Rd sharma class 9 solutions chapter 16 circles pdf

16'a"'Vs
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16.L.
Sottrtion-o I :- ,
Zh en th^L
Ro.diu5 of ci{cte c o Al 36(rn
cho{d[AB) = IrcrD
D-rA(,) oc lAt3.
rnk1,
ThpreY-t"o tt'r a e nwe to c r.rota1
tisecig th€ ctord
.'. Ac:ga = !? = 6cv,.t.
{- l2_ cv6 __ _ _}-
Nop in a ocA,bg pstr.,agcrvas theovera
Acp* oc--64-
=i 62 +ocl=RL
d 86roit=5,
>) oca:611 {g
z) ocL = 2G
--, crc,=0)_€
=) c)c sg:.2'glcr",
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5 olut(oD -o2:-
Griven on ql
D i5 tqn ce Le = qcy)
R ad ius oc c ycle ( oA ) :-lo(h .
rn A ocA69 PV +ho garoa.st.) €ro l€,m
A.t + OcL=64:-
:) ftc1a!!51q|
i A"t:too-as
5 tc = 51;
UJK T, Ehe
bne cr' Ovd
A
-fhe n
p €r PeydJ- L,.t a.r {rovn a e nt.r< t()c ho.rd 6 t.se<
C;B c
ch()rd e -- 8.(6 +8.66
:l1.Blcrn.
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6olLLiDo-o3
Rqdus oF circle coA) = 6crn
oiStan toc):qcm
Jn aocA 6c pgtJ_,ogoyeSt666y4,ya
Ac> +ocL: oAL
) A;L+ q )-=
Gt-
=) Ac2 =]6_t (
-- -_, A c: )>.) : 9 9tcrn -
r^'KT., g1-)
"
per pe f!d,ic-0 c v 6Ltstance fxoxl Cent-r€ t-6
ci.,o{ d. t,isects tr'e c horcl .
ft c --G
T1.r €n
c-:q. q1 cy.'
Aa-q,qr-1q,q.l
:.- 6" g ", C.. -
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SoruFion +:-
C D
cicnstruction t- Dyqr^1 op 1 c D
<hovd AA =s(m
Ch ord c11 = I 16 ,r1
Dfstc,hce pe - 3c"q
LeE op:y.rD
And o ( :oA =Tcr
t k 1 Perpen ct kq-l qy fxor-o ce ntye. co cho 6.d[ bise.ts
ir:
.". cp : pD_-
$ cm
Anc A4: 86r- S crn
Tn aocP,Lry pgtr-os oycis ...,eorela
n-2
-^aIr +cp.
=) 12- 12i C!.,-ft . . C ,l
?
.ry
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T n ao 6,A, b9 Pvtr-ag oTo-s h.,eorern
oA.=-o6a+AoL
-) >: L"t tg?t (
! )._. . r >t
co- P<,.r" e?b c,) qn&>t
Ldrl)>+ r F r>' _; ) -_-r2.(
| f
-t x3+9 -t6 -r t
' i(t+ 61 -"{L
fS
-
j-ll.lr€l
(t
l! ->sqq -Y
11 - -
:) -J. : ]56
l;o tqt iob -S 1-
st€Pts of constvqctrbn.,_
Ll ) ro. Ee "fl.rree Point A/B o, nd c
on tt-,c S.rven ciycte
{
C2).To ib ABqnd gc.
G t pto.- tj-,e perpe411ldol g
r€.tor'
inter5ect eq crr ot €y a t- 6 .
) poi n to uti be ree&ireol
tok.r, jn" gi9€.kov o €<hoycl
c irC le tecqrl6e
ct-[uro*i < Pcs<es th'voc€i''
( € 4ts{e
22,
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SDtqti on -o 6 i-
o
qiven :- c iS vi- e r^icl - pc;ht of chof,d. AG
-ro -PYoV€ i- D i5 tt€ rnid poi^to.c c yc AB
P-{oo f
Tn A 0A6 6n""a[ z:OBc
o tr' 'oB fR<ad iu.s 6C c;y.le]
O t-Oc t.ornr On)
Ac:6. f ciS b.e ",uici
po i ru 6p A6l
-i-€ n 6 oA.g^ oBc CBv ss5 condiEr.6nl
.'. LAoc ::Lgo.' f c.p.. t_l
5 rn cn-b) =ro c6pJ
j,.) [o : Bo
Her e, bi<tr.e roid poio{ 61 .1y. A(a
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S Dlulron- Oa l-
SLePS o€ .crrls kqct i on t-
ci) ro,,ke Lirxz e p6ie.,1
L!] ) f Oir, AB o.n6| 66
LQ) 56in oo .
L<) tcitr-, cenrye
c ib,d e .
A,8 q.g .. on th€ giVe a 4".
(3 ) f>Y6 !o tLe Il e {s lr i.s< c to ry of c h o.cl s
Lc t^-' c!. rlt€-.sects e.r,cl-, othe y a t lroin
o u-:ir bf tr, e r€qL1vqJ centJ f of
cifde.
A3 crtd c c
FO Tr^"en
Lne rr <9,
o qrd vqa[,'u5 oA, .o.optete h^q
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#
'soL.tioh-3:-
SLPPoS€ LLoo ciycl<<, ioLeygp6 t in tr,r< e pot n tsA,g,
f-e n A t B,C arQ Dob -€Dt[iheq. .StJ ,A uh ig q q
ai.{Ce Po'sse < th"ro '-L€i}r
u^,ese B...rv <e poibS ThrS
iS contra A[cUion to rhe Face 65 <l* tu:6g ly"r.,
Cifct€ E ate pqs sing tr-rroutg t', A,t3rc , .ne D<e )
E roo circ e9 c.^nnot inkrse cf << C^^, ., th€y c L-_
'h of < tlqn tc^.1o poiDts.
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Sotrrtion -to:-
tL{) toin 66
[st rPi bi' centY€ o
(l ) cxo d 4 Lir'e gigvnent
AB o{ A _
tii I Drauc rhe re.,
>(m
bisectovs of A6.
[ii ) Loi$ aenLre A cand yarrr.u<
^"
Lc!..rcr" 'rnbersqc.6
1
Li
'i'' clYqo q}rqrc
Poin e o u1e . ::, ::":"J;:";;::.... o
a,d tadi,-rs OA dra*o.;yct€.
No/(!? .anhot d^carc q cir-cte of rc.dJLts 2airt
Passibg U.{oush A and B bercrus€ !.oEen
AT crc O f r qcl it"s zcro Co itr-r cenhre A r
coit yo t lrrbe'(6€ct u"'e PelpeDclic.,la1
a
. nd' uJe 63 it not {ird the .ent'e .
Lo e al.raqJ
tr., e 6,y6-
b i 6ec tov
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5ol(,{.Eioh-l:-
LaE Ae c be clo e9(ito t€l.<:.j Lric^n3le or Sideg._
o.nd ;p1 6o Dne 65 its l-., eJ[ ici?) s Le t G ic€ (:hc
6'-46tvoicl of .aABc. TAen Ag 9 eo:-98 t
G) k T i^ qr^, e?u_itq l€ro.J 4.te C€ntror.d coirrrd
g5iYr, tne circLro c enLr-e.
Tkter€ for€ rG iS Lhe 5€,n66q
,3 i:n crcq.'o Rqcf iU S q A .
Nso q is kne cent<v cir-rd Gn -I-Bc.Ther€fure,
r n {'ighk tri<n6€ noB, Lr€ hqve
A 13 L: Aoa +o g>
=) q-=Ag>r-DB-
f+D:
- (rh
A D :.3i4 < m
Of L-r.re c i rcqy,rf<ren"
.. Ra.dius --
^a
-8.1
I
3
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Solqgi66-123-
Ste Ps OF constyqction;-
(i) rqke tl-,'(ee point A, (3,c or,, ine giy.r,
t2 ) J-o b AB q1-, d g c .
o ornd rad i r-.s o a -_,./ Cor.oFlefr
L3J D Yo r.:) b,1 e P er p,
FB an{ g c tlhic,i*t..*o
t'i. e<*ors o{ c horcls
^t€rse<t eq-c h o{ler qt po,nr
o Tbe. o c,rt be Fr,e regt"r.reJ 2e1. i
rrequjvcd circe
. u -sY tfe oS che
LQf :oi.., oR.
Cs) LJi b^ c€ ntr e
Ul.r e c irc te -
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sio teti oD -13 -
€acl-' PdiY OS crcleS . bave
fh e rn<la.r rn * t'"r y-,q.nLev
5 o lurtio n -tq l-
SLePS o{ coo €tytr.ction,_
0 ) io z g LL..r-€ e point
A,,G qnd c tt egiver)
Ci-cte-
t>l to,n A€qnd oc
La) D-y4vJ trre pelp erd,c,^tar
1a isect:"s o{- chord Aqq"d ac
ugich intgrsecL €".ch o t-,ev"..t o .
9) P oi nt. o toi I I t€ L|,eTeaLiYaJ
C€ ntre ol O.,e circte b ecqc{s€
(Dq k boco Cr.,.* Lr1 e rl€Y
bfse.to" ol&-e cJ-,o{c) OJro-gg
Pa9 S eS throq5h 6re ceyrtve ,
o, il;r$ 2 P rriyA inco"n,,on
orn no ints irl (o mw16n,a
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SotLrlion - t5 l-
Dxd.LJ OM l-Aq qf.d C*J -L co join ()R d.rrd oD i
Gr'^; Aq
2,; L Per peq1,p*.1",
A ro rr) a e.y^rfr e 6 ise<ts
txe c r,ol-d, )
l.'r D -cD
L€L oN b€ v ,so oF4 Lril t e6_.{
in .l Mor],.
o *t+ n4g> =o B-
( 6^'t>z + (
i)t-oet
36+ rr2-r>y + 2E,-",i__oB.
in A N o;-;
o N2+ N62_-ooz
+3r- tjl';. o;-
_.-- cr)
12. I
q oD-.-.. (t-
2,
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Soqtion-16.1
D igtcrnc e o+
ciTcle -- 9c'n
0
P.
s; rr,cl leY chov d A B
o M :qcrv)
crorn centYe 6f
C'cyo e c i r<-ie]
9
}"^8-- A-9
.2
'tn .A or.,r g
,-3 c rr' '
ovY I* r,r er: ogL
L.l t- .. B:a= O BL
l6tq =oea
oB: {i<
oe =sc.'.n
T f! aoND
OO: 9B:5c)^n f.f adiiof
ND= C9- e
2 - ;-- 9crn
o Na.1 11 1't1 =
"or-
oNt *(9)1 - t-g )L
ON2:),9-lG:q
oN : l
So, ctis tcnce o+ L,i55€r' c ho{d FYoyo Ci r(te i3 3(t
=6
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S ol{tioryt6
SoluLiob-oj-
Let R,S or.'d Fl be !r^'e
Nasha resP ec bev€ tg .
AR: As= 2q _ 2 Cm.
oR:os :o^1 : 2oro c.radiiof
f, r) 1)AR
OAa-f AR>- =oez
oh> + Ltt) -j'-- c'o-i
- ^)-
'1^
OP- : l qoo-tqy)y6-= r < 6rn-
O A :6rn
rn ts r, in qn is oscet q trgqnge alti<a& cl_i viJE
tw' € base' soin ARsna Lpcs r.ill 6e g 3 q.})d
R c = cr.4.
A^rea o{ aDAs=L voAxRs
= 4 xRc yos = jx/6 v>q
=) Rcvro =le yzg =)eC= li. I -1 prv-- 2Lrqa)
po€ibio1^, o{ as itq i s qqq)".|
t'r Sr^-)
c t'rcle )
=:3 '9
So,C.'Star,.e LlL. ishr tQ Nr sh<^- i5 3 g .,n-
Lir '
5r ^)
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'I
t1
5o l.rtio n -9 :-
:-) o D > LC,.-r .
.'- RD =6A *o D -nt o t!o) w)
=6O./^''
e aA*; r)
6iven t-'q lc ABi'Bc = CA
5 o r A B C iS q-t., e?oitoterq,l tr ic,n 5i e
o A C y6diuS) ,- tlgrn .
Mecj.ions o4 e9-&itaL€ Tol trio.nsle F c".ss b_, rt,c^9.1"
tb'e circ.rr-n c e nt-r€ (o) O.p €h,e e gl,-iq (< )..J
LriGngle sAac
(J€ d-tl'o crz knor^r t},q b cn ecd,^,a n inber5q66- goc,"
Oth€y a. E t2€ e Sl As eo f S U-,e rn e.cltcz n O f €_q*,O,r*
tr ra,nge ABc, k)e aa,
".,gJyi L-6, r
OA
ol)
-)
; 2--
-T
L{ Aift
.D
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In
B9
A ADC
u.snS ? 9 tl"q
Ac? = Ao> t
9 oyou s
D cL
bxeov e m
Ac-= Aot-+ D.L
4 .t _- 16 o)-+
A c>r 36Do +
N.> =.86o0
lAqL
4l-
Y
c>-qBoe)
4.=9oJ?r",
Ol Str ing OC
=)3q
:a
go, Leniu. €a.cur pl.,q".ro, -,. r,,,.*'.'. -,<!o6
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5 o tuLi on -1 3"
UeB = 56
89 deg'fee rne q.sqre t-eo{em.
LAOB=2LAPB.
=) LAPB = 9x53 -tooo
Sihce oA-oB f R qli€ 9{ cVclel
_fl.,en LoA 6 _-LoBA .f nngle,s oPp6sit€ to €9 rr"al
L€L LoA /3=.u .
s id esl
an A o Aa,!,g angle svtnpro6ertv
LbAB +LoBA +LAo8: tBd
=) -'+'/. { loo :tBc,o
=) >d+roo = t6f
1.'L = 8d'
:t@
L,a.st LgA6: LoB A =go"
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s olqtiorF"z 3-
uoe ho.ve LtA o r = $f
LAoc + fef te{Aoc >360" Sro'-opler. arSleJ
z: t56 +vef le',r, X. LJiot =3et
,=) {€ € er L}oc ;' 3 6,f - r<o"
=) ) g{l et LAo':-rlo
= t€+l-& LA o. -- I rd
=)>
gLAgc= 21d f Bg degree ,ne.s eJlL
:) LA o ( -- 9Jo -, ^,
,
lt-'s o't s'n1
>- ',"
50tlrbion -o33-
uJ€ hqve Laoa = 3;
And LA oc :rl o'
- LA oB I r_A o. + Le o. .-36Do [-rompleK o'agl"]
=) B-3 r ttoo I t!3o. = 368
=) LBoc . ZaS -g]-4-t-o"
BC degtee r eqsu) e {teorem
t-Boc 'etgRc
--) l-r 3 ;Q LG Ac
:) LBhc _ r.-l 8
; =8s
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Solq tioY) -q ci ) :-
lhoc = t 6a
' LAoc-9 L.tGo cr-LG ij f Li neq y p c.iy of a nq I esl
9 3 s"-l ;B or = 16,,o
) LBo. =iB8 -135" ::v i"
Ag d,efl"e € rn e asLq € Tt eot e vr)
L3ic. .' ? t-co A
==) !<" _. z-r,.
z: rt - 51
= S), -L'
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5otrrtion -q ciii )
tD€ hAV e
LA oc - lzJ
Bv le3xee measqye tt.,eoyeyyl .
[-A oc _-r LApc
.--t r:.j = QIAP c
_ - L^p r__ t:d _."
; .-60
' Ln P c f LAB( i- - "tlsD I oppoSrbc ah3teS of
cgct ie 2tl. od-, i 1.. p6o11
'=) 63 f l-qBc -l8.f
4- 63 a t6ci s2ne 6
=) ;_A B c -1>.i
LAB6 rL1D 66 -t c,j t L t ned,'l Pczir o€
:) ?o *-t= tgdo @( .:€ i es]
--) -1. j rBd _f .2 o" -_6g
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6 olcrLo n -q ci v) :-
Lae bcve
L-cDo=65"
' '' LAB c J Lc g D --t *.f Ct-in eqv pqiv ot a.q3le3l
:) LAac+6<o _- L6.i
=i LAc c =ls 3 -6so =trS"
f 83 .{eg r<6 beqgL_,r
,' . RePre.t fL&or = >L^ ga
th eovg ,.y13
, 1 : lXltSo
o
_r n =?30
5o tc{tio n-q Cvl :-
(^)e ho.ve
' Loaa -a<"
-fhen/ LoBA --1oAB-- 3d f Anglej oFposite to € 9-qo.l
radi i l
lrn A A06, t,g Ar,Sl€ suio proFertlj
LA o B t LC
^B 1l_o6n =- 1go"
.:> LADB t 3 {i35" :t5;
d LAo B --{go" -3s" -' f6o =rroo
-". LAo g +n€e c,t LlAo6 _3aj 5 r6'"'ple a",3). J
+ trJ +'r€f len LAoB r86.9
't Re{l<r LAoB
=36grr,i -:so.
g9 de3rree rn €46qfe t!re6"e' Re{l e r LAo g: )LACA
:) 2soo:zx
"=1 r. .. 2!d
=1>s" .
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3olrrEi o rFCl cv i) l-
(re hqv e
L-A o e =e 3
Bb d degTe€ '',eqs q-6'€ tleore-.
LAol3 :z+-A.tz
'> a3 : Q I-AcB -
=) LAce : € =a S f Ansles: bpposit< bo
eZqoJ rcd iil
3 *=93.
50 Lution q cvrr):-
Ure hAv e,
18 Ac :<oo
ond LDBc:1oo
SBoc = 1p lc = 6-c," CAr'5e ln Sq"ne Segroent]
f . A BDc, bY qYg€ S qYr) Proeet E9 ' .
L8 oc f L8 cD tt-o s. = L6.i
=) 5:d +-r r-ro =tG""
=) 1 :tB3 --l .3 - sci t 6o" .
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sot(,tiof)-q1^f0
o)e - qV e
r-aB6.q3
LD B c--9J C ang[e ?hse,^i ci]-cte-l
:) LDBo+UoAc:!ri
--) qi + loa c :.1,!
1=t VC Q.c = go" -yJ =aoo
B9 de gre a ,-relsq-re tbeo r e.o
LAoc => 1cr B c
-)
)t --2)( 6 6o =1o|)o
.
Soeiu-,5t, { tf x-l
e D .AD AB , ts rarge Srr.a prrop€.,.q, .
LADB *LbAB TLABD -_t3o.
:) 3:-o + Lo As +5d = lgo'
:> Lo AA .-*18 3 -3r"_srr"
:) LoAg:9e".
1_rr3ti,
LoAB +LDcB:16,3 f o ppos,be <ingles r:+
ca< tt< 9 uqAritq i< r.-rl
=) 98'{a.tgJ
i) -x:tGo-9t o
- c6"" ,
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S oluttion -t{ Lt r'J3-
Gl€ hav €
LB A cy 35o
uBEc-uBA c:350 f AnSle frr Sovne se;vneap
Tn ABcD g9 af)3e Sur1.fl pyeps119
LeD.+LgcD {-1JpBc = 183
=) ? S+.t t6d = 16 j
:. -n =6J-3s'-<<'.G3.
i:-olu
u]
9n -q c{i) 3-
r.oe ho,v e ,
l-AB D =qo'
. LAc D =LABD =_qJ f org" in .5qy,.e se3rne.0
rn Ap.D, B9 qnge -gq,n pto pe.rt_9
LPCD +LcPo+LPDc = !6.j
*, qo 1-llo t r :-lGc,"
) 1:-6o-Sri
-) -y. .-33 -
Sotq tlon -eiarr) i-
qiV€r, tt,c.-t lB A c:5>o
The n, Ll3 pC = ;B A c __ s:o f nngle i n lsqr
shce oD:oc
r
" <5qbe S$t"e"r]
The.n oD =-o<_
Tl-: € r'r r L6Dc.Loco f oePosite q'D91e) r_ o eet,<J vqdiil
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sotugion{ }
G-liv€n o i5 the ci-fctt./o cen tye orc a ABc qnd onlec
to P.6ove LB on --r r.ic
Pra o { l-
In aoBD qnd A oco
L oDf}: LoDc Ceo.cb 9JJ
DB=oL c Rad.ii o{ civclq
On :6D fc orr, monl
,^(n a oeD ?60.o f 89 p Hs conditiu,J
.cr) Ccp.cT)LB oo = LCo D". -
h,, ^r ^"7 (-'c lree -€Gs.q"e t l-€ oY?
',r
LB oc =: Ll:^A.c.
=) Q LB6I) : I L.B+c
n LaoD>u3nc
I e9 i{s iyrs (tJ
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ggUI:LC:
(riv< n , e o iq the t,isec ror o{ LA B c
1-o Pl-ove A8:6a
f -ro o€
Sin.e,
Tb€ o,
Sinc e,
l-Fr€ n
5in
1l--, e
ce og
LCBo::
:: oa
L6 CB
B o is tn e t,i6ec Lo" o+LAB._
LROorLcBo._-.Cr)
o B:oA f Ro,&.u 9 oF ci rclel
r LABo: Lbna...421
(>
^
foepos,.Le
engrel to_ zq{ sid.€sJ
F R a di.^.- Oc cr rc tel
' '(3) topposirp ^-,,. c F€ lej
to e 9t1 ^1 ^,
aovi-) pqrn, e?b-s 6) (2) $ g)
*" -t stdg']
Lo*a =L_cctz -_.,c9)
r-n aoAr3 4nd aocB,
L OAB.'-Lo<.6 Lfxo"o f .{)l
Lo B A
=Lo?,c z ftlivenl
OB =66 [- rornwr66]
Tl^. € o, A oAr .w
" =_ ocB t Bg nRc _
" LO nq Ftonl
. AB ;1f,6 L cp cul.
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S o[.r-ti o n -.1 )
de l^tav e I [3 = L4 €e.ngeg in Sorne Apegrr.ro.F)
i l:^ = > L3 l-Bg degyee me4sL.>e theodeyq
:) ;x = IZ -t tB.
=) J ^4{L3 I L-l -. Cl) f ya:yal
Bht, L9: L3 tL! Fey ea€rior <ingle p 6q'l
a L3 = LY-'Lt -' L2)
{to- 19 .4nd. a>)
L',. : LV -Lr + L9
t la: Ly lLl -D
=) t-'^ = L9 + Lz, yg/ y' ,eV€rLel.riy
4 & = Ly + tz- .
ens l€ PraPJ
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sot!^t'ion -c6 ?-
89 deSr€ e rr1 ectgu'{€ tb e of € }n
LAD13= ! LAcB
=) r33 3A yA cB
--) LA( B = r3:-'
--ed
-" LA.BI L@cO:-1sc" LL-i ne,',-1 prc,it og <]n,
=) &" l& cp :163
-l L,B cD =L€o" - dj' =rr{o
Qz deree Dea.sl{ t Q l7e6y 1-
R<-Pl<r LBoo :2Lf.tn.
=) Ref l€,1 L€ dld
!lgo
) --r jl€ 6 : a 3o"
NoN, Pell ea L/3 afl6 . , n I
' Lta a'D:a6J
f (orople)t
-) 23Do yt t36as 491 eJ
4 -t- _. Z6.j -a.a..l
: /5D
4:1.40
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6 olcr bio a -? 3*
5in.e pe is d,io"r€i€l
Th€ t / ll"Ro = 9D" C Ang le ir,s&niciycl€J
lP eq + LTRo ' ,"re f l. inacq pa.iv of o.vrglel
Lqo" + LTR& - tS j,
LTRQ ' t6; -1f, :g 3 .
BV .l.6ree m€q.sqre Theo{ern
LRos,2LRAc
? Lt ()- --9rRqs
:) LRss -d:pj
!n NRq r / 89 o'Yro€ 5
'''{Y'l PYaPQ{tsi
LRe't+.-dRT t- LRTs = Itf
a e-S *aS + LR rs __r8,9
d Ler 5 . 1s,l - lJ *
1o, _- .r oo
roqti o n -to l-
LJe ba.V€ l
L^ca = rrJ ' uPf e =rz'i
:. LheB.-L-NLg = 4oo C An3e i n sq,yn€ segmq6ll
E n A PoA, LY qn3l€ 5!rr) PYoPq'LcY
LFbfr.+LPBofLBFD
= lr .j
a
,) 1 o + LpBO r12u = )to"
4 LP B D -- lg-" -v,j - rr.i
=_ yp Bo =2u.
LcB D =2n
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a (fl! Efon-I t "":
Ul€ rqve /
R a,cli u 5 DA :.chofd AA
=> oA :oB = Ag e.
Tlren66 AD iS q. e9L!ra ky..l tric,hSle
.'. LA oa :6S fone qy1g16, oF ee(,.llo.t< rql 6)
A9 d,e3r€ e rneqsqre g^eotero
l-A o E: l-t-AP B
--)
(6o;a IAPB
=a 6ri _ er-ApE
-_J tApB: 69 -- o
> - a)
r -otF I LAPA +Dg8--lgo f o pposiL< 6.nglas of
(gciic 9qqd1-a 33 +LAqg :ls"'
=) LAq0 _-re .f -a,S = tsd,
"
n rr9i l€ by Chord ABc_t rru. noy
An3ie b9 cborcl Ao q mojov
94
qrc
Gtrc
=rgd
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.so(u.llon- O I :-
since, 44ga is an e.q+ila.tera-l +rlanu[3.
Then, L.6Ac = 5c,"
'. r--BA c+ ;-BEc:1gj
9 63+ Lg6.: tB
*-) tBE
" =
Solu+io4- o2 .,-
t^re, have t-_9s e --3 4
9in. e, Nt5q p rs an i50sretes tvicrbgle
fhen fee " rfRo -35a.
:1. A PqR 89 <rvrgle 5."{m Ptop< rrs
r-f + r-PqB + l-[' Rg:re.i
_, U+zs' t3q" :tgi
,) LP : tB o. _ s<" _3d
=) Lf =rrf,. fAngles i.,
No ur, L-q s R + L6r€ _ t<-
a ,'J r Lqrp *r8i
>) Lffi = r6d, -tld'
) r-A rP : -7i
Co ?posi+-e G,3les of
ct q-llaari t^te'a,t1l
LcE c= rt3_a.i
ooi iJ. PA :pp
sq,De 6e5_n
eD _d.
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<;ot^tion-o3:-
C'iv€r tj^ot o is tbe <entre of
rJ€ h6V€l tBoO : t63
Bg d,es-(ee rneqs,4-re
tf€o"r €m
IJZoO: &LB aD
Dx^
t5i = e.9
28 r' o 1;Bco : tG,9
Y*r. = t€ cP
=) .y13! = rgce
:) "l = rgci_s8
-J
,l:
tt, e ci ycle
f oPeosl6.
ag ct i.
G.r'gleE
q/^ad1 .
D{
5 olqt'i on - of j-
u)e haVe,
LBco : to d <t rcl r_Aao: .16
.'. LDAB + LIB(D =lGS
I LtAA+ 16,3 = 16"'
--*) LDAB ITtS *t od ; 3j
fopposiLe q egles
cg ct,t 9r.al ]
ol
'-) t-?A B =gS
:f 4 aD AB ,69 q ra I e S.{vD p-!.op ertj
l-h o 6l LDAa +l.E Bo :_ I *.i
=.: tb4a *x3+-l ,.,o.1goo
*._____ - . ?*eg?:].{,-,,.jjJ€
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Sottltion-os''
s i n.e A B co is *t'i'litod r itq.tevc1 co itA AD ll B< .
-Ibe., Lq1r_. __tcd - _,c,) f opP osiue a.gt€s of a di_.
AYd , Ll+LBr =r€d "--.tr-)
q'u.tJ
;At;€ - 1gj f co i'"'5q"1." q-g tesl
coFlPqYe (r r a.nd ())
"gL{o.t
ion s.
;€:tc
Solqtion -06.-
Gtiverr tr-,a F LAoc =166,
Bv desr ee rreqsq{e b^poye,,,>
LAoc,_ 2tbpc
_> to,i = 2 .:4-p<
_J Ln p. = ,r9o"
="s
.._ 1A pc.1-g14Ct
= reJ lopposi t_€ a.6le< e4 61
=) sd t LABc :.16; (9ctic A,, o4 o,.tate,a.l]
9 lABe- -isa'^srj
=l3oo.
-. Lta( +LceD : t<o [- I i',g.l.u pdr-(6f a"rglqsl
:) t3dlL.gD 2)6ios
4 LQB D > e;o'
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5oqt,ion -c)-1 3-
'--'-#
6tver Lhat,
L-o B o = 5 j
sirrce ) AB a.rrd aD arre Er"e dio"oek v
o is tbe cenL(e ol tt.p < ixcte
_,_ LpBc =gu" t AnSte
LoB/u^ + LD Bc :- 4o'
sd +p Bc = 1,9
a4 LDBc -Cd-cf =q,i.
Bg d€gree rn€q,s(..ce tje o.r€"rrj
1-Aoc,g6rr6a
,5ot.rE1or"-O"
Loe hove , LcAa:-g j,
LAcA --g 3 f Aflgleihg,rf,icirclel
of cl rcte L:+,e n
rn S er4iciy. lel
=) LAoc = R*q,j - go".
f-n A Asc, 89
LtAB *LAcB
>J 3.3 r go"
ti LAAc
aDg le .5,-{t) prc,pey 19
+ l}ac i- taci
t LAAc -- tei
-- IfD -l 20
- 60
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SoLuEion r o1l,
Gri ve n tlq t LB
S in gge AB cD
?-r,tqdx itobcvql
=?o"
isq
= lo.
cS c tic
-Fh€n LB tLD =teoe
=) -l Do tLD . t6o.
-', Lo = l8d
@
*1 o" = 11af
slnce Ag lt pc
-Then LA+ Lc : l$ d,
4 .1.,t f 1-c = 1g';
-J Lc -- tea" -td
o
,tlo
NOur- LA+ Lc - l€ oo Dopp66ire
4 L*+l)ci =tBJ
:) uA =163 -ttrj
>)lg;1o".
f' .of nbe rioy zrng l € sJ
o..gle5 Of C,ctrz
?aqd r alo,terall
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SotqtioG-lo "-
(^)e F cVe / LA
L€t Lc -
Th€ r) A
.. LA+LC
-3Lc
-{
:- 31
= l€ oo fopposiLe cng,es o f cgcri"
9ctad?
=, 31 ta
=-) q{ = l6oa :) ,lt.=. l8o
T : 9s'
LA:3a
= 3:rq4
= t3Go
..LA:Bs'
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Sotulion 'l l:-
c^]e ho.ve LDA B = sc'
89 aegree rDeasu^re Lbeo-rern
L-f3OD: QLg^D
-) a --p,.s*3 =rod.
$rre, q6.p 'tg c{ cacticgL{,.d.rito.t€y.u
-fh€n LA r Lr -- l€ ,3
:-J SO -t9 - r€ r.o
:) v -t6o" -5rr"
- ,2,o
Satqtion - I f, "-
89 qsiYg q.r,gle s.qyDpTopey 19 ia AAB c,
tD --r g.,'- C6B r);J =t6.,"
jvr cy ctic a q6d riLqtera_l AGCO, LJ€ have j
L-B+rP.6""
Lo = t 6 o'-r ooo =63.
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q/
5 o I qLio r,- 13 :-
Since 4Aaa iS qn €g u.it4t€ ra.l ]-T;q hgl e
Th€ n LaAc =63.
r- LGDC - LsA. =63 f Angl 6s S < Yne
S e 3v-en t')6 i h.e, g qqd
Th€ n Lg
A BE a is c cvcr;c
Ac + Lll Ec = t€?
?"todrito. tta,t _
) C3+ gBE c-16f
-) l9 Ec = rco' * 6 6e :- r)6o
SorqEion-tq:-
Co€ h6.!€ , LAEc =gJ
€lnce,.)uadt i iqte(ar, Fe ctr
The
',, LSA c {-rjl Fc -teoo
:t +3S =te o.
:- ?,. : I B.i -Bo" =tsoo
Bv 61egr-ee Fr€a.gqy6 t1.eo{e rn
LAoc* :2 UEc
:) g = Qx6 o" __
5 q cA cti c ?.1(qd,y i tqt{n r
- '. LAD c LAE. f nn916. i.6^n_e
5 e 3>ae",r:1
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SolqtioB -rsi-
(re ho.va , Lir A D =18", u) ca = f c. rrj
ti rce, ABc D iS q cac(ic g, qoJr_itqLt ya._l
-roQ n t' Ll3A D + LBco = t*o.
--o f€ + LBCD -t*oo
LD€tr=V"
NOG),
:) LBcD:
Lgc D + LD cF
-- = it
ir." -t
".
:l6ij
C
-l Eo'
= lo:-o
Li neo, v
P*iv ot qY'S | €sl
3 i nr e,
Tl-.,e
:) -r-
=
b.EF

i
€ o"-L o,o _- r?"
s a qyrr C aqoO vi to [e r..l
n / - 19 =l*j
1f -v = te u"
9 --ref -r *" :lo2o
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Sotqti ar)- l5 i-
(ne hq ve
tJ -19:63 ...c)
since , Ae,cD ;g at .gcLic ?qodr i LqteycJ
Th€n LA tLc =r€c," . -.. L)-
Add egqqbion< cr) .l!-,d cz).
LA -tc + [A+G = 6J +red
sos rJr ,- D(
Y o
=) 1A - &9f
& : l&3
PqJg Vq.{e oI cA in e?,qa tion Lg ).
L{o + LC :-lsd
:i 1_A =1goo..,$ro -.6o,
Sotution,tt.'-
L€D c t L.bA =t*f [iuinc,., p aiy of orgles]
+ gd t L.o A = l€oo
=) LcoA .163 €,o" =1e;
9incp, AG"o is 4 aycti c g,ua&i Lo. te,6-[
LAo . rUB c -- le c"
i toJr f Ba = 16"o0
,> LAAc = t$o- _l oo" = t;
r-](7Gr / LABc + LA B€ - r6; IlLihe(v ?o,ttY gt er;
d 8o t-a' : I €o"
-:) ^d: reo" _9.,. =torj._
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SotqEion -16 c;) :-
sin.€, AB Cb 5 q.
-rh€r) , Llec + LAoc
cvcLic ?uad.T itcrbe y.rl
9 LAec+ld
5 ;)8c:
=
: t(l50
163 -ud
'1 .)
€inco. AollE c
THer) LDAA + LAsc - €;.' Ci-o_
=) IJP A c t 5d+1o" :.1 €S
qne l" )l
) LoAc -- l6s -l2o' = es
Sot,.r-tion - t6 1i 11 .-
LgAc:- LeDc ,_qj fa^3le in
.34 rr e
Se3fi€'"ll
Jn a g D c, by er-e a 6u, m propeytr .
LDr3 c tLBro tL.Boc
= l€f
Bo +Llj4o+,,1 d = teJ
'' LB(D : lQti -,.16i -g u'
-) [q(D :6d
in er iov
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Solrrtion -t 9 :-
LetABcD b€ o cactic cfuod.r itqt-<ya.r, qr.rd Le t_. <)
be Lae .eytLre of h.e co-ryelpaY.dih5 ci{cte.
Tt €n, €ach side oF tLr€ e? klq-teyal AgcD is <r <l"ord
D+ t'.e circl€ &nd tbe perpendrcr,Jar t isectoy.of
a cbord a,.l r,oa{ 5 P osse_6 tr-,rotr5)" lh€ aenLfe of Che
ciYcle.
6'5, ^( igKt biseceors of t_r"e 6ides 6p
AG to, r,.:tr pc16s thyoq6 br,e re,..,1r"
c oYY e6; ponding civcle .
.'olulion - po :-
Leb () be tr€ circl € c lrtue€<gqOg ciTcdYisc.ritihg
ture cgcl{c .f ectan5l e ABco.Sibc€ L*Bz _-- qF qnd
Ac i5 q cL'o{d of tJ.' e ciT.l €, So A c i S a d i.. u",e t€
cl{ q ci-f,cr< S tr",ilclvte goig q diqr-ret-1y
9uqy c,Li [<nlr y6]
O of c?' e
l-1Qo<e po"r nt o t i
^L€l
qQcLion of Acc^rcl B,)
ae^le O! Vne ( iy(l(.
(s t +r{-
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1,4
Sot!!bion
L€ t A AcD b€ or -r l.ovrrloo Lrs Sq.b tr, q t its d. i o g55..tr
Ac crn6l go intergect ato.
Sin c €rExe d ia3onaJ.: OF zi vhovot o,,is d irrkrsg.l
o.}. r r5l^.} qng l€
.'"LAc13 :,-13 Oc:-LgoD,_ LgoA =9S
No e,:, LAo4 :q,f + ciycle d_rpscrr.bed onA6 c^s &q,",€q
cDi PAs s broqgh .o.
6 ir.frqr 9 q,l_t l:ne ciycle<
co 4S <LioryoeKy pqs.-s r".'.;;:''ted
oqBc,ADqbd
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Sotul.io^ - Q E:-
C1 rV€o ABcD i< a c:,ctic ?r^a ?r<li{c.tc ycJ in n r.i(r.
TtPrrov € )- (i) Aoll Bc Ol EB:Fc. FA-E.O.
Propf,)- 1t) Sin.e E A=gp.
Then IJ:^D , L€ DA f on.posi g a.nglea to egl^oJ
Since,'qa.o is q e,.{i. s ioe sl
rh€,^ LAac+LAD<
=,1 "l'nt*"-AJ A'3. L€ @. :lEo" CLihecy
.pc-ir o f cyb i <sltlnr n
LADc -- EG. .--..(>)
Co,.?< -6e €2qc rions cr) qncl (2)
LE: AD_ Lc€ c __- c3),
Since, cor-t€Spo oclio ans le a*n
The n Bc ll Ao
cii I f{ orn ez, - @
LcAD j-LErlc..- (3)
3 i.."i1 61v19 L'F o4 : LF( B r . ---(q
eey.?a6e egl(drr.ons ci), t3 I o.,od Cq) LeG
=)[4-E< copPos, e r-.-.ter tb < iJ es
€21rc.1 .
)
4 - Ltr <E
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.5otution - 9q l-
sihce A6 is q d.iayn eten.
-rhe n 1a,o A =9J ...-- CiJ
€nre A. iS q diabeFy
-rhe n , Laoc _9d ..._(>)
Add e2uo tioD cr) a nd .>)
L3D B + LAoc =g6 +g,3
-. ) LBD c -_ rgo" .
fte ., Bo. is or-ine .
Her,.e,the cir616'a oD qh.g t(^)o
o+6er ai the F!,iid gid.e,
f Ansle i. Se,".i d,rcte-l
f nngt e in Se
'nic irc t€]
sr'de s inte rsect eo.cb
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5o t qti on -t 6:-
4!':-'-
LncA ts 61r, qngle in
'Io Pnove Ln cA 49g
'Tl^' € n LA +LB -- t*"-
lA . rroo
.S i ri ee AB cP i S q q:,<ti
=> lto"+r".)Bj
maj or 6eg ryre n t .
C-toof j B9 deg-r"e r.r e qs !r e ti-,e o 16, F4
LAoB=zLAcB.
And tA oB zrcf
Th€n, s LAca -ta f
LAca z g3
SolL^-€io n- g -1.'-
6 ive" !r"o.
Ae.D i3 q .9cri < Tr.,lP€ziu,,
u)iLt'ADIJ gc qnd LB =lo"
6111 tP, A3<o is < gq4rr, L<@ t oJ
n-€n ;g +Lb : rgf
+ r,3+1o.16;
=) [b : ig3 - td __11f,
liince A D tec
i iAt r3-6; f cr: i n teniov .rng I es]
. ? t' o.ol r it*t<'.J Th€ n LA+ r::ttJ
:) L< :- 163 -l lo'='l;i
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5o(r^*'ior, -eA t-
Le( A Ar). be 61 rrgl qnc"tp,.
' . r { rc.ngleJ c-t qng le B
Let P Be t.e ,iid poiht of brg potenqs€ Ac.
Drcr ro o. ci)rcle io ibr centre
6i A r s, LA Bc --9oo, -rhere{ore
tl-f o (^51' 6.
P qrd Ac -.." qct iq rne(t
The circie FqGSes
" 6P = Fo.dit^g
l Lso Ap=cF -_ Rad-ict<
.'. Ap- Bp.<p
Hen.e, Bp; r o-
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