- cbse labs · pdf file16'a"'vs rd sharma class 9 solutions circles rd...

16'a"'Vs

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RD Sharma Class 9 Solutions Circleswww.LearnCBSE.in

RD Sharma Class 9 Solutions Circleswww.LearnCBSE.in

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,Thpr\eY-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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6olL\LiDo-o3

Rqd\us oF circle coA) = 6crn

oiStan toc):qcmJn 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

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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.rDAnd o ( :oA =Tcr

t k 1 Perpen ct kq-l qy fxor-o ce ntye. co cho 6.d[ bise.tsir:.". cp : pD_- $ cm

Anc\ A4: 86r- S crn

Tn aocP,Lry pgtr-os oycis ...,eorelan-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&>tLdrl)>+ 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 con 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&ireoltok.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 fTn 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 sLc t^-\' c!.\ \rlt€-.sects e.r,cl-, othe y a t lroino u-:ir \ bf tr, e r€qL1vqJ centJ f ofcifde.

A3 crtd c c

FO Tr^"en

Lne rr <9,

o qrd vqa[,'u5 oA, .o.optete h^q

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#'so\L.tioh-3:-SL\PPoS€ 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 qai.{C\e Po'sse < th"ro '-L€i}r u^,ese B...rv <e poi\bS ThrSiS 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 ) c\xo d 4 Lir'e gigvnent AB o{ A _

tii I Drauc rhe r\e., >(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 2airtPassibg U.{oush A and B bercrus€ !.oEen

AT\ c\rc O f r qcl it"s zcro Co itr-r cenhre A r

coi\t y\o t lrrbe'(6€ct u"'e PelpeDclic.,la1a.\ nd' uJe 63 it\ not {ird the .ent'e

.

Lo e al.raqJ

tr., e 6,y6-

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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€ (:hc6'-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 coirrrdg5iYr, tne circL\ro c enLr-e.

Tkter€ for€ rG iS Lhe 5€,n66q

,3 i\:n c\rcq.'o Rqcf iU S q A .

N\so 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

-8.1I

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,nro Tbe. o c,r\t\ be Fr,e regt"r.reJ 2e1. irrequjvcd circ\e

. 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-' Pd\iY OS c\rcleS . bavefh 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 ocLa) D-y4vJ trre pelp erd,c,^tar

1a isect:"s o{- chord Aqq"d ac

ug\ich 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-ggPa9 S eS throq5h 6re ceyrtve ,

o, il;r$ 2 P rriy\A inco"n,,on

orn no ints irl (o mw16n,a

'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'^; Aq2,; 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 Lri\l 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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So\qtion-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'c\yo e c i r<-ie]

9

}"^8-- A-9.2

'tn .A or.,r g

,-3 c rr' '

ovY I* r,r er: ogLL.l t- .. B:a= O BL

l6tq =oeaoB: {i<

oe =sc.'.nT f! aoND

OO: 9B:5c)^n f.f adiiof

ND= C9- e2 - ;-- 9crn

o Na.1 11 1't1 = "or-

oNt *(9)1 - t-g )L

ON2:),9-lG:qoN : l

So, ctis tc\nce o+ L,i55€r' c ho{d FYoyo Ci r(te i3 3(t

=6

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S ol\{tioryt6

SoluLiob-o\j-

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.radiioff, 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 trgqng\e 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 '9So,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.io\ns 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,-i\q (< )..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,ng\e ABc, k)e aa,

".,gJyi L-6, r

OAol)

-)

; 2---T

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In

B9

A ADC

u.s\nS ? 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

lAq\L

4l-Y

\c>-qBoe)

4.=9oJ?r",

Ol Str ing OC

=)3q

:ago, Leniu.

€a.cur pl.,q".ro, -,. r,,,.*'.'. -,<!o6

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5 o tuLi on -1 3"

U\eB = 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 - lzJBv 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 ofcgct 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 olcrL\o n -q ci v) :-Lae bc\veL-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 : lXltSoo_r n =?30

5o tc{tio n-q Cvl :-(^)e ho.ve' Loaa -a<"

-fhen/ LoBA --1oAB-- 3d f Anglej oFposite to € 9-qo.lradi 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< boeZqoJ rcd iil3 *=93.

50 Lution q cvrr):-

Ure hAv e,

18 Ac :<oo

o\nd LDBc:1oo

SBoc = 1p lc = 6-c," CAr'5\e ln Sq"ne Segroent]

f \. A BDc, bY qY\g\€ S qYr) Proeet E9 ' .

L8 oc f L8 cD tt-o s. = L6.i

=) 5:d +-r r-ro =tG""

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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 .

So\eiu-,5t, { tf x-l

e D .AD AB , ts rarg\e Srr.a prrop€.,.q, .

LADB *LbAB TLABD -_t3o.

:) 3:-o + Lo As +5d = lgo'

:> Lo AA .-*18 3 -3r"_srr":) LoAg:9e".

1_r\r3ti,

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)3\e 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 qng\e -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>oThe n, Ll3 pC = ;B A c __ s:o f nngle i n lsqrs\hce 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 onlecto 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? ',rLB 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 A8:6af -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 rclelr 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 .{)lLo 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.ng\eg 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 de\ree 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 4\91 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€lTh€ 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 = Itfa e-S *aS + LR rs __r8,9

d Ler 5 . 1s,l - lJ * 1o, _- .r oo

ro\qti o n -to l-

LJe ba.V€ lL^ca = rrJ ' uPf e =rz'i

:. LheB.-L-NLg = 4oo C An3\e 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€ r\qve /

R a,cli u 5 DA :.chofd AA

=> oA :oB = Ag e.

Tlren66 AD iS q. e9L!r\a 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 49in. 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 ofct 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€ h6\V€l tBoO : t63Bg 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 eglescg ct,t 9r.al ]

ol

'-) t-?A B =gS

:f 4 aD AB ,69 q ra I e S.{vD p-!.op ertjl-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_.

AY\d , 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.

;€:tcSolqtion -06.-

Gtiverr tr-,a F LAoc =166,Bv desr ee rr\eqsq{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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5o\qt,ion -c)-1 3-'--'-#

6t\ver Lhat,L-o B o = 5 j

sirrce ) AB a.rrd aD arre Er"e dio"oek vo 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 tj\e 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" -tdo,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,terallwww.Le

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SotqtioG-lo "-

(^)e F\ c\Ve / 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.=. l8oT : 9s'

LA:3a

= 3:rq4= t3Go

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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 qsiY\g q.r,gle s.qyDpTopey 19 i\a 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 eTh€ 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, F\e 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 U\Ec

:) g = Qx6 o" __

5 q cA cti c ?.1(qd,y i tqt{n r

-\ '. LAD c LAE. f nn916. i.6^n_e5 e 3>ae",r:1

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SolqtioB -rsi-

(re ho.va , Lir A D =18", u) ca = f c. rrjti 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*j1f -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 . rU\B 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\ eSe3fi€'"ll

Jn a g D c, by er-e\ a 6u, m propeytr .

LDr3 c tLBro tL.Boc = l€fBo +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? k\lq-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 o\sse_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.ritihgture 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^l\e 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..trAc crn6l go intergect ato.

Sin c €rExe d ia3onaJ.: OF zi vhovot o,,is d irrkrsg.lo.}. r r5l^.} qng l€

.'"LAc13 :,-13 Oc:-LgoD,_ LgoA =9S

No e,:, LAo4 :q,f + ciycle d_rpscrr.bed onA6 c^s &q,",€qcDi \\ PAs s \broqgh .o.

6 ir.fr\qr \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 io\e sl

rh€,^\ LAac+LAD< =,1 "l'nt*"-A\J A'3. \ L€ @. :lEo" CLihec\y

.pc-ir o f c\yb 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*nThe n Bc ll Ao

cii I f{ orn ez, - @LcAD j-LErlc..- (3)

3 i.."i1 61v19 L'F o4 : LF( B r . ---(qeey.?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(^)oo+6er ai the F!,iid gid.e,

f Ansle i. Se,".i d,rcte-l

f nngt e in Se 'nic irc t€]

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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 zrcfTh€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 oJn-€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 rrg\l qnc"tp,.' . r { rc.ngleJ c-t qng le BLet 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

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