Ganit (10) - वृत्त

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File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-10\Chap-10.PM65

v;k; 10

o`k

10.1 Hkwfedk vki vius nS fud thou esa ] cgqr lh ,s lh oLrq vks a os Q laioZ Q es a vo'; vk, gksa xs ftuos Q vkdkj xks y gks a % tS ls fdlh xkM+ h dk ifg;k] pwfM+;k] dbZ ?kfM+;ksa os Q Mk;y] 50 iSls] ,d

#i;k vkSj ik p #i, ew Y; osQ flDosQ] pkch os Q xq PNs] deht os Q cVu vkfn (nsf[k, vko`Qfr 10-1)A ?kM+h es a vkius ;ku fn;k gksxk fd ls osa QM dh lqbZ ?kM+h osQ Mk;y os Q ij

tYnh&tYnh pDdj yxkrh gS rFkk bldk ,d fljk ,d xksy iFk esa pyrk gSA lsos aQM dh lq bZ os Q fljs ls curk gq vk iFk ,d o k(circle ) dgykrk gSA bl v;k; esa] vki o`k] blls lacaf/r vU; inks a rFkk o`k os Q oqQN xq.kks a os Q ckjs es a v;;u djsa xsA

vko` Qfr 10.1
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10.2 o` k vkSj blls la caf/r in % ,d iq ujkoyksdu ,d ijdkj yhft, rFkk bles a ,d isa fly yxkb,A bldk uqdhyk fljk ,d dkxt os Q i` "B os Q ,d fcUnq ij jf[k,A nwljh Hkq tk dks oqQN nw jh rd [kksfy,A uq dhys fljs dks mlh fcUnq ij fLFkj dj nw ljh Hkq tk dks ,d pDdj ?kq ekb,A isa fly ls dkxt ij cuh vko` Qfr D;k gS\ tS lk fd vki tkurs gS a fd ;g ,d o`k gS(nsf[k, vko`Qfr 10-2)A vkius o` k oS Qls iz kIr fd;k\vkius ,d fcUnq A dks fLFkj j[kk rFkk os lHkh fcUnq

cuk, tks A ls ,d fLFkj nw jh ij gS aA bl iz dkj] gesa fuEu ifjHkk"kk iz kIr gqbZ %,d ry ij mu lHkh fcUnqvksa dk lewg] tks ry

os Q ,d fLFkj fcUnq ls ,d fLFkj nwjh ij fLFkr gks a] ,d ok dgykrk gSA

fLFkj fcUnq dks o`k dkos QUnz(centre ) dgrs gS a rFkk fLFkj nwjh dks o` k dhf=kT;k(radius ) dgrs gS aAvko` Qfr10.3 es a] O o` k dk os QUnz rFkk yEckbZ OP o` k

dh f=kT;k gSA

fVIi.kh : ;ku nhft, fd os QUnz dks o` k os Q fdlh fcUnq ls feykus okyk js[kk[ka M Hkh o` k dh f=kT;k dgykrk gSAvFkkZ r~ ^f=kT;k* dks nks vFkks esa iz ;ksx fd;k tkrk gS &

js [kk[ka M osQ :i es a rFkk bldh yEckbZ os Q :i es aAvkidks d{kk 6 ls fuEu esa ls oq QN vo/kj.kkvksa

dk Kku gSA ge os Qoy mudk iq u% Lej.k djrs gS aA ,d o`k ml ry dks] ftl ij og fLFkr gS] rhu

Hkkxks a es a foHkkftr djrk gSA ;s gS a % (i) o` k osQ vUnj dk Hkkx] ftls vH;a rj (interior ) Hkh dgrs gSa ] (ii) o`k ,oa

(iii) o` k os Q ckgj dk Hkkx] ft ls cfgHkkZ x (exterior ) Hkh dgrs gS a (nsf[k, vko`Qfr 10-4)A o`k rFkk bldk vH;arj feydjo`kh; {ks =k (circular region ) cukrs

gSa A

vko` Qfr 10.2

vko` Qfr 10.3

vko` Qfr 10.4

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;fn ,d o`k ij nks fcUnq P rFkk Q ysa ] rks js [kk[kaM PQ o`k dh ,dthok dg ykrk gSA (nsf[k, vko` Qfr 10-5)A ml thok dks tks o` k os Q osQUnz ls gksdj tkrh

gS] o` k dkO;kl dgrs gS aA f=kT;k os Q leku 'kCn ^O;kl* dks Hkh nks vFkks esa iz ;q Dr fd;k tkrk gS] vFkkZr~ ,d js[kk[kaM osQ :i esa rFkk bldh yEckbZ osQ :i esa A D;k vkidks o` k es a O;kl ls cM+h dksbZ vkS j thok iz kIr gks

ldrh gS\ ughaA vki ns[k ldrs gS a fd O;kl o` k dh lcls yEch thok gks rh gS rFkk lHkh O;klks a dh yEckbZ leku gksrh gS tks f=kT;k dh nks xq uh gks rh gSA vko`Qfr 10-5 es a] AOB o`k dk ,d O;kl gSA ,d o` k esa

fdrus O;kl gks ldrs gS a\ ,d o` k [khafp, vkSj ns f[k, fd vki fdrus O;kl cuk ldrs gS a\

nks fcUnqvks a os Q chp osQ o` k osQ Hkkx dks ,dpki (arc ) dgrs gS a A vko`Qfr 10-6 esa ] fcUnqvks a P rFkk Q os Q

chp osQ o` k osQ Hkkxks a dks ns f[k,A vki ik,xs fd nksuks a Hkkxks a es a ls ,d cM+ k gS rFkk ,d Nks Vk gS (nsf[k, vko`Qfr 10-7)A cM+s Hkkx dksnh?kZ pki (major arc )PQ dgrs gSa rFkk Nks Vs Hkkx dksy?kq pki (minor arc )

dgrs gS a A y?kq pki PQ dks

P ls O;Dr djrs gS a rFkk nh?kZ pki PQ dks P R ls] tgk R pki ij P rFkk Q os Q chp esa dksbZ fcUnq gSA tc rd vU;Fkk dgk u tk,] pki PQ ;k P y?kq pki dks iz n'kr djrk gSA tc P vkSj

Q ,d O;kl osQ fljs gks a] rks nksuksa pk i cjkcj gks tkrs gSa vkS j iz R;sd pki dks v/Z o`k( semicircle ) dgrs gSa A

la iw .kZ o`k dh yEckbZ dks mldhifjf/ (circumference ) dgrs gSa A thok rFkk izR;s d pki os Q e; {ks=k dks o` kh; {ks =k dk [ka M ;k ljy 'kCnksa es ao`k[kaM dgrs gS a A vki ik,xs fd nks iz dkj osQ o` k[kaM gksrs gS aA ;s gS a%nh?kZ o`k[kaM (major segment ) rFkky?kq o`k[kaM (minor segment )(nsf[k, vko`Qfr 10-8)A osQUnz dks ,d pki os Q fljksa ls feykus okyh f=kT;kvks a ,oa pki os Q chp

os Q {ks =k dksf=kT;[ka M ( sector ) dgrs gS aa A o`k[ka M dh rjg] vki ikrs gS a fd y?kq pki y?kq f=kT;[kaM os Q rFkk nh?kZ pki nh?kZ f=kT;[ka M os Q la xr gSA vko`Qfr 10-9 esa ] {ks =k OPQ y?kq f=kT;[ka M (minor sector ) rFkk 'ks "k o` kh; {ks=knh?kZ f=kT;[kaM (major sector ) gSA tc nks uks a pki cjkcj gks tkrs gS a] vFkkZ r~ iz R;s d v/Zo` k gks rk gS] rks nks uksa o`k[kaM rFkk nksuks a f=kT;[ka M ,d leku gks tkrs gS a vkS j izR;sd

dksv/Z o`kh; {ks=k( semi circular region ) dgrs gS aA

vko` Qfr 10.7

vko` Qfr 10.6

vko` Qfr 10.5

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vko` Qfr 10.8 vko`Qfr 10.9

iz'ukoyh 10.11. [kkyh LFkku Hkfj, %

(i) o` k dk osQUnz o`k os Q esa fLFkr gS (cfgHkkZx@vH;arj)A(ii) ,d fcUnq ] ftldh o`k os Q osQUnz ls nw jh f=kT;k ls vf/d gks] o`k os Q

esa fLFkr gksrk gS (cfgHkkZx@vH;arj)A(iii) o` k dh lcls cM+ h thok o` k dk gksrk gSA(iv) ,d pki gks rk gS] tc blos Q fljs ,d O;kl os Q fljs gks aA(v) o` k[kaM ,d pki rFkk osQ chp dk Hkkx gks rk gSA

(vi) ,d o`k] ftl ry ij fLFkr gS] mls Hkkxks a es a foHkkftr djrk gSA2. fyf[k,] lR; ;k vlR;A vius mkj os Q dkj.k nhft,A

(i) os QUnz dks o`k ij fdlh fcUnq ls feykus okyk js [kk[kaM o`k dh f=kT;k gks rh gSA(ii) ,d o`k es a leku yackbZ dh ifjfer thok, gksrh gS aA(iii) ;fn , d o k dks rhu cjkcj pkiksa es a ck V fn;k tk,] rks iz R;sd Hkkx nh?kZ pki gksrk gSA(iv) o` k dh ,d thok] ftldh yEckbZ f=kT;k ls nks xquh gks] o`k dk O;kl gSA(v) f=kT;[ka M] thok ,oa la xr pki os Q chp dk {ks =k gksrk gSA(vi) o` k ,d lery vko`Qfr gSA

10.3 thok }kjk ,d fcUnq ij varfjr dks .k ,d js[kk[kaM PQ rFkk ,d f cUnq R, tks js [kk PQ ij fLFkr u gks] yhft,A PR rFkk QR dks feykb, (nsf[k, vko` Qfr 10-10)A rc dks .k PRQ, js [kk[ka M PQ }kjk fcUnqR ij

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varjfr dks .k dgykrk gSA vko` Qfr 10-11 es a dks.k POQ, PRQ rFkkPSQ D;k dgykrs gS a\ POQ thok PQ }kjk os QUnz O ij varfjr dks .k gS] PRQ rFkk PSQ e'k% PQ }kjk

nh?kZ pki PQ rFkk y?kq pki PQ ij fLFkr fcUnq vksaR vkSjS ij varfjr dks .k gS aA

vko` Qfr 10.10 vko` Qfr 10.11

vkb, ge thok dh eki rFkk mlos Q }kjk osQUnz ij va rfjr dks .k es a la ca/ dh tk p djsaA vki ,d o`k es a

fofHkUu thok, [kha pdj rFkk muos Q }kjk osQUnz ij va rfjr dks .kksa dks cukdj ns [k ldrs gSa fd thok ;fn

cM+ h gksxh] rks mlos Q }kjk os QUnz ij va rfjr dks .k Hkh cM+ k gks xkA D;k gks xk ;fn vki nks cjkcj thok, ys axs\ D;k osQUnz ij varfjr dks.k leku gksa xs ;k ugha\

,d o`k dh nks ;k vf/d cjkcj thok, [khafp, rFkk os QUnz ij muos Q }kjk va rfjr dks .kksa dks ekfi, (ns f[k, vko`Qfr 10-12)A vki ik, xs fd muosQ }kjk os QUnz ij va rfjr dks.k cjkcj gS aA vkb, bl rF; dh ge miifk nsa A

iz es ; 10.1 : o`k dh cjkcj thok, os QUnz ij cjkcj dks .k varfjr djrh gS aA miifk : vkidks ,d o`k] ftldk os QUnzO gS] dh nks cjkcj thok, AB vkS j CD nh gq bZ gSa (nsf[k, vko`Qfr 10-13) rFkk vki fl djuk

pkgrs gSa fd AOB = COD gSA f=kHkq tks a AOB rFkk COD esa ]

OA = OC (,d o`k dh f=kT;k,)OB = OD (,d o`k dh f=kT;k,)AB = CD (fn;k gS)

vko` Qfr 10.12

vko` Qfr 10.13

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vr%] D AOB @ D COD (SSS fu;e) bl izdkj] ge ikrs gSa fd AOB = COD (lok xle f=kHkq tks a os Q laxr Hkkx) n

fVIi.kh : lqfo/k osQ fy, ^lok xle f=kHkq tks a os Q la xr Hkkx* os Q LFkku ij la{ks i esa CPCT dk iz;ksx fd;k tk,xk] D;ks afd tSlk fd vki ns [ks axs fd bldk ge cgq/k iz;ks x

djrs gSa Avc ;fn ,d o k dh nks thok, osQUnz ij cjkcj dks.k va rfjr djsa ] rks mu thokvkssa

os Q ckjs esa vki D;k dg ldrs gSa \ D;k os cjkcj gSa vFkok ugha\ vkb, ge bldh fuEu f;kdyki }kjk tkp djsa A

,d vDl dkxt (tracing paper) yhft, vkSj bl ij ,d o k [kha fp,A bls o`k osQ vuq fn'k dkVdj ,d pdrh (disc) izkIr dhft,A blos Q osQUnz O ij ,d dks.k AOB cukb,] tgkA, B o`k ij fLFkr fcUnq gS aA ds U ij] ,d nwljk dks.k POQ dks .k AOB osQ cjkcj cukb,A pdrh dks bu dks.kksa os Q fljksa dks feykus okyh thokvksa osQ vuqfn'k dkVs a (nsf[k, vko` Qfr 10-14)Avki nks o` k[ka M ACB rFkk PRQ izkIr djs axsA ;fn vki

,d dks nw ljs os Q ij j[ksa xs] rks vki D;k vuq Hko djs axs\ os ,d nwljs dks iw .kZ r;k

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iz'ukoyh 10.21. ;kn dhft, fd nks o` k lok xle gks rs gSa ] ;fn mudh f=kT;k, cjkcj gksa A fl dhft, fd

lokxle o` kksa dh cjkcj thok, muosQ os QUnz ksa ij cjkcj dks .k varfjr djrh gS aA2. fl dhft, fd ;fn lok xle o`kks a dh thok, muosQ os QUnz ksa ij cjkcj dks.k va rfjr djsa ]

rks thok, cjkcj gks rh gS aA

10.4 os QUnz ls thok ij yEc f;kdyki : ,d vDl dkxt ij ,d o` k [khafp,A

ekuk bldk osQUnz O gSA ,d thok AB [kha fp,A dkxt dks O ls tkus okyh ,d js [kk osQ vuqfn'k bl iz dkj eksfM+, fd thok dk ,d Hkkx nw ljs Hkkx ij iM+sA eku yhft, fd eksM+ dk fu'kku AB dks M ij dkVrk gSA rc

OMA = OMB = 90 vFkok OM, AB ij yEc gS(nsf[k, vko`Qfr 10-15)A D;k fcUnq B, A osQ la ikrh

gks rk gS\ gk] ;g gks xkA blfy,MA = MB gSA

OA vkS j OB dks feykdj rFkk ledks.k f=kHkqtks a OMA vkSj OMB dks lok xle fl dj bldh miifk Lo;a nhft,A ;g mnkgj.k fuEu ifj.kke dk fo'ks "k n`"Vka r gS% iz es ; 10.3 : ,d o`k os Q osQUnz ls ,d thok ij Mkyk x;k yEc thok dks lef}Hkkftr djrk gSA

bl iz es ; dk foykse D;k gS\ bldks fy[kus os Q fy,] loZ izFke gesa Li"V gksuk gS fd iz es ; 10-3 es a D;k fn;k x;k gS vkS j D;k fl djuk gSA fn;k gS fd osQUnz ls thok ij yac [kha pk x;k gS vkS j fl djuk gS fd og thok dks lef}Hkkftr djrk gSA vr% foykse esa ifjdYiuk gS ^;fn ,d os QUnz ls tkus okyh js[kk o` k dh ,d thok dks lef}Hkkftr djs* vkS j fl djuk gS ^js[kk thok ij yEc gS*A bl iz dkj] foyks e gS%

iz es ; 10.4 : ,d o`k os Q osQUnz ls ,d thok dks lef}Hkkftr djus osQ fy, [kha ph xbZ js[kk thok ij yac gksrh gS A

D;k ;g lR; gS\ bldks oqQN fLFkfr;ks a es a iz ;Ru djosQ ns f[k,A vki ns [ksa xs fd ;g bu lHkh fLFkfr;ks a es a lR; gSA fuEu vH;kl djos Q ns f[k, fd D;k ;g dFku O;kid :i esa lR;

vko` Qfr 10.15

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gSA ge blos Q oq QN dFku nsa xs vkS j vki buos Q dkj.k nhft,A eku yhft, fd ,d o k] ftldk os QUnzO gS] dh

AB ,d thok gS vkS j O dks AB os Q e;&fcUnq M ls feyk;k x;k gSA vkidks fl djuk gS fd OM ^ AB gSA OA vkSj OB dks feykb, (nsf[k, vko` Qfr 10-16)A f=kHkq tks a OAM rFkk OBM esa ]

OA = OB (D;ks a \)A M = BM (D;ks a \)OM = OM (mHk;fu"B)

vr%] D OAM @ D OBM (D;ks a \) blls iz kIr gksrk gS% OMA = OMB = 90 (D;ks a\)

10.5 rhu fcUnqvksa ls tkus okyk o` k v;k; 6 esa vkius i

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D;k vki rhu lajs[kh fcUnqvks a ls ,d o`k [kha p ldrs gS a \ ugha A ;fn fcUnq ,d js[kk ij fLFkr gksa ] rks rhljk fcUnq nks fcUnqvksa ls gks dj tkus okys o`k os Q vanj ;k ckgj gks xk (nsf[k, vko`Qfr 10-18)A

vr% vkb, ge rhu fcUnq A, B vkSj C ys a] tks ,d js[kk ij fLFkr u gksa ;k nw ljs 'kCnks a esa ] os lajs[kh u gks a[ ns f[k, vko` Qfr 10.19(i)]A AB rFkk BC os Q e'k% yEc lef}Hkktd

PQ vkS j RS [kha fp,A eku yhft, ;s yEc lef}Hkktd ,d fcUnq O ij iz frPNsn djrs gS a (;ku nhft, fd PQ vkS j RS ijLij iz frPNsn djsa xs] D;ks afd os lekarj ugha gS a)[ ns f[k, vko` Qfr 10.19(ii)]A

(i) (ii)

vko` Qfr 10.19

vc D;ks afd O, AB osQ yEc lef}Hkktd PQ ij fLFkr gS] blfy, OA = OB gSA;ku nhft, fd v;k; 7 es a fl fd;k x;k gS fd js[kk[kaM osQ yEc lef}Hkktd dk

iz R;sd fcUnq mlosQ va r fcUnq vks a ls cjkcj nwjh ij gksrk gSA blh izdkj] D;ks afd O, BC osQ yEc lef}Hkktd RS ij fLFkr gS a] blfy, vki ikrs

gS a fd OB = OC

blhfy, OA = OB = OC gS, ftldk vFkZ gS fd fcUnq A, B vkSj C fcUnq O ls leku nwjh ij gSa A vr% ;fn vki O dks osQUnz rFkk OA f=kT;k ys dj ,d o` k [kha ps] rks og

B vkS j C ls Hkh gks dj tk,xkA ;g n'kkZ rk gS fd rhu fcUnq vks a A, B vkS j C ls gks dj tkus okyk ,d o` k gSA vki tkurs gS a fd nks js [kk, (yEc lef}Hkktd) osQoy ,d fcUnq ij

vko` Qfr 10.18

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iz frPNs n dj ldrh gSa A nwljs 'kCnksa es a] A, B vkS j C ls gksdj tkus okyk ,d vf}rh; o` k gSA vkius vc fuEu izes; dks fl dj fy;k gS% iz es ; 10.5 : rhu fn, gq, vlajs [kh fcUnqvksa }kjk gks dj tkus okyk ,d vkS j osQoy ,d o`k gS A

fVIi.kh : ;fn ABC ,d f=kHkqt gks] rks izes ; 10-5 ls 'kh"kks A, B vkS j C ls gksdj ,d vf}rh; o` k [khapk tk ldrk gSA bl o`k dks D ABC dk ifjo`k dgrs gSa A bldk osQUnz

rFkk f=kT;k e'k% f=kHkqt osQifjosQUnz rFkkifjf=kT;k dgykrs gSa A

mnkgj.k 1 : ,d o`k dk pki fn;k gq vk gSA bl o`k dks iwjk dhft,A gy : eku yhft, ,d o`k dk pki PQ fn;k gqvk gSA gesa o` k dks iwjk djuk gSA bldk vFkZ gS fd gesa bldk os QUnz ,oa f=kT;k Kkr djuh gSA pki ij ,d fcUnq R yhft,A PR rFkk RQ dks feykb,A izes; 10-5 dks fl djus osQ fy, dh xbZ jpuk dk mi;ksx djosQ osQUnz rFkk f=kT;k Kkr dhft,A

bUgh osQUnz rFkk f=kT;k dks ys dj o`k dks iw jk dhft, (ns f[k, vko` Qfr 10-20)A

iz'ukoyh 10.31. o`kks a os Q dbZ tks M+s (;qXe) [kha fp,A izR;s d tks M+s es a fdrus fcUnq mHk;fu"B gS a\ mHk;fu"B

fcUnq vks a dh vf/dre la[;k D;k gS\2. eku yhft, vkidks ,d o`k fn;k gSA ,d jpuk blos Q os QUnz dks Kkr djus os Q fy, nhft,A3. ;fn nks o` k ijLij nks fcUnqvksa ij iz frPNsn djs a] rks fl dhft, fd muos Q osQUnz mHk;fu"B

thok osQ yEc lef}Hkktd ij fLFkr gS aA

10.6 leku thok, vkS j mudh osQUnz ls nwfj;k eku yhft, AB ,d js [kk gS vkSj P dksbZ fcUnq gSA D;ks afd ,d js [kk ij vla [; fcUnq gks rs gS a] blfy, ;fn vki bu lHkh dks P ls feyk, rks vkidks vla [;

js [kk[kaM PL 1, PL2, PM, PL3, PL4, vkfn feysa xsA bueaa s ls dkS u lh fcUnqP lsAB dh nw jh gS\ vki FkksM+k lkspdj

vko` Qfr 10.20

vko` Qfr 10.21

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bldk mkj izkIr dj ldrs gS aA bu js [kk[kaMksa ] es a lsP ls AB ij yEc js[kk[kaM vFkkZr~ vko`Qfr 10-21 esa PM lcls NksVk gks xkA xf.kr esa bl lcls NksVh yEckbZ PM dks P lsAB dh nwjh os Q :i esa ifjHkkf"kr djrs gS aA vr%] vki dg ldrs gS a fd %

,d fcUnq ls ,d js [kk ij yEc dh yEckbZ js [kk dh fcUnq ls nw jh gks rh gS A ;ku nhft, fd ;fn fcUnq js[kk ij fLFkr gS] rks js [kk dh blls nw jh 'kwU; gSA

,d o k es a vla[; thok, gks ldrh gS aA vki ,d o` k es a thok, [kha pdj tkp dj ldrs gS a fd yach thok] NksVh thok dh rqyuk es a osQUnz os Q fudV gksrh gSA bldh vki fofHkUu yEckbZ dh dbZ thok, dh [kha pdj rFkk mudh osQUnz ls nw fj;k ekidj tkp dj

ldrs gS aA O;kl] tks o`k dh lcls cM+ h thok gS] dh osQUnz ls D;k nw jh gS\ D;ksa fd os QUnz bl ij fLFkr gS] vr% bldh nw jh 'kwU; gSA D;k vki lksprs gS a fd thok dh yEckbZ vkS j mldh osQUnz ls nw jh esa dks bZ laca / gS\ vkb, ns [ks a fd D;k ,slk gSA

vko` Qfr 10.22

f;kdyki : fdlh f=kT;k dk vDl dkxt ij ,d o` k [kha fp,A bldh nks cjkcj thok, AB rFkkCD [khafp, rFkk bu ij os QUnz O ls yEc OM rFkk ON Hkh cukb,A vko` Qfr

dks bl iz dkj eksfM+, fd D, B ij rFkk C, A ij iM+s [ ns f[k, vko`Qfr 10-22 (i)]A vki ik,xs fd O eks M+ os Q fu'kku ij iM+ rk gS vkSj N, M ij iM+rk gSA vr%] OM = ON gSA bl f;kdyki dks osQUnz ks aO rFkk O osQ lokxle o`k [kha pdj vkSj vyx&vyx

cjkcj thok, AB rFkk CD ysdj nksgjk, A mu ij yEc OM rFkk O N [khafp, [ nsf[k, vko`Qfr 10-22 10.22(ii)]A bues a ls ,d o` kkdkj pdrh dks dkVdj nw ljs o`k ij

bl izdkj j[ks a fd AB, CD dks iw .kZ :i ls

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iz es ; 10.6 : ,d o`k dh(;k lokxle o`kks a dh ) cjkcj thok, osQUnz ls(;k osQUnzksa ls )leku nw jh ij gksrh gS A

vc ;g ns [kk tk, fd D;k bldk foyks e lR; gS vFkok ugha A blosQ fy, osQUnz O okyk ,d o`k [khafp,A osQUnz O ls o` k os Q Hkhrj jgus okys nks cjkcj yEckbZ osQ js[kk[kaM OL

rFkk OM [kha fp, [ ns f[k, vko`Qfr 10-23 (i)]A vc e'k% nks thok, PQ vkSj RS [kha fp, tksOL vkS j OM ij yEc gks a [ ns f[k, vko`Qfr10.23(ii)]A PQ vkSj RS dh yEckb;k ekfi,A D;k ;s vleku gS a \ ugha] nks uksa cjkcj gS a A f;kdyki dks vkSj vf/d leku js [kk[ka Mksa rFkk mu ij yEc thok, [khapdj nks gjkb,A bl izdkj] izes; 10-6 dk foyks e

vko` Qfr 10.23

lR;kfir gks tkrk gS] ftldk dFku uhps fn;k x;k gS% iz es ; 10.7 : ,d o`k osQ osQUnz ls lenwjLFk thok, yEckbZ esa leku gksrh gSa A

vc ge mi;q ZDr ifj.kkeksa ij vk/kfjr ,d mnkgj.k ys rs gS aA

mnkgj.k 2 : ;fn ,d o`k dh nks iz frPNsnh thok, iz frPNsn fcUnq ls tkus okys O;kl ls leku dks .k cuk,] rks fl dhft, fd os thok, cjkcj gS aA gy : fn;k gS fd ,d o` k] ftldk osQUnz O gS] dh nks thok, AB vkSj CD fcUnq E ij iz frPNsn djrh gS aA

E ls tkus okyk PQ ,d ,s lk O;kl gS fd AEQ = DEQ gS (nsf[k, vko`Qfr 10-24)Avkidks fl djuk gS fd AB = CD gSA thokvks a ABvkSj CD ij e'k% OL rFkk OM yEc [kha fp,A vc]

vko` Qfr10.24

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LOE = 180 90 LEO = 90 LEO (f=kHkqt os Q dks.kks a osQ ;ksx dk xq.k)= 90 AEQ = 90 DEQ

= 90 MEO = MOE

f=kHkq tks a OLE rFkk OME es a] LEO = MEO (fn;k gS) LOE = MOE (ij fl fd;k gS)

EO = EO (mHk;fu"B)vr%] D OLE @ D OM E (D;ksa \)

blls iz kIr gksrk gS% OL = OM (CPCT) blfy,] AB = CD (D;ksa \)

iz'ukoyh 10.41. 5 cm rFkk 3cm f=kT;k okys nks o` k nks fcUnq vksa ij izfrPNsn djrs gSa rFkk muos Q osQUnz ks a

osQ chp dh nwjh 4cm gSA mHk;fu"B thok dh yEckbZ Kkr dhft,A2. ;fn ,d o`k dh nks leku thok, o` k os Q vUnj izfrPNsn djsa ] rks fl dhft, fd ,d

thok osQ [ka M nwljh thok osQ laxr [ka Mksa osQ cjkcj gSa A3. ;fn ,d o`k dh nks leku thok, o`k osQ vUnj iz frPNs n djs a] rks fl dhft, fd

izfrPNs n fcUnq dks osQUnz ls feykus okyh js [kk thokvks a ls cjkcj dks.k cukrh gSA4. ;fn ,d js[kk nks la osQUnz h o` kksa (,d gh os QUnz okys

o` k) dks] ftudk os QUnz O gS] A, B, C vkSj D ij iz frPNs n djs] rks fl dhft, AB = CD gS(nsf[k, vko`Qfr 10-25)A

5. ,d ikoZQ esa cus 5 m f=kT;k okys o`k ij [kM+h rhu yM+ fd;k js'kek] lyek ,oa eunhi [ksy jgh gSa A js'kek ,d xs a n dks lyek os Q ikl] lyek eunhi os Q ikl rFkk eunhi js 'kek osQ ikl isa Qdrh gSA ;fn js'kek rFkk lyek os Q chp vkSj lyek rFkk eunhi os Q chp dh iz R;sd nw jh 6 m gks] rks js 'kek vkSj eun hi os Q chp dh nw jh D;k gS\

vko` Qfr 10.25

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6. 20 m f=kT;k dk ,d xks y ikoZ Q (o`kkdkj) ,d dkyksuh es a fLFkr gSA rhu yM+ os Q va oq Qj] lS ;n rFkk MsfoM bldh ifjlhek ij cjkcj nw jh ij cSBs gS a vkSj iz R;sd osQ

gkFk esa ,d f[kykS uk Vs yhiQks u vkil esa ckr djus os Q fy, gSA iz R;sd iQks u dh Mksjh dh yEckbZ Kkr dhft,A

10.7 ,d o`k osQ pki }kjk varfjr dks .k vkius ns [kk gS fd ,d thok osQ var fcUnq (O;kl os Q vfrfjDr) o`k dks nks pkiksa es a ,d (nh?kZ rFkk nwljk y?kq ) foHkkftr djrs gS aA ;fn vki cjkcj thok, ys a] rks vki mu pkiks a

dh ekiks a osQ ckjs es a D;k dg ldrs gS a\ D;k ,d thok }kjk cuk pki nwljh thok osQ }kjk cus pki ds cjkcj gS\ okLro es a] ;s cjkcj yEckbZ ls Hkh oqQN vf/d gSA ;g bl vFkZ es a] fd ;fn ,d pki dks nw ljs pki os Q ij j[kk tk,] rks fcuk ,sa Bs ;k eks M+s os ,d nw ljs dks iw.kZ r;k

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mijksDr xq.k ,oa iz es ; 10-1 os Q lanHkZ esa fuEu ifj.kke lR; gS%fdlh o`k osQ lokxle pki(;k cjkcj pki ) os QUnz ij cjkcj dks.k va rfjr djrs

gS aA vr%] fdlh o`k dh thok }kjk osQUnz ij va rfjr dks.k la xr (y?kq) pki }kjk osQUnz

ij varfjr dks .k os Q cjkcj gksrk gSA fuEu iz es ; ,d pki }kjk osQUnz ij varfjr dks .k rFkk o` k os Q fdlh fcUnq ij varfjr dks .k es a la ca/ nsrh gSA

iz es ; 10.8 : ,d pki }kjk os QUnz ij varfjr dks.k o`k osQ 'ks "k Hkkx osQ fdlh fcUnq ij varfjr dks.k dk nq xq uk gksrk gSA

miifk : ,d o`k dk pki PQ fn;k gS] tks osQUnz O ij POQ rFkk o` k osQ 'ks"k Hkkx osQ ,d fcUnqA ij PAQ va rfjr djrk gSA ges a fl djuk gS fd POQ = 2 PAQ gSA

vko` Qfr 10.28

vko`Qfr 10-28 es a nh xbZ rhu fofHkUu fLFkfr;ks a ij fopkj dhft,A(i) es a pki PQ y?kq gS] (ii) es a pki PQ v/Z o` k gS rFkk (iii) es a pki PQ nh?kZ gSAvkb, ge AO dks feykdj ,d fcUnq B rd c

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blls iz kIr gksrk gS% BOQ = 2 OAQ (1) blh iz dkj] BOP = 2 OAP (2)

(1) vkSj (2) ls] BOP + BOQ = 2( OAP + OAQ)vFkkZr~ ] POQ = 2 PAQ (3)

fLFkfr (iii) osQ fy,] tgk PQ nh?kZ pki gS] (3) osQ LFkku ij iz frorhZ dks .k POQ = 2 PAQ gksxkA n

fVIi.kh : eku yhft, fd mi;q ZDr vko` Qfr;ks a es a ge P vkS j Q dks feykdj thok PQ cukrs gS aA rc] PAQ dks o`k[ka M PAQP es a cuk dks .k Hkh dgrs gSa A

iz es ; 10-8 es a o` k osQ 'ks "k Hkkx ij dksbZ Hkh fcUnqA gks ldrk gSA blfy, ;fn vki o` k

os Q 'ks "k Hkkx ij ,d vkS j fcUnq C ysa (nsf[k, vko`Qfr 10-29)] rks vki ik, xs%

POQ = 2 PCQ = 2 PAQ

vr%] PCQ = PAQ ;g fuEu dks fl djrk gS%

iz es ; 10.9 : ,d gh o`k[kaM os Q dks .k cjkcj gks rs gSa A vkb, vc iz es; 10-8 dh fLFkfr (ii) dh vyx ls foospuk djs aA ;gk PAQ ml

o` k[kaM esa ,d dks .k gS tks v/Z o`k gSA lkFk gh] PAQ = 12

POQ =12

180 = 90

gSA ;fn vki dksbZ vkS j fcUnq C v/Z o`k ij ysa ] rks Hkh vki ikrs gS a fd PCQ = 90

bl izdkj] vki o` k dk ,d vkSj xq.k ikrs gS a tks fuEu gS%v/Z o`k dk dks .k ledks .k gks rk gSA

iz es ; 10-9 dk foyks e Hkh lR; gS] ftldk bl iz dkj dFku fn;k tk ldrk gS%

vko` Qfr 10.29

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iz es ; 10.10 : ;fn nks fcUnqvks a dks feykus okyk js [kk[kaM] mldks va ro"V djus okyh js[kk os Q ,d gh vksj fLFkr nks vU; fcUnqvksa ij leku dks.k va rfjr djs] rks pkjksa fcUnq ,d ok ij fLFkr gks rs gS a (vFkkZ r~ os ph; gks rs gS a )Avki bl dFku dh lR;rk fuEu iz dkj ls ns[k ldrs gS a %vko`Qfr 10-30 esa AB ,d js [kk[ka M gS] tks nks fcUnq vks a C vkS j D ij leku dks .k varfjr

djrk gSA vFkkZ r~ ACB = ADB

;g n'kkZ us os Q fy, fd fcUnq A, B, CvkSj D ,d o k ij

fLFkr gSa ] fcUnq vksa A, C vkSj B ls tkus okyk ,d o` k [khafp,A eku yhft, fd ogD ls gksdj ugha tkrk gSA rc] og AD ( vFkok c

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prqHkq Zt dh e la [;k A B C D A + C B + D1.

2.

3.

4.

5.

6.

bl lkj.kh ls vki D;k fu"d"kZ fudkyrs gS a\

;fn ekius esa dksbZ =kq fV u gq bZ gks] rks ;g fuEu dks lR;kfir djrk gS% iz es ; 10.11 : ph; prq Hkq Z t os Q lEeq[k dks.kksa os Q izR;s d ;q Xe dk ;ksx 180 gksrk gS A

okLro es a bl izes; dk foyks e] ftldk dFku fuEu izdkj ls gS] Hkh lR; gS%

iz es ; 10.12 : ;fn fdlh prqHkqZ t osQ lEeq[k dks .kks a os Q ,d ;qXe dk ;ks x 180 gks] rks prq Hkq Zt ph; gksrk gS A

bl iz es ; dh lR;rk vki izes; 10-10 esa nh xbZ fof/ dh rjg ls tkp ldrs gSa A

mnkgj.k 3 : vko` Qfr 10-32 esa ] AB o` k dk ,d O;kl gS vkS j CD f=kT;k osQ cjkcj ,d thok gSA AC vkSj BD c

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ftlls CEB =90 30 = 60,vFkkZ r~ AEB = 60 iz kIr gksrk gSA mnkgj.k 4 : vko`Qfr 10-33 esa ] ABCD ,d ph; prq HkqZ t gS] ftles a AC vkSj BD fod.kZ gSa A ;fn

DBC = 55 rFkk BAC = 45 gks] rks BCD Kkr dhft,A gy : CAD = DBC = 55 (,d o k[ka M osQ dks .k)vr%] DAB = CAD + BAC

= 55 + 45 = 100

ijUrq] DAB + BCD = 180 (ph; prqHkq Zt osQ lEeq [k dks.k) blfy,] BCD = 180 100 = 80

mnkgj.k 5 : nks o` k nks fcUnqvks a A vkS j B ij iz frPNsn djrs gSa A AD vkSj AC nksuksa o`kksa osQ O;kl gS a (nsf[k, vko`Qfr 10-34)A fl dhft, fd B js [kk[ka M DC ij

fLFkr gS aA gy : AB dks feykb,A vc]

ABD = 90 (v/Zo` k dk dks .k) ABC = 90 (v/Zo` k dk dks .k)

blfy,] ABD + ABC = 90 + 90 = 180vr%] DBC ,d js [kk gSA vFkkZ r~ B js [kk[ka M DC ij fLFkr gSA

mnkgj.k 6 : fl dhft, fd fdlh prqHkq Zt osQ var% dks .kksa osQ lef}Hkktdks aa ls cuk prqHkq Zt (;fn la Hko gks) ph; gksrk gSA gy : vko` Qfr 10-35 esa ] ABCD ,d prqHkqZ t gS ftlosQ var%dks .kksa A, B, C vkS j D osQ e'k% dks.k lef}HkktdAH, BF, CF vkSj DH ,d prqHkqZ t EFGH cukrs gS aAvc] FEH = AEB = 180 EAB EBA (D;ksa \)

vko`Qfr10.33

vko` Qfr 10.34

vko`Qfr10.35

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= 180 12

( A + B)

rFkk FGH = CGD = 180 GCD GDC (D;ks a\)

= 180 12

( C + D)

vr%] FEH + FGH = 180 12

( A + B) + 180 12

( C + D)

= 360 12

( A+ B + C + D) = 360 12

360

= 360 180 = 180

blfy,] iz es ; 10-12 ls prqHkqZ t EFGH ph; gSA

iz 'ukoyh 10.51. vko` Qfr 10-36 esa ] os QUnz O okys ,d o` k ij rhu fcUnq A,B vkS j C bl iz dkj gS a fd

BOC = 30 rFkk AOB = 60 gSA ;fn pkiABC os Q vfrfjDr o`k ijD ,d fcUnq gS] rks ADC Kkr dhft,A

2. fdlh o` k dh ,d thok o`k dh f=kT;k os Q cjkcj gSA thok }kjk y?kq pki osQ fdlh fcUnq ij va rfjr dks .k Kkr dhft, rFkk nh?kZ pki os Q fdlh fcUnq ij Hkh va rfjr dks.k Kkr dhft,A

3. vko`Qfr 10-37 es a] PQR = 100 gS, tgk P, Q rFkk R, os QUnz O okys ,d o` k ij fLFkr fcUnq gS aA OPR Kkr dhft,A

vko`Qfr 10.36 vko` Qfr 10.37

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4. vko` Qfr 10-38 esa ] ABC = 69vkSj ACB = 31 gks] rks BDC Kkr dhft,A

5. vko`Qfr 10-39 es a] ,d o k ij A, B, C vkS j D pkj fcUnq gSa AAC vkS j BD ,d fcUnq E ij bl iz dkj iz frPNs n djrs gS a fd BEC = 130 rFkk

ECD = 20 gSA BAC Kkr dhft,A

6. ABCD ,d ph; prq Hkq Z t gS ftlos Q fod.kZ ,d fcUnqE ij izfrPNsn djrs gSa A ;fn DBC = 70 vkS j BAC = 30 gks, rks BCD Kkr dhft,A iq u% ;fnAB = BC gks, rks ECD Kkr dhft,A

7. ;fn ,d ph; prqHkqZ t osQ fod.kZ mlosQ 'kh"kks ls tkus okys o`k osQ O;kl gks a] rks fl dhft, fd og ,d vk;r gS A

8. ;fn ,d leya c dh vlekarj Hkq tk, cjkcj gks a] rks fl dhft, fd og ph; gSA9. nks o` k nks fcUnqvksa B vkS j C ij iz frPNs n djrs gS aA

B ls tkus okys nks js[kk[kaM ABD vkSjPBQ o` kksa dks A, D vkS j P, Q ij e'k% izfrPNs n djrs gq ,

[kha ps x, gSa (nsf[k, vko` Qfr 10-40)A fl dhft, fd ACP = QCD gSA

vko` Qfr 10.38

vko` Qfr 10.39

vko`Qfr10.40

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o` k 223


10. ;fn fdlh f=kHkq t dh nks Hkq tkvksa dks O;kl ekudj o`k [khaps tk,] rks fl dhft, fd bu o`kks a dk izfrPNsn fcUnq rhljh Hkqtk ij fLFkr gSA

11. mHk;fu"B d.kZAC okys nks ledks.k f=kHkq t ABC vkS j ADC gSa A fl dhft, fd CAD = CBD gSA

12. fl dhft, fd p h; leka rj prq Hkq Z t vk;r gks rk gSA

iz 'ukoyh 10.6 ( ,sfPNd)*1. fl dhft, fd nks izfrPNsn djrs gq , o kksa dh os QUnz ksa dh js [kk nksukas izfrPNsn fcUnq vksa ij

leku dks.k va rfjr djrh gSA2. ,d o k dh5 cm rFkk11 cm yEch nks thok, AB vkS j CD leka rj gS a vkS j osQUnz dh

foijhr fn'kk esa fLFkr gS a A ;fnAB vkSjCD os Q chp dh nw jh 6 cm gks] rks o` k dh f=kT;k Kkr dhft,A

3. fdlh o`k dh nks leka rj thokvksa dh yEckb;k6 cm vkSj8 cm gS a A ;fn Nks Vh thok osQUnz ls 4 cm dh nw jh ij gks] rks nwljh thok os QUnz ls fdruh nw j gS\

4. eku yhft, fd dks.k ABC dk 'kh"kZ ,d o k osQ ckgj fLFkr gS vkS j dks.k dh Hkqtk, o` k ls cjkcj thok, AD vkS j CE dkVrh gSa A fl dhft, fd ABC thokvks a AC rFkkDE }kjk osQUnz ij va rfjr dks.kks a os Q va rj dk vk/k gSA

5. fl dhft, fd fdlh leprqHkq Z t dh fdlh Hkq tk dks O;kl ekudj [khapk x;k o`k mlos Q fod.kks osQ izfrPNs n fcUnq ls gksdj tkrk gSA6. ABCD ,d leka rj prq Hkq Zt gSAA, B vkSjC ls tkus okyk o`kCD (;fn vko';d gks

rks c

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PAQ bl izdkj [kha pk x;k gS fdP vkS j Q nks uks a o` kksa ij fLFkr gS aA fl dhft, fd BP = BQ gSA

10. fdlh f=kHkqt ABC esa ] ;fn A dk lef}Hkktd rFkkBC dk yEc lef}Hkktd izfrPNs n djsa ] rks fl dhft, fd os D ABC osQ ifjo`k ij izfrPNs n djsa xsA

10.9 lkjka'k bl v;k; ea s] vkius fuEu fcUnqvksa dk v;;u fd;k gS%

1. ,d o`k fdlh ry os Q mu lHkh fcUnqvksa dk lew g gks rk gS] tks ry osQ ,d fLFkj fcUnq ls leku nw jh ij gksa A

2. ,d o k dh (;k lokxle o`kks a dh) cjkcj thok, osQUnz (;k laxr os QUnz ks a) ij cjkcj dks .k va rfjr djrh gSa A

3. ;fn fdlh o` k dh (;k lokxle o` kksa dh) nks thok, osQUnz ij (;k la xr os QUnz ksa ij) cjkcj dks .k va rfjr djsa ] rks thok, cjkcj gksrh gSa A

4. fdlh o` k osQ osQUnz ls fdlh thok ij Mkyk x;k yEc mls lef}Hkkftr djrk gSA5. os QUnz ls gksdj tkus okyh vkS j fdlh thok dks lef}Hkkftr djus okyh js[kk thok ij yEc

gks rh gSA6. rhu vla js[kh; fcUnq vksa ls tkus okyk ,d vkSj os Qoy ,d o`k gks rk gSA7. ,d o k dh (;k lokxle o` kks a dh) cjkcj thok, osQUnz ls (;k laxr os QUnz ksa ls) leku nw jh

ij gksrh gS aA8. ,d o k osQ osQUnz (;k lok xle o`kksa osQ os QUnz ksa ) ls leku nwjh ij fLFkr thok,a cjkcj gksrh gS a A9. ;fn fdlh o` k osQ nks pki lok xle gksa ] rks mudh la xr thok, cjkcj gksrh gSa vkS j foyks er%

;fn fdlh o`k dh nks thok, cjkcj gksa ] rks muos Q la xr pki (y?kq] nh?kZ) lok xle gksrs gS aA10. fdlh o`k dh lokxle pki os QUnz ij cjkcj dks.k va rfjr djrs gS aA11. fdlh pki }kjk os QUnz ij varfjr dks .k mlosQ }kjk o`k osQ 'ks"k Hkkx osQ fdlh fcUnq ij varfjr

dks.k dk nqxq uk gksrk gSA12. ,d o k[ka M es a cus dks .k cjkcj gksrs gS aA13. v/Z o`k dk dks.k ledks .k gksrk gSA14. ;fn nks fcUnq vks a dks feykus okyk js [kk[ka M mldks va ro"V djus okyh js [kk osQ ,d gh vksj

fLFkr nks vU; fcUnqvksa ij leku dks.k va rfjr djs] rks pkjksa fcUnq ,d o k ij fLFkr gksrs gSa A15. ph; prqHkq Zt os Q lEeq[k dks.kks a os Q izR;s d ;q Xe dk ;ksx 1800 gks rk gSA16. ;fn fdlh prqHkq Zt os Q lEeq [k dks.kks a osQ fdlh ,d ;qXe dk ;ks x 1800 gks] rks prqHkqZ t ph;

gksrk gSA

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Ganit (10) - वृत्त