI.UO 6iYIHa CTOpoHa Tpaneuil € n: BHCOTOIO, TO TfilY Tpane-
uiro Ha3HBaIOTh np.HMOKYTHOIO (pnc. 66).
B C B_.....,____
A--------n A~-----n
PHc. 65 Pnc. 66
46
50. 8. Tpaneui~
Pnc. 67
Tpaneui.H - u;e OKpeMHH BH,n 'llOTHPHKYTHHKa. 3a'H30K Mim
110THPHKYTHHKaMH Ta ix oKpeMHMH BH.n;aMH iJIIOCTpye cxeMa,
ao6pameHa Ha pucyHKY 67.
0 3 H a 11 e H H .H. ce p e ,ll H L 0 JO JI i H i € JO Tpaneu;ii H83HBa-
JOTL BiJJ;pi30K, JIKHH CDOJiyq:a£ cepe,n;HHH i9i 6i'IHHX CTOpiH.
Ha pncyHKY 68 ai.n;piaoK MN - cepe,ZJ;HH JiiHiH TpaneuJ1
ABCD.
Teo p e Ma 8.1. CepeiJna J£inia mpanef.fii napaJ£eJ£bna oc-
no6aM i iJopiBnroc ix niBcyMi.
a0 6 e i) e fl fl .fl. e Hexaif MN - cepe.n;HH JiiHi.H Tpaneu;ii
II
. BC+ AD
ABCD (plic. 69). ,lloBe,n;eMo, m;o MN AD 1 MN =
2
•
IIpoae.n;eMo npHMY BN i T011KY iI nepeTHHY a npHMOIO AD
noaHa1111Mo E.
OcKiJihKH N - cepeAHHa ai.n;piaKa CD, To CN =ND. KpiM
TOI'O, KYTH 1 i 2 piBHi HK aepTHKaJibHi, a KyTH 3 i 4 piaHi HK
piaHocTopoHHi npH napaJieJibHHX np.HMHX BC iAE i ci11Hiif CD,
B_____ ~--rl""'c
Pnc. 68 Pnc. 69
47
51. .-2_§ 1. Yornp11KyTH11K11
Orme, ~ BCN =~ EDN 3a JJ;pyroro 03HaKoIO piBHOCTi TPHKYTHH-
KiB. 3Bi~cH BC= DE i BN =NE. ToJJ;i Bi,npi3oK MN - cepeJJ;-
H.H JiiHi.H TPHKYTHHKa ABE. 3 u;horo BHITJIHBae, rn;o MN 11 AE,
T06To MN II AD, i MN= !AE. MaeMo:
2
MN= !_AE = !(AD+DE) = !(AD+BC). •
2 2 2
0-- 3 a .IJ; a q a. l(oeeiJimb, w,,o B piBHo6i1lHiu mpaneu,.fi:
1) -,cymu npu "ICOJICHiu OCHOBi piBHi;
2) OiaZOHllJti piBHi;
3) eucoma mpaneu,.i"i, npoeeiJena 3 eepwuHu mynozo -,cyma,
noiJiJtJI,£ OCHOBY mpaneu,.il Ha OBa BiiJpi3"1CU, MeHlUUU 3 JIXUX
iJopiBH10€ niBpi3HUU,.i OCHOB, a 6iJtblUUU - niecyMi OCHOB
( cepeiJHiu JtiHfi mpaneqil).
P o3 e' a 3 a H H a. Po3rJI.HHeMo piBHo6i"tIHY TParreu;i10 ABCD
(AB= CD).
1) IlpoBe~eMo BHCOTH BM i CK (pnc. 70). OcKiJihKH AB=
=CD i BM= CK, To np.HMOKyTHi TPHKYTHHKH AMB i DKC
piBHi 3a KaTeToM i rinoTeHy3010. To.ni L A = L D.
MaeMo: L A= L D, L A + L ABC= 180°, L D + L DCB =
= 180°. 0Tme, L ABC= L DCB.
2) Po3rJI.HHeMo ~ ACD i ~ DBA (pnc. 71).
MaeMo: AB= CD, AD - cniJihHa cTopoHa, L BAD i L CDA
piBm .HK KY™ npn ocHoBi piBHo6i"tIHo1 Tpaneu;il. Orme, D.. ACD =
= ~ DBA aa ~BoMa CTOpoHaMH i KYTOM Mi.m HHMH. Toro_ AC= BD.
3) ¥ ~OTHPHKYTHHKY BMKC (pnc. 70) BM II CK, BC II MK,
L BMK - np.HMHH. 0Tme, u;eli ~OTHPHKYTHHK e np.HMOKYT-
HHKOM. 3Bi~CH MK= BC.
3 piBHOCTi TPHKYTHHKiB AMB i DKC BHITJIHBae, rn;o AM=
=KD. ToJJ;i
B C
A
PHc. 70 PHc. 71
48
52. 8. Tpaneui~
AM = AD - 11K = AD - BC ;
2 2
MD= AD-AM= AD- AD - BC= 2AD-AD + BC= AD+ BC.
2 2 2
i» ~ --- - - - ------- - -
• r: - -
• 1. flKl-'1111 LiOTlt1plt1KYTHl!1KHa3lt1Bal0Tb rpaneu,iEIO?
2. S1Ki CTOpOHlt1 rpaneu,i"I Ha3lt1Bal0Tb OCHOBaMlt1? 6iYHlt1Mlt1 CTOpoHaMlt1?
3. Ll.lo Ha3l!1Bal0Tb Blt1COTOIO rpaneu,i'i?
4. YoMy AopiBHIOE cyMa Kyris Tpaneu,i·I, nplt1nenMx AO Ti 6iYHa°i aopoHlt1?
5. S1Ki iCHYIOTb OKpeMi Bl!1)J,l!1 Tpane14iilt?
6. S1KY Tpaneu,iIO Ha3lt1BalOTb pisH06iYHOIO?
7. S1Ky rpaneu,iio Ha3lt1Bal0Tb nps:iMoKyTHOIO?
8. Ll.lo Ha3lt1Bal0Tb cepeAHbOIO niHiEIO Tpaneu,iP
9. CcpopMynioiltTe TeopeMy npo snaCTlt1BOCTi cepeAHba°i niHi°I Tpaneu,iL
10. CcpopMynioiltre snacrl!1socri pisHo6iYHa·i rpaneu,iL
~nPAKTM~HllAB,QAHH~ -
• ·, _.. 1,:-.1 ~ .
216.
0
HaKpecJiiTb, BHKopncT0By1011n KJiiTHHKH 30IIIHTa, Tpa-
rren;iro:
1) piBH06i11Hy;
2) IIp.HMOKYTHy;
3) .HKa Hee Hi rrp.HMOKYTHOIO, Hi piBH06iqH010;
4) y .HKOI O~HH 3 KyTiB IIPH OCHOBi rOCTPHM, a ,n;pyrHM - TY-
..,
IIHH.
217.
0
IIepepncynTe B 30IIIHT pncyHoK 72, rrpoBe,n;iTb BHCOTH
TpanenSi, o~HHM 3 KiHIJ;iB .HKHX e BiArroBiAHo ToqKn B, M, K i D.
~ -- ~. --..--
I ff I I k4!B 1 !fl I C i I D l
' l
1 /l I
~I I
'
l I I
1'
'!~!
'-
I
~ I l I I
I""•
' • •
---J
I
~ L' '
W. l . l 11 1'.1I
- "' I I
I ~i
- I , I
r.A. '
- --<
I I K I IJJ I c NJ. BI
I I I 111 I I ! 1 I I I 1 1 1 1 1 ! l 1 i I 1JPMc. 72
49
53. .~ § 1. Yornp'1KYTH'1K'1
- , Jo -- ~
- - --- - - -
218." ITepnMeTp piBH06il:!H01 Tpaneu,ii ,n.opiBHIDe 52 CM, oc-
HOBH - 13 CM i 21 CM. 3HaH~iTh 6i'tJHY CTOPOHY Tparreuil.
219.
0
ITepnMeTp Tparreu,ii ,n;opiBHIOe 49 cM, 6i'llHi cTopoHH -
5,6 CM i 7,8 CM. 3Haii,n;iTb OCHOBl1 Tparre:a;il, .HKUJ;O o,n;Ha 3 HHX
na 7,4 CM 6i.rrI>IIIa 3a ,n;pyry.
220."' 3Hali,niTb Ha PHCYHKY 73 Tpane~il, yKaJKiTb lX OCHOBH
i 6i'tJHi CTOpOHH.
A c
a)
·n
6)
P1:1c. 73
B)
221.
0
l.J:n e "CJOTHPHKYTHHK ABCD, 3o6pameHHH Ha pncyH-
KY 74, Tpaneu,ie10? ¥ BlIIIa'AKY II03llTHBHOl Bi,n;noBi,ni BKamiTb
OCHOBH i 6iqHi CTOPOHH Tparre:a;il.
222.
0
3Hali,n;iTI> KYTH Ai C Tparre~il ABCD 3 ocHoBaMH AD
i BC, .HKID;O LB= 132°, L D = 24°.
B
A
A D
a) D 6)
E
B C E
L130vA D A D
B) i)
PRC. 74
50
54. 8. Tpanewis:i
223.
0
3HaiiJJ;iTh KYTll Tparreu.il ABCD, rrpJiner.rri JJ;O 6iqHo1
CTOPOHll AB, .HK~O KYT A MeHIIIlIH Bi,n: KyTa B Ha 38°.
224.
0
3HaiiAiTb KYTll Tparreu,i1 4BCD, rrpJiner.rri )lO 6iqHo1
CTOpOHll CD, .HKIII;O L c : L D = 8 : 7.
225.
0
0,ZJ;JIH 3 KYTiB piBHo6iqHo1 Tpane:u;il ,ZJ;opiBHIO€ 46°.
3Haii,l(iTb pernTy iI KyTiB.
226.
0
3Ha:liJJ;iTb KYTll piBHooiqHoi Tpane:a;il, .HKIII;o pi3HJIII;.H
11 rrpOTll.Jie»HlIX KyTiB ,l(OpiBHl0€ 20°.
227.
0
y piBHo6iqHiH Tparre:u;il KyT Mim 6iqHOIO CTOpoHOIO i Bll-
COT.OlO, rrpoBe,ZJ;eHOlO 3 BepIIIl!Hll Tynoro KyTa, ,ZJ;OpiBHl0€ 23°.
3Haii,z:dTb KYTll Tpanerdi:.
228.<! "llll MO»YTb y Tparreu;il 6yTJI:
1) TPll rrp.HMJIX KYTJI;
2) Tpll rocTpHx KYTll;
3) ,ZJ;Ba rrpoTJIJiemHJIX KYTll TYillIMJI;
4) ,n:Ba npoTJIJieIBHJIX KYTll np.HMJIMn;
5) ,ZJ;Ba npoTJIJieJitHHX KYTJI piBHJIMJI?
229.
0
"llJI MomyTb:
1) OCHOBll Tparre:u;il 6yTM piBHl!Mll;
2) )liaroHaJii Tparre:u;il TOqKoro nepeTJIHY ,ZJ;i.JIJITJIC.H HaBni.rr?
230.
0
,IJ;oBe,ZJ;iTh, r:u;o KOJill KYTll IlPll O,ZJ;Hiii 3 OCHOB Tpane:u;il
piBHi, TO ,n:aHa Tparreu;i.H e piBH06iqHOIO.
231.
0
,IJ;oBe,ZJ;iTb, r:u;o cyMa npoTJIJiemHlIX .KYTiB piBHo6iqHoi
Tparreuil .n:opiBHroe 180°. l:JJI e npaBJIJibHJIM o6epHeHe TBep,ZJ;-
meHH.H: .HKIII;O cyMa rrpOTJIJieIBHlIX KyTiB Tpaneul1 ,ZJ;OpiBHIOe
180°' TO ,ZJ;aHa Tparre:a;i.H € piBH06iqH010?
232.° Cepe,ZJ;H.H JiiHi.H piBHocTopoHHboro TPlIKYTHJIKa 3i cTo-
poHoIO 6 CM po36JIBa€ :Horo Ha TPlIKYTHlIK i qoTlIPlIKYTHlIK.
BM3HaqTe BlIJJ; qoTlIPlIKYTHJIKa Ta 3HaiiJJ;iTb :Horo rrepJIMeTp.
233.
0
BHCOTa piBH06iqHol TparreuJi, rrpoBe,ZJ;eHa 3 KiHIJ..H MeH-
IllOl OCHOBll, ,n:iJIJITb 6iJibIIIY OCHOBY Ha Bi,n:pi3Kll 3aB,ZJ;OB»Kll
6 CM i 10 CM. 3Haii,ZJ;iTb OCHOBll Tpaneu,il.
234.
0
0,ZJ;JIH 3 KyTiB piBHo6iqHOl Tpane:a;il ,n:opiBHIO€ 60°'
6iqHa CTOPOHa - 18 CM, a cyMa OCHOB - 50 CM. 3Haii.n;iTb
OCHOBll Tparren;il.
235.
0
OcHOBll rrp.HMORYTHoi: Tpaneuii ,ZJ;opiBHIOIOTh 10 CM
i 24 CM, a 0,ZJ;l!H 3 KyTiB - 45°. 3HaH)1iTb MeHIIIY 6iqny CTOPO-
HY Tparren;iL
51
55. ~~ § 1. Yornp'1KffH'1K'1
236.
0
OcHOBH rrp.HMOKYTHoi'. TparreuJI ,n;opiBHIOIOTh 7 cM
~ 15 CM, a o,n;HH 3 RyTiB - 60°. 3Hali,n;iTh 6iJihlllY 6iqHy CTopo-
HY Tparreuil.
237.
0
Y Tparreuil ABCD AB= CD, L BAC = 20°, L CAD=
= 50°. 3Hali,n;iTh KYTH ACB i ACD.
238.
0
Y Tparreuii: ABCD BC II AD, AB ..l AD, BC= CD,
L ABD =80°. 3Hali,n;iTh KYTH 'fparreuiL
239.
0
Y Tparreuil ABCD MeHma ocHoBa BC ,n;opiBHIOC 6 CM.
"l!epea BepmHHy B rrpoBe,n;eHo np.HMY, .HKa rrapaJieJihHa cTopoHi
CD i rrepeTJIHae cTopoHy AD y Toqui M. 3Hali.n;iTh rrepHMeTp
Tpaneuil, HKIUO rrepHMeTP TPHKYTHHKa ABM ,nopiBHIOe 16 cM.
240.
0
Y Tpaneuil ABCD qepea BepIIIHHY C rrpoBe,n;eHo rrpHMY,
.HKa rrapaJieJihHa 6iqHiii CTOpOHi AB i rrepeTJIHae 6iJihIIIY OCHO-
BY AD y Toqui E. 3Ha:M:,n;iTI> KYTH Tpaneuil, HKIUO L D = 35°,
L DCE =65°.
241.-o OcHOBH Tparreuil ,n;opiBHIOIOTh 9 CM i 15 CM. "l!oMy
,n;opiBHIOe iI cepe,n;HH JiiHi.H?
242.° Cepe,n;H.H JiiHiH Tparreuil ,n;opiBHIOe 8 CM, a o,n;H~ a oc-
HOB - 5 cM. SHali,n;iTI> ,n;pyry ocHoBy Tparreuil.
243.
0
0,n;Ha a ocHOB Tparreuil Ha 8 CM 6i.rrhma aa ,n;pyry, a ce-
pe,n;HH .rri Hi.H ,n;opiBHIOe 17 CM. 3Hali,n;iTh OCHOBH Tparreuii'..
244.
0
OcHOBH Tpaneuil Bi,n;HOC.HTI>C.R .HK 3 : 4, a cepe,n;HH .rriHi.H
,n;opiBHIOe 14 CM. 3Hali,n;iTh OCHOBH Tparreuii:.
245.° KomHy a 6iqHJIX cTopiH Tparreuii ABCD (pHc. 75)
rroAi.rreHo Ha qOTHpH pinni qaCT11H11: AE=EF= FK=KB, DN=NM=
=MP=PC. 3naimiTh BmpiaKH EN, FM i KP, HKruoAD= 19 cM,
BC= 11 CM.
246.
0
BJicoTa rrp.HMOKYTHoi: Tparreuil, rrpoBe,n;eHa a BepIIIHHH
Tyrroro KyTa, rro,n;iJI.He 6iJihlllY OCHOBY Ha Bi,n;pi3KH aaB,n;OB:'>KKH
7 CM i 5 CM, paxy10qn Bi,n; BepmHHlI rrp.HMOro KyTa. 3Hail,n;iTh
cepe,n;Hro JiiHiro Tparreuil.
247.° Cepe,n;HH JiiHi.H rrp.HMOKYTHOl
Tpaneun ,nopiBHIOC 9 CM, a BHCOTa, npo-
Be,n;eHa a BepmHHH Tyrroro KyTa, no.n;i-
.rr.He 6i.rromy OCHOBY Ha Bmpi3Kll, o,n;HH
3 .HKI1X y 2 pa3H 6iJihilllIH aa ,n;pyrHH'
paxy10qJI Bi,n; BepmHHll rrp.HMoro KyTa.
3Hali,n;iTh OCHOBH Tparreuil.
52
P11c. 75
56. . 8. Tpane4ifl
0-- 248: ,D,iaroHa.rri piBHo6iqHoi Tparreuil ABCD (AB= CD)
rrepeTHHaIOThCSI B Toqui 0. ,D,oBe,niTh, rn;o AO= OD i BO= OC.
249: BHcoTa piBHo6iqHo1 Tpaneuil ,nopiBHIO£ h, a 6iqHy
CTOPOHY BH)lHO 3 TOqKH nepeTHHY ,niaroHaJieii ni.n; KYTOM1
60c.
3Haii,niTh ,niaroHaJih Tpaneuil.
250: 0CHOBH piBHo6iqHo1 Tpaneuil Bi)lHOCSITbC.H SIK 2 : 5,
a ,niaroHa.nb ,ni.nHTh TynHii KYT Tpane:u;il HaBni.n. 3Haii,niTh
CTOPOHH Tpaneu;il, SIKill;O 11 nepHMeTp ~opiBHIOe 68 CM.
251: Y Tparre:u;i1ABCDAB=CD,AD=24 CM, LADB=L CDB,
a nepHMeTp ,nopiBHIO£ 60 CM. 3Haii.n;iTh HeBi,noMi CTOPOHH Tpa-
neu;fi.
252: CTopoHH Tpaneuil ,nopiBHIOIOTh a, a, a i 2a. 3Haii,n;iTh
KyTH Tpane:QiL
253.• Y Tpaneuil ABCD ,n;iaroHaJib AC nepneH,nHKY.n.HpHa ,no
6iqHoi cTopoHM CD i £ 6iceKTpHcoro KyTa BAD, L D = 60°, ne-
PHMeTp Tpaneuil ,nopiBHIO£ 40 CM. 3Haii,niTh OCHOBH TpaneuiL
254: ,Z:IJaroHaJih piBHo6iqHo1 Tpane:u;i:i nepneH,nHKYJI.HpHa ,n;o
6iqHoi CTOpOHH, a MeHIIIa OCHOBa ,nopiBHIOe 6iqHiii CTopoHi.
3Haii,niTh KYTH TpaneuiL
255.• 3a SIKOl YMOBH BHCOTa piBH06iqHo1 Tparreuil ,n;opiBHIOe
niBpi3HHD;i OCHOB?
256: Ilo6y,nyiiTe piBH06iqHy Tpaneuiro aa OCHOBOIO, 6iqHOIO
CTOpOHOIO i KYTOM Mim HHMH.
257: Ilo6y,nyiiTe np.HMOKYTHY Tpaneuiro 3a OCHOBaMH i MeH-
IIIOIO 6iqHOIO CTOpOHOIO.
258: Ilo6y.n;yiiTe piBH06iqHy Tpaneuiro aa OCHOBOIO, 6iqHoro
CTOPOHOIO l ,n1aroHaJI.nro.
259: BiqHa cTopoHa piBHo6iqHo1 Tpaneuil ,n;opiBHIO£ 6 CM,
6i.nhIIIa OCHOBa - 10 CM. 3Haii,niTh cepe,nHIO JiiHiIO Tpaneuil,
.HKill;O O,lIHH 3 11 KyTiB ,nopiBHIOe 60°.
260.· ,UiaroHaJih piBH06i"llHOl Tparre:u;il ,nopiBHIO£ 14 CM i YTBO-
proe 3 OCHOBOIO KYT 60°. 3Haii,n;iTb cepe.n;Hro .niHiIO TparreuiL
261: Cepe~HSI .niHi.H Tpaneuii ABCD poa6HBae 11 Ha ,n;Bi
Tpaneuil, cepe,nHi JiiHil .HKHX .uopiBHIOIOTb 15 CM i 19 CM. 3Haii-
~iTh ocHOBH Tpaneuii ABCD.
1 Hexaii .z:i:aHo Bi;:i;piaoK AB i ToqKy M noaa npSIMOIO AB TaKy, mo
L. .A.."flvfB =a. Y TaKoMy BHnaJlKY KamyTb, mo Bi}lpiaoK AB BH;:i:Ho a ToqKH M
rri.zi KYTOM a.
53
57. 262:· ,UoBeniTh, ru;o Ko.nn .n;iaroHa.ni piBHo6iqHo1 Tpanerdl
nepneH,n;HKYJI.HpHi, TO ii BHCOTa .n;opiBHIO€ cepe,n;HiM JiiHil Tpa-
ne:u;iL
26a:· ,UoBe,n;iTh, :w;o RO.nH BHCOTa piBHo6iqHOI Tpane:u;il
AopiBHIO€ 11 cepe,n;Hili .niHil, To .n;iaroHani Tparreuil nepneH,uH-
KYJI.HpH1.
264.•• ,l]JaroHaJih np.HMOKyTHoI Tpaneu;il poa6nBae li Ha ,n;Ba
TPHKYTHHKH, o,nHH a .HKHX e piBHOcTopoHHiM ai cTopoHoro a.
3Hali,n;iTh cepe,n;Hro JiiHiro Tpaneu;il.
265.•• ,UiaroHaJih piBHo6iqHol Tpaneu;il poa6nBae 11 Ha ,n;Ba
piBHo6e.n;pennx TPMKYTHMKH. 3Haii,n;iTh .KyTM Tparre:u;il.
266:· Y Tpane:u;il ABCD (BC II AD) AC l. BD, L CAD::::: 30°,
BD =8 CM. 3Hali.n;iTh cepeAHIO JiiHiro Tparre:u;ii.
261:· ,UoBe,n;iTh, ru;o TOqKa nepeTHHY 6iceKTPHC KyTiB, npH-
.ner.nnx .n;o 6iqHol CTOpOHH Tpane:u;il, HaJie)KHTh np.HMiH, .HKa
MiCTHTh Ii cepe,nHIO JiiHiIO.
268.•• IIo6y.n;yliTe Tpane:u;iro:
1) aa OCHOBaMH M 6iqHHMH CTOPOHa.MH;
2) aa OCHOBOIO, BHCOTOIO i .n;iaroHaJI.HMH;
3) 38 pi3HHD;eIO OCHOB, 6iqHHMH CTOPOHaMH :H ,z:i;iaroHaJIJIIO.
269:· Ilo6y.n;yliTe piBHo6itIHY Tpaneu.iro aa OCHOBOIO, BHCO-
TOIO i 6iqHOIO CTOpOHOIO.
210:· Ilo6y.n;yliTe Tpaneu;iro:
1) aa ocHoBaMH li .n;iaroHaJI.HMH;
2) 3a 6iqHHMH CTOpoHaMH, cepe,n;HhOIO JiiHiero i BHCOToro;
3) aa OCHOBOIO, npnJieI'JIHM .n;o He! KYTOM i 6iqHHMH CTOPO-
HaMH;
4) aa 6i"CJHHMH CTopoHaMH, BHCOTOIO i ,niaroHa.nJIIO.
271.* l:fepea BepIIIHHY B napaJie.norpaMa ABCD rrpoBe,n;eHo
np.HMY' .HKa He Mae 3 napaJieJiorpaMOM iHIIIHX cniJihHHX ToqoR.
BeprnHHH A i C Bi.n;JianeHi Bi.n; u.iel npHMOI Ha Bi,n;cTaHi a i b
Bi,n;noBi,n;Ho. 3Hali,n;iTh Bi,ncTaHh Bi.n; ToqKn D .n;o u;iei np.HMol.
,..CH roTY€MOCSI JlO 81118'-'EHHSI H080f TEMlll -.
272. Y Ko.ni a n;eHTpoM 0 rrpoBeAeHo .n;iaMeTpH AB i CD.
,UoBe,z:i;iTh, ru;o AC= BD i AC I! BD.
54
58. 273. Y Ko.r,d 3 :a;eHTPOM 0 npoBe.n;eHo .n;iaMeTp AB i xop.n;y
AC. ,IJ;oBe.n;iTh, ru;o L BOC= 2 L BAC.
274. Ilp.HMa AB )J;OTHKa€TbC.H .n;o KO.Tia 3 :a;eHTPOM 0 B TOq-
:a;i C, AC= BC. ,UoBe.n;iTh, mo OA =OB.
275. Xop.n;aAB Ko.rra 3 :a;eHTpoM 0 neprreH,n;HKYJI.HpHa .n;o pa,n;iy-
ca OC i .n;iJIHTh lioro HaBniJI. 3HaJ:miTb: 1) L AOB; 2) L ACB.
276. CKi.rrhKH crri.rrhHHx ToqoK MaIOTh .n;Ba KOJia 3 pa,n;iycaMH
6 CM i 8 CM, .HKIIJ;O Bi.n;cTaHb Mim i:x :a;eHTpaMH .n;opiBHIO€:
1) 15 CM; 2) 14 CM; 3) 10 CM; 4) 2 CM?
IloHOBi-rh y naM'stti aMicT nyn:wria 19, 20. 21, 22 Ha c. 199-
200.
diecnOCTEPlfAj1TE. P111CYlllTE.-
~ KOHCTPYt0111TE, 4>AHTA3Yl'1TE
277. MHoroKyTHHK po36HTO .n;iaroHaJI.HMH Ha TPHKYTHHKH,
.HKi rrocpap6oBaHO y qopHHll Ta 6iJIHll KOJibOPli TaK, ru;o 6y.n;h-
.HKi .n;Ba TPHKYTHHKH' ru;o MalOTb CIIlJibHY CTOpOHY' rrocpap6oBa-
HO B pi3Hi KOJibOpli. ,IJ;oBe.n;iTh, ru;o KiJihKiCTb qopHHX TPHKYT-
HHKiB He 6iJihllia aa IlOTpoeHy KiJihKiCTb 6innx TPHKYTHHKiB.
9.U.eHTpanbHiienMCaHiKyTM
0 3 H a q e H H .H. ue HT p a JI L H HM Ky T 0 M K 0 JI a Ha3H-
BalOTL KYT 3 BepmHHOIO B u;eHTpi KOJia.
Ha PHCYHKY 76 KYT AOB - :a;eHTpa.rrh-
HHli. CTopoHn :a;I>oro KyTa rrepeTHHaIOTh
KOJIO B ToqKax A i B. U:i TOTIKH ,n;i.rr.HTh
KO.TIO Ha ,D;Bi ~yrH, .HKi BH.n;i.rreHo Ha PHCYH-
Ky 76 pi3HHM KOJihopoM. ToqKH A i B
Ha3HBaIOTb KiHIJ;.HMH .n;yrH' i BOHM HaJie-
maTh KomHiii 3 BH.n;i.rreHHX .n;yr. KomHy
3 D;HX .n;yr MO»Ha II03HaqHTli TaK: uAB
(qHTaIOTh: «.n;yra AB»).
Ar.re 3a 3aIIHCOM uAB He MOJKHa po3pi3-
HHTH .n;yrH Ha pHcyHKY 76. HKID;O Ha
.HKiMCb 3 .ZJ;BOX ,z::i;yr II03HaqHTli TOqKy (Ha
PHCYHKY 77 :a;e ToqKa M), TO 3p03YMiJio,
55
..
PHC. 76
PHc. 77
59. § 1. LlOT111pV1KYTHV1KV1
m;o rroaHaCJeHH.H uAMB Bi,n;HocHThC.H .n;o «CHHhoi » ,n;yrn. HKm;o
Ha o,n;Hi:ll a nBox .n;yr AB Bi,n:MiCJeHo TOqKy, TO ,n;oMOBHMOC.H, mo
Il03Hat.IeHH.H uAB Bi,llHOCllThC.H ,n;o ,n;yrn, .HKill ~.H TOt.IKa He
Ha.rremHTh (Ha pnc. 77 ne «ae.rreHa» ,n:yra).
)lyra AB HaJiemHTh neHTpaJihHOMY KYTY AOB (pnc. 77).
Y UhOMY Bnrra,n;Ky KamyTh, m;o neHTpaJihHH:ll KYT AOB cnH-
pacTI>CH Ha )lyry AB.
KomHa ,n;yra KOJia, .HK i Bee KOJIO, Mae rpa,n;ycHy Mipy. rpa,n;yc-
HY Mipy BChOro KOJia BBamaIOTb piBHOIO 360°. HKIIJ;O neHTpaJib-
HMR KYT MON crrnpaeThC.H Ha ,n;yry MN (pnc. 78), To rpa,n;ycHy
Mipy ,n;yr:n MN BBamaIOTh piBHoro rpa,n;ycHiii Mipi KyTa MON
i aarrncyroTh uMN = L MON (qnTaIOTh: «rpa,n;ycHa Mipa ,n:yrn
MN ,n;opiBHIOe rpa,n;ycHili Mipi KyTa MON»). rpa,n;ycHy Mipy
,n;yrn MEN (pnc. 78) BBamaIOTh piBHoro 360° - L MON.
Ha pncyHKY 79 ao6paJKeHo ,n;Ba rreprreHJIHKy.rr.HpHHX ,n;iaMeTpH
AB i CD. To,n;i uAMD= 90°, uACD= 360° - 90°= 270°, uACB=
:::: uADB =180°. KomHy a ,n;yr ACB i ADB HaanBaIOTh niBKOJJOM.
Ha pncyHKY 79 rriBKo.rraMH e TaKom ,n;yrn CAD i CBD.
Ilpo xop,n;y, .HKa a'e,n;Hye KiHui ,n;yrn, RamyTh, m;o BOHa CTH-
ryc ,n;yry. Ha pncyHKY 80 xop,n;a AB CT.R:rye KomHy a ,n;yr AB
iAKB.
QqeBH.n;Ho, ~o 6y,n;h-.HKa xop,n;a cT.Hrye ,n;Bi ,n;yrn, cyMa
rpa,n;yCHHX Mip .HKHX ,n;opiBHIOe 360°.
0 3 H a q e H H .H. B n H c a H H M Ky T 0 M KOJla Ha3HBalOTL
KYT' BepIUHHa HKOrO Jie:tKHTL Ha KOJJi, a CTOpOHH nepeTHHaIOTb
KOJIO.
Ha pncyHKY 81 KYT ABC - BnncaHnii. ,Uyra AC HaJiemHTh
UhOMY KYTY, a ,n;yra ABC - He Ha.;"IemHTh. Y TaKoMy BHrra,n;Ky
KamyTh, 1140 BrrncaHnli KYT ABC cnHpaeTLCH Ha ~yry AC. Ta-
E
PHC. 78
c
D
PHc. 79
56
B
PHc. 80
B
c
PHC. 81
60. 9. UeHTpanbHi i sn111caHi Kyn1
KOJK MO.iKHa cKaaaTH, ro;o BnncaHnli KYT ABC cn11paeTLCJ1 ua
xop,lly AC.
T e o p e M a 9.1. Bnucanuii Kym BuMiproembcR nono6unoro
zpaiJycnoi Mipu iJyzu, na RKY 6in cnupaembcR.
a0 8 e ae H Ii Jl. e Ha PHCYHKY 81 KYT ABC - BIIHCaHHH.
,IJ;oBe,n;eMo, ro;o L ABC=! uAC.
2
PoarJI.JIHeMo TPH BHIIa,z:(KH poaTamyBaHH.H u;eHTpa 0 KoJia
Bi,n;HocHo BnncaHoro KyTa ABC: ,
1) ueHTp 0 HaJie.iKHTb o,n;Hili ia cTopiH KyTa, HarrpHKJia,n;, BC
(pnc. 82);
2) n;eHTP 0 HaJie.iKHTb KYTY, rrpoTe He HaJie.iKHTb .mo,n;Hili a :Ho-
ro cTopiH (pnc. 83);
3) n;eHTP 0 He HaJie.iKHTb KYTY (pnc. 84).
Ha pncyHKY 82 rrpoBe,n;eMo pa,nJyc AO. II;eHTpaJibHHH KYT ·
AOC - aoBHiIIIHili KYT piBHo6e,n;peHoro TPMKYTHHKa ABO (OA =
=OB .HK pa,n;iycn). To,n;i L AOC= LA+ LB. IlpoTe LA= LB.
3Bi,n;cn L ABC = ! L AOC=! uAC.
2 2
Ha pncyHKY 83 rrpoBe,n;eMo .n;iaMeTp BK. 3riAHo a ,n;oBe,n;eHHM
L ABK=! uAK, L KBC=! uKC. MaeMo:
2 2
L ABC= L ABK +L KBC= !uAK +!uKC = !uAKC.
2 2 2
,[{JI.H TpeTboro BHrra,n;Ky (pnc. 84) rrpoBe,n;iTb ,n;oBe,n;eHH.H ca-
MocTiliHo. •
Ha c JI i ,n; o K 1. Bnucani Kymu, RKi cnuparombcR na oiJny
ii my caMy iJyzy, pi6ni (pnc. 85).
B
c
PHC. 82
B
KC
PHC. 83
57
B
A
c
PHC. 84 PHC. 85
61. ~-~._____§_1_._Y_o_T~_P_~_K_Y_TH_~_K_~-------------------------------------D
M N
Puc. 86 PMc. 87
Hac.Jii,n;oK 2. Bnucanuii Kym, siKuii cnupaembcsi na iJia-
Memp (ni6KOJ1,0) - npRMUU (p1:1c. 86).
,lJ;oBe,n;eHH.H IJ;l:IX HaCJii,n;KiB rrpoBe,z::t;iTb caMOCTiMHO.
0-- 3a;:i;aqa (BJiaCTMBiCTb KyTa Mim ,ll;OTM"CJ-
HOIO i xop,n;oro). HexauAB - xopaa KOJla 3 qenmpoM 0
(pMC. 87). l/epe3 mo1lKY A npoeeiJeno aomU1lHY MN. l(oeeiJimb,
uto L MAB= ! u AB i L NAB= !uAKB.
2 2
Po 3 e' Jl 3 a H H Jl. IIpoBe,n;eMo ,n;iaMeTp AD (pMc. 87). OcKi.Jih-
KM MN - ,n;oTM"l!Ha, TO L DAM= 90°. TaKom L B = 90° .HK
BIIMCaHMM, ~o crrMpaeT.hC.H Ha ,n;iaMeTp AD. To,n;i L 2 + L 3 =
= 90°. Arre L 1+L3=90°. 3Bi,n;cM L 1 = L 2.
To,n;i L MAB= L BDA =! uAB.
2 1
MaeMo: L NAB .= 180° - L MAB = 180° - - u AB =
2
= 180° _ !(360° - uAKB) =180° -180° + ! uAKB = ! uAKB.
2 2 2
II pH K JI a .n;. IIo6y,n;yiiTe .n;oTM"l!HY ,n;o ,n;aHoro KOJia, .HKa
rrpoXO)l;MTb "l!epea ,n;aHy TO"l!KY, ~o Jie:>KMTb II03a KOJIOM.
P o 3 e' Jl 3 a H H Jl. Ha pMCYHKY 88 ao6pameHo KOJIO a u;eHT-
poM B TO"llu;i 0 i TO"l!KY M, .HKa JiemHTh noaa IJ;MM KOJIOM.
PMc. 88 PMC. 89
58
62. 9. LieHTpanbHi i snvicaHi KyTvi
Po3.ZliJIHMO Bi,n;pi30R MO HaBniJI (pnc. 89). Hexaif TOt.IKa K -
lioro cepe,n;nHa. Ilo6y,n;yeMo Ro.no 3 n;eHTPOM y Tot.In;i K pa,zdyca
KO. Ilo3Hat.IHMO TOt.IKH E i F nepeTHHy no6y,n;oBaHoro i .naHoro
KiJI. To,n;i KOmHa 3 rrpflMHx 1t!E i MF e myKaHOIO JJ;OTHl.IHOIO.
,IliiicHo, L MEO = goc RR BnncaHnii, ~o cnnpaeThC.H Ha
,z:ciaMeTp MO. Bi,z:cpiaoK OE - pa.niyc ,naHoro Ko.,i:a. To,n;i aa
03HaKOIO JJ;OTH"'IHOI rrp.HMa ME - myKaHa ,noTH"CIHa.
))
• 1. 5iKV1ill KyT Ha3111Bal0Tb LleHTpailbHV1M KYTOM KOJla?
2. 51K Ha3v1Bal0Tb YaCTll1HV1 KOJla, Ha s:!Ki AiJls:JTb illoro ABi TOYKV1?
3. ~KlllM Cll1MBOJ10M n03HaYalOTb JJ,yry KOJla7
4. y s:!KOMY s111naAKY no3HaYeHHs:! Ayrvi ABOMa 6yKBaMll1 OAH03HaYHO fl
Bll13HaLlaE:?
5. y s:!KOMY BlllnaJJ,KY Ka:>t<yTb, w.o weHTpaJlbHll1ill KYT cn111paETbCs:! Ha
Ayry?
6. YoMy BBa.>KalOTb piBHOIO rpaAYCHY Mipy KOila?
7. 51K nos's:i3aHi rpaAycHi Mipvi weHTpanbHoro Kyra Kona i ,rwr111, Ha s:iKy
weill KYT cn111paabcs:i?
8. CKiflbK111,Qyr crs:iryE KO:>t<Ha xopJJ,a? YoMy AOpiBHIOE cyMa ·ix rpa,qycH111x
Mip?
9. ~Kiii~ KYT Ha3111Ba10Tb sn111caH111M KYTOM Kona?
10. Y s:iKoMy s111naAKY Ka>Kyrb, ~o sn111caH111ill KYT cn111paETbcs:i Ha Ayry?
11. ~K Bll1MiplOETbCs:! BnV1caH111ill Kyr?
12. ~KY BJlacnmiCTb MalOTb BnvicaHi KYT111, w.o cn111pa10TbCs:! Ha OJJ.HY ill Ty
caMy Ayry?
13. 51Kll1ill B111A MaE KyT, Ll.lO cn111paETbCs:! Ha AiaMerp?
- - - - -
- - - -- ~
278.b CKiJihKll rpa,n;yciB MicTHTh u;eHTpaJihHHH KYT KOJia,
.HKllM cnnpaeThC.H Ha ,n;yry' ~o CTaHOBllTh:
1) ! Ko.rra; 2) _!_ Ko.rra; 3) ! Ko.rra; 4) 2 Ko.rra?
6 10 2 9
279.Q 3Haft,z:ciTh rpa,z:cycHi Mipll ,ZlBOX ,n;yr KOJia, Ha .HRi lioro
no~iJI.HIOTh ,n;Bi TOl.IKll, .HK~O rpa,n;ycHa Mipa o,n;Hiel 3 HllX Ha 80°
6i.rrhIIIa aa rpa,n;ycHy Mipy ,n;pyrol. ·
59
63. B
A
E E
PMC. 90 PMC. 91
280.
0
3Han.n;iTh rpa,n;ycHi Mipn .n;Box ,uyr KOJia, Ha sud :Hom
. . . . .
IIO,ll;lJIHlOTh ,ll;Bl TO"l!Kll, HKID;O rpa,n;ycHI Mlpll IJ;llX .n;yr Bl.D;HO-
CHThCH HK 7 : 11.
281.
0
3Han,n;iTh rpa,n;ycHy Mipy ,n;yrn, HKY orrncye KiHe:a;i,
ro,n;nHHoi cTpiJIKn: 1) aa 2 ro,n;; 2) aa 5 ro,n;; 3) aa 8 ro,n;; 4) aa
30 xB; 5) aa 12 ro,u.
282.
0
.f!Ki 3 KYTiB, ao6pameHMX Ha pncyHKY 90, e BIIHCaHHMll?
Ha HKY ,n;yry crrnpaeThCH KomHHH ia BrrncaHnx KyTiB?
283.
0
Ha pllCYHKY 91 ao6pameHO KOJIO 3 :a;eHTpoM 0. 3Ha.H,zti.Tb:
1) L BDC, HK~o L BAC = 40°; 3) uCE, HKID;O L CDE = 80°;
2) L BEC, HK~o L BOC=70°; 4) L DBA, HKID;o uDBA=300°.
284.
0
qll e IIOMHJIKll Ha pncyHKY 92?
ToqKa 0 -
:a;eHTp Ko.rra
a)
6)
0
6)
:a;eHTP KO~i:a
P11c. 92
Z)
60
64. 9. UeHTpanbHi i Bnl!lCaHi KyTVl
B 1f--
A11------"""B
K
Pile. 93 Pile. 94 Pile. 95
285.
0
3HaH,ZliTh BIIHCaHHll KyT, .HKIQO rpa,n;ycHa Mipa ,n;yrH,
Ha .HKY BiH CIIHpaeThC.H, ,n;opiBHIO€: 1) 84°; 2) 110°; 3) 230°;
4) 340°.
286.
0
Ha pHCYHKY 93 uAB = 74°, L ABC= 68°. 3Haii,n;iTh
uBC.
287.
0
Ha pHCYHKY 93 uAB=64°, uBC=92°. 3HaiW.Th LABC.
288.
0
UeHTpaJihHHli KYT AOC Ha 25° 6iJihIIIHli aa BIIHCaHHli
KYT ABC, ~o crrHpaeThC.H Ha ,n;yry AC (pHc. 94). 3Hali,n;iTh
LAOC i LABC.
289.° KiHa;i xop,n;H AB ,n;iJI.HTh KOJIO Ha ,n;Bi .ZlYrH, rpa,ZJ;ycHi
MipH .HKHX Bi,ZJ;HOC.HThC.H .HK 3 : 7. flip; .HKHMH KYTaMH BH,ZlHO
uro xop,n;y a TO"t!OK M i K (pHc. 95)?
0-- 290.
0
Ha pHcyHKY 96 xop,n;H AB i CD piBHi. ,ll;oBe,n;iTh,
~o uAMB = uCND.
0-- 291.
0
,ll;oBe,n;iTh, l.QO KOJIH ,ZlBi ,n;yrH K0.118 piBHi, TO piB-
Hi i xop,n;H, HKi !x CTHryroTh.
292.
0
TotIKHA, Bi C ,zti.JISITI> KOJio Ha 3 .eyrH TaK, ~o uAB: uBC:
: uAC =1 : 2 : 3. 3Hali,n;iTh KYTH TPHKYTHHKa ABC.
293.
0
Bepnn-IHH piBHo6eApeHoro TpHKYTHHKa ABC (AB= BC)
,n;iJI.HTh OilHCaHe HaBKOJIO HhOI'O KOJIO Ha 3 ,n;yrH, ripH"t!OMy uAB =
= 70°. 3Hali,ZliTh KyTH TPHKyTHHKa ABC.
294.° KiH1.i;i ,n;iaMeTpiB AC i BD KOJia rro-
c.,r1i,n;oBHo crroJiy"t!eHo TaK, IQO YTBopHBCSI
"t!OTHpHKYTHHK ABCD.
1) BHaHa"t!Te BH,n; "t!OTHpHKyrHHKa ABCD.
2) 3Hali,n;iTh uAB, uBC, uCD i uAD,
.HKl~O L ABD = 80°.
295.
0
rocTpHH KYT rrp.HMOKYTHOI'O TpH-
KYTHHKa ,n;opiBHIOe 32°. 3Hali,n;iTh rpa,ZJ;ycHi
61
M
D
c
Pile. 96
65. § 1. YOT111pV1KYTHlllKlll
Mipn .n;yr, Ha .HKi BepIIIHHll TpHKyTHHK8 .n;i.rr.HTb KO.no, orrncaHe
HaBKOJIO Hhoro, Ta pa.n;iyc :u;horo KOJia, .HKI.IJ;O rirroTeHyaa i:i;aHo-
ro TPHKYTHMKa ,nopiBHIOe 12 CM.
0-- 296: ,lloBe,n;iTh, m:o KO.JIM BIIHCaHHll KyT €, rrp.HMHM, TO
. .BIH crrnpaeThC.H Ha .UlaMeTp.
0-- 297.• Xop,n;n AB i CD KOJia rrepeTHHaJOThC.H B To11:u;i M
(pnc. 97), uAC =50:), uBD = 70°.
1) 3Haft.n;iTb L AMC.
2) ,Il;oBe,n;iTb, m;o L AMC = .!_ (uAC + uBD).
2
0-- 298: Xop.n;n AB i CD KOJia He rrepeTHHaIOThC.H, a rrp.H-
Mi AB i CD rrepeTnHaIOThC.H B To11:u;i M (pnc. 98), uAC = 100°,
uBD= 30°.
1) 3HaiiJ~iTh L AMC.
2) ,Il;oBe,niTh, m;o L AMC =.!_ (uAC - uBD).
2
299.· qepea T011Ky A, .HKa Jie:>KHTh rroaa KOJIOM 3 ueHTpoM 0,
rrpoBe.n;eHO ,n;Bi rrp.HMi, o.n;a:a 3 .HKHX ,llOTMK8€ThC.H KOJia B T011-
ui B, a .n;pyra npoxo,n;HTh 11epea ftoro neHTp (pnc. 99). Bi.zi;oMo,
m;o uBMC = 100°. 3Haif.n;iTh L BAC.
300.• BiceKTpnca KyTa B TPHKYTHnKa ABC rrepeTHHae KOJIO,
OIIHCaHe HaBKOJIO UhOrO TPHKyTHHKa, y T011Ui D. 3Haif.n;iTh
KYTH TpHKYTHHKa ADC, .HKIIl;o L ABC = 80°.
ao1: Ha .n;yai AC KOJia, OIIHCaHOro HaBKOJIO piBHOCTOpOH-
Hboro TpHKYTHHKa ABC, noaHatJeHo T011KY M TaK, m;o uAM:::::
=2 uCM. 3Haii.n;iTh KYTH TPHKYTHHKa AMC.
302.• KoJio, rro6y.n;oBaHe Ha CTopoHi AB TpHKYTHHKa ABC
.HK Ha .n;iaMeTpi, nepeTnHa€ CTopoHM AC i BC y TOtJKax M i K
Bi.n;rroBi,n;Ho. ,IJ;oBe.n;iTh, I.IJ;O Bi.n;piaKH AK i BM - BHCOTH Tpn-
KYTHnKa ABC.
Puc. 97 Puc. 98 P11c. 99
62
66. 303: KoJIO, no6y,n;oBaHe Ha CTOpoHi AC TpHRYTHHKa ABC
.HR Ha .n;iaMeTpi, nepeTHHae cTopoHy AB y TO"ll1.i;i K TaR, I.Qo
LACK= L BCK. JloBe.n;iTh, ~o !). ABC - piBHo6e.n;peHHH.
0-- 304: ,lJ;oBe,n;iTh, ~o rpa,n;ycHi Mipn .n;yr ROJia, I.QO Mic-
T.HThC.H Mim ,l.l;BOMa napaJieJihHHMH xop,n;aMH, piBHi.
305.· BepIIIHHH RBa,n;paTaABCD Jie.maTh Ha ROJii. Ha ,n;y3i AB
no3Ha"CJeHo ,n;oBiJihHY TO"CJKY M. ,lJ;ore,n;iTh, ~o L A.J.fD=L CMD =
=L CMB.
306.• KyT npH BepillHHi piBHo6e.n;peHoro TPHKYTHHRa .n;opiB-
HID€ 56°. Ha 6iqHili CTopoHi TPHKYTHHKa HK Ha ;oJaMeTpi II0-
6y,n;oBaHO niBKOJIO, .HKe iHIIIi CTOPOHH TpHRYTHHKa ,n;iJI.HTh :e:a
3 ,n;yrH. 3Hali,n;iTh rpa,n;ycHi MipH YTBOpeHHX ,n;yr.
307.•• HR, RopHcTyIO"CJHch JIHI.IIe RocHHQeM, 3HaliTH ueHTP
,n;aHOI'O KOJia?
308... .U.aHo KOJIO, y .HKOMY npoBe,n;eHo ,nia-
MeTp AB, i TO"IIRY C no3a ROJIOM (pnc. 100).
HR, RopHCTYIO"l!HCh JIHille JiiHi:liRoro, npo-
BecTH -qepe3 TO"l!RY C np.HMy, .HRa nepneH- Ai-------1B
,n;HRYJI.HpHa .n;o np.HMOI AB?
309... qepe3 TO"IIKY ,l.l;OTHRY ,ZJ;BOX RiJI rrpo-
Be.n;eHo ,n;Bi ci"CJHi. To"CJRH ix rrepeTHHy 3 Ko-
JiaMH 3'€,n;HaHO xop,n;aMH. ,lJ;oBe,n;iTh, IlO ui
xop,n;H napaJieJihHi.
Pttc. 100
310:· ,lJ;o ROJia, onncaHoro HaBROJIO TpHRYTHHRa ABC, npo-
Be,n;eHo y TO"l!Qi B ,n;oTH"llHY, .HKa nepeTHHae np.HMY AC y TO"l!Qi D,
BM - 6iceRTpnca TPHRYTHHRa ABC. ,lJ;oBe,nJTh, IlO BD =MD.
0-- 311.* ,lJ;aHo Bi.n;pi30R AB i RYT a. 3Hali,n;iTh reoMeTpH"ll-
He Micue TO"l!OK X TaRnx, lllO L AXB == a.
312.* Ilo6y,n;yliTe TPHKYTHHK 3a CTOpOHOIO, npOTHJie:IBHHM
RYTOM i BHCOTOIO, npoBe,n;eHOIO .n;o ,l.l;aHOl CTOPOHH.
313.· Ilo6y,n;yftTe TPHKYTHliK 38 CTOpOHOIO, rrpoTHJie:IBHllM
RYTOM i MetdaHOIO, npose,n;eHOIO ,no ,n;aHOl CTOPOHH.
314.* Ilo6y,n;yftTe napaJieJIC?I'PaM 3a .D;BOMa CTOPOHaMH i RY-
TOM Mim .n;iaroHaJI.HMH.
315.*Ilo6y,n;yliTe napaJieJiorpaM 3a RyToM i .n;BOMa .n;iaroHa-
JI.HMH.
316.* ,lJ;aHo Bi,n;pi3oR AB. 3Hali,n;iTh reoMeTpH"l!He Miene To-
qoR X TaRHX, I.Qo TPHKYTHHR AXB - np.HMORYTHHH.
63
67. 317.* BiceKTpHca KyTaA TPHKyTHH-
Ka ABC rrepeTHHae orrttcaHe HaBKOJro
HhOro KOJIO B TO"t.IUi D. To"CJKa 0 -
P11c. 101
a;eHTp BIIHCaHoro KOJia TpHKYTHHKa
ABC. ,IJ;oBe,n;iTh, ~o DO= DB= DC.
318.* EiceKTpHcH KyTiB A, B i C
TpHKYTHHKa ABC rrepeTHHaIOTh orrHcaHe HaBKOJIO Hhoro KOJio
B ToqKax Al, Bl i cl Bi,n;rroBi,n;Ho. ,IJ;oBe)liTh, m;o AlBl J_ CCI.
319.ft Ha pHcyHKY 101 ao6pameHo ,n;Ba KoJia a ueHTpaMH 0 1
i 0 2
• ITo6YAYHTe npHMY l, HRa ,11,0TRRaeTbCH AO ll,RX Rin TaK,
~o TO"'C!KH ,l.l;OTHKY Jie:>KaTh B O,n;Hiii rriBIIJIO~HHi Bi,D;HOCHO rrpJI-
MOl 0102 (TaRY rrp.HMY l Ha3l1BaIOTh 30BHiIIIHbOIO cniJihHOIO
,UOTH'IHOIO ,l.l;BOX ,l.l;aHHX KiJI).
320.ft Ilo6y,n;yiiTe TPHKYTHHK aa CTOpOHOIO, IIpOTHJieJKHHM
,n;o Het KYTOM i pa,n;iycoM BIIHCaHOI'O KOJia.
321.ft IIo6y,n;yliTe TPMKyTHHK aa cTopoHoro, rrpoTHJie:>KHHM
,n;o Hel KYTOM i Me,n;iaHOIO, rrpoBe,n;eHOIO ,n;o ,n;pyro! CTOPOHH.
82£
X7!8roTYf:1
MOC$1 AO BM8fEHHS1 HOBOi TEMM
322. IlepMMeTp TpHKYTHHKa ABC ,n;opiBHroe 30 cM. ToqKa
,n;OTHKY BIIHCaHOI'O KOJia ai CTOpOHOIO AB no,n;iJI.He Ii y Bi,n;Ho-
rueHHi 3 : 2, paxyroqn Bi.n; BepIIIMHM A, a TO"CJKa noTMKY ai cTo-
poHoro BC ei,n;,n;aneHa Bi,n; BepIIIRHH C Ha 5 cM. 3Hali,n;iTh CTo-
poHM TpMKYTHHKa.
B
323. .n;o KOJia, BilHCaIIOI'O B TpliKyTHliK
ABC, rrpoBe,n;eHo 3 .n;oTH"CJHi (pHc. 102).
IIepMMeTpM TPMKYTHMKiB, .HKi ui ,n;o-
TH"CJHi Bi,11,THHalOTh Bi,n; ,n;aHOI'O TpHKYT-
HHKa, ,n;opiBHIOIOTh P1
, P2
i P3
• 3Haii,n;iTh
rrepHMeTp TPHKYTHHKa ABC.
AL..-__,)r,~~-1---~ C 324. YcTaHOBiTh BM,n; TPMKYTHMKa,
P11c. 102 y .HKOro ueHTP OilHCaHOI'O KOJia HaJie-
mHTh Me,n;iaHi.
lloHOBiTb y naM..ffTl 3MiCT nyH1<Ty 23 ltn c. 201.
64
68. 1 0 . Bn~caHiion~caHi YOT~p~ KYTH~K~
c:.f~ cnOCTEPlrA_VITE, PIACYVITE,-
~ KOHCTPYtOlllTE, CbAHTA3YV1TE
~
325. KJiiTHHKII KBa,n;paTa poaMipoM 100 x 100 KJiiTIIHOK
poa<}Jap6oBaHo y rnaxoBoMy rrop.H,n;Ky. KBa,n;paT poapi3aJI11 Ha
KBa,n;paTII, CTOpOHII .HKIIX MiCT.HTb HerrapHy KiJil>KiCTb KJiiTII-
HOK, i B KO:>KHOMY TaKOMy KBa,n;paTi II03Hffl!IIJIII u;eHTpaJibHY
KJiiTHHKy. ,Il;oBe,n;iTl>, ~o 6iJIIIX i "l!OpHIIX KJiiTIIHOK II03Ha"CieHO
IIOplBHy.
10.BnMCaHiionMCaHi~OTMpMKyrH~K~
0 3 H a q e H H .H. qOTHpHKYTHHK Ha3HBa- B
~---
IOTb B n H c a H H M' SIK~O icHyE KOJIO, SIKOMy
uanemaTL yci iioro BepmHHH.
Ha pIIcyHKY 103 ao6pa.meHo BIIIICaHIItt
"CIOTIIPIIKYTHIIK ABCD. Y IJ;l>OMY BIIrra,n;Ky
TaKO:>K Ka.iKyTI>, m;o KOJIO OilHCaHe HaBKOJIO
"l!OTIIPIIKYTHIIKa. PMc. 103
TeopeMa 10.1. HKUfO 11,omupuKymnuK € BnucanuM, mo
cyMa UOZ-0 npomune:HCUUX KymiB iJopi6llHJ€ 180°.
po 6 e iJ e tt tt Jl. 0 KyTII A i C - rrpoTIIJiemHi KYTII BIIIIca-
Horo "l!OTIIPIIKYTHIIKa ABCD (pIIc. 103). OcKiJII>KH n;i KYTII
BIIIIcaHi, TO L A = ~ u BCD i L C =! u DAB. Pille uBCD +
2 2
+ uDAB = 360°. 3Bi,n;cII L A + L C = 180°. AHaJiori"CIHO M omHa
rroKaaaTII, ~o L B + L D = 180°. •
Ha Bi,n;MiHy Bi,n; TpHKYTHIIKa, KOJIO MO:>KHa OIIIICaTII He HaB-
KOJIO 6y,n;I>-.HKoro "l!OTIIPIIKYTHIIKa. PoarriaHaBaTII BIIIIcaHi "l!O-
TIIpIIKYTHIIKII ,ll;03BOJI.H€ TaKa TeopeMa.
T e o p e M a 10.2 (o 6 e p He H a ,n; o T e o p e M II 10.1). HKUfO
6 11,omupuKymnuKy cyMa npomune:»enux KymiB iJopiBnHJ€
180°, mo Biu € BnucanuM.
lJo 6 e iJ e tt tt Jl. @ PoarJI.HHeMo "l!OTIIpIIKYTHHK ABCD, y .HKO-
MY LA+ LC= 180°. IlpIIrrycTIIMo, ~o HaBKono n;horo "CIOTII-
65
69. § 1. LIOT111pV1KYTHll1Klt1
D
PRC. 104
c
D
PRC. 105
pllKYTHHKa He MOmHa OilHCaTH KOJIO. TO}J;i OilIUlleMo KOJIO Ha-
BKOJIO TpHKY'fHHKa ABD. 3a rrpnny:w;eHH.HM wqKa C He HaJiemHTh
D;hoMy KOJiy. ToMy MO:>KJIHBi JlBa BHrra,n;Kn:
1) TO"t!Ka C JiemHTh rro3a orrncaHHM KOJIOM TpHKYTHHKa ABD
(pnc. 104);
2) TO"t!Ka c JiemHTb Bcepe.n;nHi OIIHCaHoro KOJia Tpl:IKYTHl:IKa
ABD (pHc. 105).
Ha pl:ICYHKY 104 CTOpoHa BC nepeTl:IHae KOJIO B TO'Clll,i cl.
qOTHPHKYTHHK ABC1
D - BrrncaHHH. To.n;i 3a TeopeMoro 10.1
LA + L BC1
D = 180°. Arre 3a yMoBoro LA + L C = 180°. 3Bi.n;-
cn L BC1
D = L C. Ilp<;>Te n;.H piBHiCTh cyrrepe"CIJIHBa, ocKiJihKH
3a BJiaCTHBicTro 30BHiIIIHhoro KyTa TpRKYTHHRa L BC1
D =L C +
+ L CDC1
•
,Il;JI.H .n;pyroro BHna,n;Ky (pnc. 105) npoBe,n;iTh .n;oBe,n;eHH.H ca-
MoeTiRHO. £.
TeopeMy 10.2 MomHa BBamaTH 03HaKoro HaJiemHOCTi "CIO-
THpbOX TO"t!OK 0,1.J;HOMY KOJIY•
.HKru;o "t!OTllpllKYTHllK e BilHCaHHM, TO icHye TO"t!Ka, piBHOBi,.n-
,n;aJieHa Bi,.n ycix :Moro BepIIIHH (u;eHTP onHcaHoro KoJia). ID;o6
3Ha:aTl:I IJ;IO TO"t!Ky, .n;ocTaTHbO 3HaMTR TO"t!KY nepeTHHy cepe,n;HH-
HllX nepneH,n;HKYJI.HpiB ,1.J;BOX cyci.n;HiX CTOpiH "llOTllpl:IKYTHllKa.
0 3 Ha "CI e H H .H. "llOTHpHKYTHHK Ha3H- B
BaIOTh 0 n H c a H HM' .HK~O icHy£ KOJIO,
HKe .IJ;OTHKa£ThC.H .no ncix iioro CTOpiH. c
Ha pl:ICYHKY 106 3o6pameHo onHcaHJili
"t!OTllpl:IKYTHllK ABCD. ¥ IJ;hOMy BllIIa,I.J;KY A
TaKO:>K KamyTh, ru;o KOJIO BilHCaHe B tIO-
Tllpl1;KYTHHK. PRc. 106
66
70. 10. Bn111caHi i on111caHi LjOT111p111KyrH111K111
Teo p e Ma 10.3. HK~o 11,omupuKymnuK c onucanuM., mo
cyMU iiozo npomunemnux cmopin piBni.
.
JI, 0 6 e iJ e 1l H Jl. e Ha pMCYHKY 107 y 'tIOTMpMKYTHMK ABCD
BIU!CaHO KOJIO. To'l!KH M, N, P, K e TO'CJKaMH ,n;OTHKY KoJia si
CTOpOHaMli 'tIOTlipMKYTHMKa.
OcKiJII>KM Bi,n;pi3KM ,n;oTM'CJHMX, rrpoBe,n:eHMX .llO Ko.Ila a 0.1lHi£1
TO'tIKM, piBHi, TO AK =AM= a, BM= BN =b, CN == CP =c, DP=
=DK== d. 3Bi.n;cM:
AB + CD= a + b + c + d,
BC + AD= b + c + a + d.
0Tme, AB + CD ==BC + AD. A
Ha Bi,n;MiHy Bi,n; TpMKYTHMKa, He B 6y,n;I>-HKMH 'tIOTMpMKYTHMK
MomHa BrrMcaTM KO.TIO. PosrrisHaBaTM orrMcaHi 'CJOTMpMKYTHMKM
,n;osBOJI.He TaKa TeopeMa.
Teo p e Ma 10.4. HK~o 6 onyKnoMy 11,omupuKymnuKy
cyMu npomunemnux cmopin piBni, mo fl-eii 11,omupuKymnuK
c onucanuM.
l( o 6 e iJ e 11, 11, R. @ PosrJI.HHeMo orryKJIMH 'CJOTMPMKYTHMK
ABCD, y .HKOMY AB + CD= BC + AD.
Hexai1: 6iceKTpMcM KyTiB A i B rrepeTMHa:EOThC.H B TO"CJD;i 0
(pMc. 108). To.lli TO"l!Ka 0 piBHOBi,n;,n;aJieHa Bi,n; cTopiH AB, BC
i AD. 0Tme, icHye KO.TIO 3 n,eHTPOM y TO'tln;i 0, HKe ,n;oTMKa€Tb-
c.H ,n;o TPI>OX a;Mx cTopiH.
IIpMrrycTMMo, II.J;O ue KOJIO He ,[(OTHKaeTI>c.H ,uo cTopoHH CD.
To.n;i MO:IBJIMBi ,n;Ba BHrra,n;Kn:
1) cTopoHa CD He Mae crri.TII>HMX TO'tIOK a rro6y,n;oBaHMM KOJIOM;
2) cTopoHa CD Mae ,n;Bi crriJII>Hi TO"CJKM 3 rro6y,n;oBaHMM KOJIOM.
PMC. 107 PMc. 108
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