ganit (9) - समान्तर चतुर्भुजों और त्रिभुजों...

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lekarj prqHkqZ tks a vkS j f=kHkq tks a os Q {ks =kiQy 183

File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-9\Chap-9.PM65

v;k; 9

leka rj prqHkqZ tksa vkSj f=kHkqtks a osQ {ks=kiQy

9.1 Hkw fedk v;k; 5 es a] vki ns [k pq osQ gS a fd T;kfefr osQ v;;u dk mn~ xe [ksrksa dh ifjlhekvksa

dks iqu%fuer djus vkS j mUgsa mi;qDr Hkkxksa es a ckVus dh izf;k esa fufgr Hkwfe ekiuks a os Q lkFk gq vkA mnkgj.kkFkZ] ,d fdlkucqf/;k osQ ikl ,d f=kHkq tkdkj [ksr Fkk vkS j og mldksviuh nks iqf=k;ks a vkSj ,d iq=k dks cjkcj&cjkcj ckVuk pkgrh FkhA mlus f=kHkq tkdkj [ksr

dk {ks=kiQy ifjdfyr fd, fcuk] os Qoy ,d Hkq tk dks rhu cjkcj Hkkxks a esa ckV fy;k vkS j bl Hkqtk dks foHkkftr djus okys nks uks a cnq vksa dks lEeq[k 'kh"kZ cnq ls feyk fn;kA bl iz dkj] [ks r rhu cjkcj Hkkxksa esa foHkkftr gks x;k vkSj mlus vius iz R;sd cPps dks ,d&,d Hkkx ns fn;kA D;k vki lks prs gSa fd bl izdkj tks mlus rhu Hkkx izkIr fd, Fks os okLro esa {ks=kiQy esa cjkcj Fks\ bl izdkj osQ iz 'uksa vkS j vU; la ca f/r leL;kvks a

os Q mkj iz kIr djus osQ fy,] ;g vko';d gS fd lery vko`Qfr;ksa osQ {ks=kiQyksa ij iq uopkj fd;k tk,] ftUgsa vki fiNyh d{kkvksa esa igys gh i

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184 xf.kr


fd fdlh vko`Qfr dk {ks =kiQy (fdlh ek=kd es a) ,d la[;k gS tks ml vko`Qfr }kjk ?ksjs x, ry osQ Hkkx ls lac (tqM+h) gks rh gSA

ge fiNyh d{kkvks a vkS j v;k; 7 osQv;;u }kjk lok xle vko` Qfr;ks a dh vo/kj.kk ls ifjfpr gS a A nks vko`Qfr;k lokxle dgh tkrh gS a] ;fn muosQ vkdkj vkS j eki leku gksa A* nw ljs 'kCnks a es a];fn nks vko`Qfr;k A vkS j B lokxle gksa (nsf[k, vko` Qfr 9-1)] rksvki ,d vDl dkxt (tracing paper) dk iz ;ks x djosQ] ,d vko` Qfr dks nwljh vko` Qfr

ij bl izdkj j[k ldrs gSa fd ,d vko`Qfr nwljh dks iwjk&iwjk

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vki vko` Qfr A os Q {ks=kiQy dks ar(A), vko` Qfr B osQ {ks =kiQy dks ar(B), vko` Qfr T osQ {ks =kiQy dks ar(T), bR;kfn ls O;Dr dj ldrs gS aA vc vki dg ldrs gSa fd fdlh vko`Qfr dk {ks =kiQy ml vko`Qfr }kjk ?ksjs x, ry osQ Hkkx ls lac (fdlh ek=kd a)uhps fn, nks xq.kks a os Q lkFk ,d la [;k gS %

(1) ;fn A vkSj B nks lokxle vko`Qfr;k gSa ] rks ar (A) = ar (B) gS rFkk (2) ;fn ,d vko`Qfr T }kjk fuer {ks =k nks vko`Qfr;ksa P vkS j Q }kjk fuer vukfrO;kih

(non-overlapping ) ryh; {ks =kksa ls fey dj cuk gS ] rks ar (T) = ar (P) + ar (Q) gks xkAvki viuh fiNyh d{kkvks a ls fofHkUu vko`Qfr;ksa ] tS ls vk;r] oxZ ] lekarj prq HkqZ t]

f=kHkqt] bR;kfn osQ {ks=kiQyksa dks Kkr djus okys oq QN lw=kksa osQ ckjs es a Hkh tkurs gSa A bl v;k; es a] bu T;kferh; vko`Qfr;ks a os Q {ks=kiQyksa osQ chp laca / dk ml izfrca / osQ va rxZ r v;;u djosQ tc ;s ,d gh vk/kj ij fLFkr gks a vkSj ,d gh lekarj js [kkvksa os Q chp

esa gksa mijks Dr lw =kks a osQ Kku dks vf/d iz cy cukus dk iz;Ru fd;k tk,xkA ;g v;;u f=kHkqtksa dh le:irk osQ oqQN ifj.kkeks a dks le>us esa Hkh cgq r mi;ksxh jgsxkA

9.2 ,d gh vk/kj ij vkSj ,d gh lekarj js[kkvks a osQ chp vko` Qfr;k uhps nh gqbZ vko` Qfr;ksa dks nsf[k, %

vko` Qfr 9.4

vko`Qfr 9.4(i) esa] leya c ABCD vkS j leka rj prqHkq Z t EFCD es a ,d Hkqtk DC mHk;fu"B gSA ge dgrs gsS a fd leyac ABCD vkS j leka rj prqHkqZ t EFCD ,d gh

vk/kj( same base

) DC ij fLFkr gS aA blh iz dkj] vko` Qfr 9.4 (ii) esa] leka rj prq Hkq Zt PQRS vkS j MNRS ,d gh vk/kj SR ij fLFkr gS a_ vko` Qfr 9.4(iii) es a] f=kHkq t ABCvkSj DBC ,d gh vk/kj BC ij fLFkr gSa rFkk vko`Qfr 9.4(iv) esa ] leka rj prq HkqZ t ABCDvkSj f=kHkqt PDC ,d gh vk/kj DC ij fLFkr gSa A
http://schools.aglasem.com/

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vc uhps nh xbZ vko`Qfr;ks a dks ns f[k, %

vko` Qfr 9.5

vko`Qfr 9.5(i) es a] Li"Vr% leyac ABCD vkS j lekarj prqHkqZ t EFCD ,d gh vk/kj DC ij fLFkr gS aA mijks Dr osQ vfrfjDr] (leyac ABCD osQ) vk/kj DC os Q

lEeq [k 'kh"kZ A vkSj B rFkk (leka rj prqHkq Zt EFCD os Q) vk/kjDC os Q lEeq [k 'kh"kZ EvkSjF, DC os Q lekarj ,d js[kk AF ij fLFkr gS aA ge dgrs gSa fd leya c ABCD vkS j

lekarj prqHkqZ t EFCD ,d gh vk/kj DC rFkk ,d gh lekarj js [kkvks a AF vkS j DCos Q chp fLFkr gS aA blh izdkj] lekarj prqHkq Zt PQRS vkSjMNRS ,d gh vk/kj SR

rFkk ,d gh leka rj js[kkvks a PN vkS j SR os Q chp fLFkr gSa [ ns f[k, vko`Qfr 9.5 (ii)], ftlesa PQRS osQ 'kh"kZ P vkS j Q rFkkMNRS os Q 'kh"kZM vkSj N vk/kjSR os Q leka rj js [kkPN

ij fLFkr gS a A blh iz dkj] f=kHkq t ABC vkS j DBC ,d gh vk/kjBC vkSj ,d gh lekarj js[kkvksa AD vkSjBC os Q chp fLFkr gS a [ ns f[k, vko`Qfr 9.5 (iii)] rFkk lekarj prq HkqZ t

ABCD vkS j f=kHkq t PCD ,d gh vk/kjDC vkS j ,d gh lekarj js[kkvks a AP vkSj DC

osQ chp fLFkr gSa [ nsf[k, vko` Qfr9.5(iv)]A blhfy,] nks vko`Qfr;k ,d gh vk/kj vkS j ,d gh lekarj js [kkvksa os Q chp fLFkr dgh tkrh gSa ] ;fn mudk ,d mHk;fu"B vk/kj(Hkq tk ) gks rFkk mHk;fu"B vk/kj osQ lEeq[k izR;sd vko`Qfr os Q 'kh"kZ (;k dk 'kh"kZ ) ml vk/kj os Q leka rj fdlh js [kk ij fLFkr gksa A

mijksDr dFku dks n`f"Vxr j[krs gq ,] vki ;g ugha dg ldrs fd vko` Qfr9.6 (i) osQD PQR vkSj D DQR ,d gh leka rj js[kkvksa l vkS j QR osQ chp fLFkr gSa A blh iz dkj] vki ;g ugha dg ldrs fd vko`Qfr9.6 (ii) osQ lekarj prq HkqZ t EFGH vkS j MNGH

vko` Qfr 9.6

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,d gh leka rj js[kkvksa EF vkSjHG os Q chp fLFkr gS a rFkk ;g fd vko`Qfr 9.6 (iii) os Q lekarj prqHkq Zt ABCDvkSj EFCD ,d gh lekarj js[kkvksa AB vkS j DC osQ chp fLFkr gS a (;|fi

buesa ,d mHk;fu"B vk/kj DC gSvkSj ;s leka rj js[kkvksa AD vkSj BC os Q chp fLFkr gS a)A vr%];g Li"V :i ls ;ku j[kuk pkfg, fd nks uksa lekarj js [kkvksa esa ls ,d mHk;fu"B vk/kj dks va ro"V djus okyh js[kk gksuh pkfg,A;ku nhft, fd vko` Qfr 9.7(i) os Q D ABC vkSj D DBE mHk;fu"B vk/kj

ij fLFkr ugha gaSA blh izdkj] vko` Qfr 9.7(ii) os QDABC vkS j leka rj prq HkqZ t PQRS ,d gh vk/kj ij fLFkr ugha ga SA

iz'ukoyh 9.11. fuEufyf[kr vko`Qfr;ksa es a ls dkSu&lh vko`Qfr;k ,d gh vk/kj vkS j ,d gh lekarj js[kkvks a

osQ chp fLFkr gS a\ ,slh fLFkfr esa ] mHk;fu"B vk/kj vkS j nksuks a leka rj js [kk, fyf[k,A

vko` Qfr 9.8

9.3 ,d gh vk/kj vkSj ,d gh leka rj js [kkvksa os Q chp lekarj prq HkqZ t vkb, vc ,d gh vk/kj vkS j ,d gh leka rj js [kkvksa os Q chp fLFkr nks leka rj prqHkq Ztksa

os Q {ks =kiQyksa osQ e; ,d la ca /] ;fn dks bZ gS rks] Kkr djus dk iz ;Ru djs aA blosQ fy,] vkb, fuEufyf[kr f;kdyki djsa %

vko` Qfr 9.7

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f;kdyki 1 : vkb, ,d vkys[k (graph) dkxt ysa vkSj ml ij vko`Qfr9.9 esa n'kkZ , vuq lkj nks leka rj prq Hkq Zt ABCD vkSj PQCD [kha psa A

vko` Qfr 9.9

mijksDr nks uksa leka rj prq HkqZ t ,d gh vk/kj DC vkS j ,d gh leka rj js[kkvksa PBvkSjDC osQ chp fLFkr gSa A vkidks ;kn gks xk fd bu lekarj prq HkqZ tksa dk {ks=kiQy oxks

dks fxudj fdl izdkj Kkr fd;k tk ldrk gSA bl fof/ esa ] nh gqbZ vko`Qfr }kjk ?ksjs x, iw .kZ oxks dh la[;k] mu oxks dh la[;k

ftldk vk/s ls vf/d Hkkx bl vko` Qfr ls f?kjk gqvk gS rFkk mu oxks dh la [;k ftudk vk/k Hkkx bl vko` Qfr ls f?kjk gqvk gS fxudj bl nh gq bZ vko` Qfr dk {ks =kiQy Kkr fd;k

tk ldrk gSA mu oxks dks NksM+ fn;k tkrk gS ftudk vk/s ls de Hkkx bl vko`Qfr ls f?kjk gqvk gSA vki ik, xs fd bu nks uks a leka rj prqHkq Ztksa dk {ks =kiQy yxHkx15 oxZ ek=kd

gSA vkys [k dkxt ij oqQN vkSj leka rj prq HkqZ t [khapdj bl f;kdyki * dks nks gjkb,Avki D;k ns[krs gS a\ D;k nksuksa leka rj prqHkq Ztks a osQ {ks=kiQy fHkUu&fHkUu gSa ;k cjkcj gS a\ okLro es a] ;s cjkcj gS aA blfy,] blls vki bl fu"d"kZ ij igq p ldrs gS a fd ,d gh vk/kj vkSj ,d gh lekarj js [kkvksa osQ chp cus lekarj prq Hkq Zt {ks =kiQy es a cjkcj gks rs gS a A

ijUrq] ;ku jf[k, ;g os Qoy ,d lR;kiu gh gSA f;kdyki 2 : dkxt dh ,d eksVh 'khV ;k xks ij ,d leka rj prq Hkq Zt ABCD [khafp,A vc] ,d js[kk[ka M

DE vko`Qfr 9.10 es a n'kkZ , vuqlkj [kha fp,Avc ,d vyx 'khV ;k xks ij ,d vDl

dkxt dh lgk;rk ls f=kHkqt A D E f=kHkq t

* bl f;kdyki dks ,d ft;kscks MZ(geoboard) dk iz ;ks x djos Q Hkh fd;k tk ldrk gSA

vko` Qfr 9.10

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ADE osQ lokxle [kha fp, vkS j 'khV es a ls bls dkV yhft,A vc D ADE dks bl izdkj jf[k, fd AD Hkq tk BC osQ la ikrh gks] tSlk fd vko` Qfr 9.11 esa n'kkZ ;k x;k gSA ;ku nhft, fd

;gk nks lekarj prq Hkq ZtABCD vkS j EECD gS a] tks ,d gh vk/kjDC vkS j ,d gh lekarj js[kkvksa AE vkS j DC os Q chp fLFkr gSa A buosQ {ks =kiQyks a os Q ckjs es a vki D;k dg ldrs gSa \

pwa fd D ADE @ D ADEvr%] ar (ADE) = ar (A DE)

lkFk gh] ar (ABCD) = ar (ADE) + ar (EBCD)= ar (A DE) + ar (EBCD)

= ar (EE CD)

vr%] nks uksa lekarj prqHkq Z t {ks =kiQy es a cjkcj gS a Avkb, vc ,s ls nks leka rj prqHkq Ztksa os Q chp es a bl la ca/ dks fl djus dk

iz ;Ru djsa A iz es ; 9.1 : ,d gh vk/kj vkSj ,d gh leka rj js [kkvks a os Q chp fLFkr leka rj prqHkqZ t {ks=kiQy es a cjkcj gksrs gS aA miifk : nks leka rj prqHkqZ t ABCD vkSj EFCD fn, gq , gS a] tks ,d gh vk/kj DC vkS j ,d gh lekarj js [kkvksa AF vkSj DC osQ chp fLFkr gS a (nsf[k, vko`Qfr9.12)A

ges a ar (ABCD) = ar (EFCD) fl djuk gSA

D ADE vkSj D BCF es sa ] DAE = CBF (AD || BC vkS j fr;Z d js [kk AF ls laxr dks .k ) (1) AED = BFC (ED || FC vkSj fr;Zd js[kk AF ls la xr dks.k ) (2)

vko` Qfr 9.11

vko` Qfr 9.12

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blfy, , ADE = BCF (f=kHkqt dk dks.k ;ksx xq.k) (3) lkFk gh] AD = BC (lekarj prqHkqZ t dh lEeq[k Hkq tk,) (4)vr%] D ADE @ D BCF [ASA fu;e rFkk (1), (3)vkSj (4) }kjk ]

blfy,] ar (ADE) = ar (BCF) (lokxle vko`Qfr;ks a os Q {ks =kiQy cjkcj gksrs gS a) (5)vc] ar (ABCD) = ar (ADE) + ar (EDCB)

= ar (BCF) + ar (EDCB) [(5) ls ]= ar (EFCD)

vr%] lekarj prq Hkq Zt ABCD vkS j EFCD {ks=kiQy esa cjkcj gS aA nvkb, vc bl iz es ; dk mi;ksx n'kkZ us osQ fy, oqQN mnkgj.k ys a % mnkgj.k 1 : vko`Qfr 9.13 es a] ABCD ,d lekarj prqHkq Zt gS vkS j EFCD ,d vk;r gSA lkFk gh]AL ^ DC gSA fl dhft, fd

(i) ar (ABCD) = ar (EFCD)

(ii) ar (ABCD) = DC AL

gy : (i) pw fd vk;r ,d lekarj prq HkqZ t Hkh gks rk gS] blfy, ar (ABCD) = ar (EFCD) ( izes; 9.1)

(ii) mijksDr ifj.kke ls] ar (ABCD) = DC FC ( vk;r dk {ks =kiQy = yEckbZ pkS M+ kbZ ) (1)

pw fd AL ^ DC gS] blfy, AFCL ,d vk;r gSAvr%] AL = FC (2)

blfy,] ar (ABCD) = DC AL [(1) vkS j (2) ls] D;k vki mijks Dr ifj.kke (ii) ls ;g ns[k ldrs gS a fd ,d leka rj prq HkqZ t dk {ks =kiQy

mldh ,d Hkqtk vkSj laxr 'kh"kZ yEc dk xq.kuiQy gksrk gS \ D;k vkidks ;kn gS fd lekarj prqHkqZ t osQ {ks=kiQy os Q bl lw =k dks vki d{kk VII es a i

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D;k vki mijksDr dFku dk foyks e fy[k ldrs gSa \ ;g bl iz dkj gS%,d gh vk/kj (;k cjkcj vk/kjks a) vkSj cjkcj {ks=kiQyks a okys lekarj prqHkq Zt ,d gh lekarj js[kkvks a os Q chp fLFkr gks rs gS aA D;k ;g foyks e lR; gS\ leka rj prq HkqZ t os Q {ks=kiQy os Q lw=k dk iz ;ks x

djosQ] bl foykse dks fl dhft,A

mnkgj.k 2 : ;fn ,d f=kHkqt vkS j ,d lekarj prqHkq Z t ,d gh vk/kj vkSj ,d gh leka rj js [kkvks a os Q chp fLFkr gks a] rks fl dhft, fd f=kHkq t dk {ks=kiQy lekarj prqHkq Zt osQ {ks =kiQy dk vk/k gksrk gSA

gy : eku yhft, D ABP vkS j lekarj prq HkqZ t ABCD ,d gh vk/kjAB vkSj ,d gh lekarj js[kkvksa AB vkS j

PC osQ chp fLFkr gSa (nsf[k, vko`Qfr 9.14)A

vki fl djuk pkgrs gSa fd ar (PAB) = 1 a r ( A B C2

gSA

,d vU; lekarj prq Hkq Z t ABQP iz kIr djus osQ fy,] BQ || AP [khafp,A vc lekarj prq HkqZ t ABQP vkS j ABCD ,d gh vk/kj AB vkS j ,d gh leka rj js [kkvksa AB vkS j PC osQ chp fLFkr gS aa A

vr%] ar (ABQP) = ar (ABCD) ( izes; 9.1 }kjk ) (1)

ijUrq D PAB @ D BQP ( fod.kZ PB leka rj prq HkqZ t ABQP dks nks lok xle f=kHkqtksa es a ckVrk gS)

vr%] ar (PAB) = ar (BQP) (2)

blfy,] ar (PAB) = 12

ar (ABQP) [(2) ls] (3)

blls iz kIr gksrk gS ar (PAB) = 1 a r ( A B C2

[(1) vkS j (3) ls ]

vko` Qfr 9.14

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iz'ukoyh 9.21. vko` Qfr9.15 esa ] ABCD ,d leka rj prq HkqZ t gS]

AE ^ DC vkS j CF ^ AD gSA ;fn AB = 16 cm,AE = 8 cm vkS j CF = 10 cm gS] rks AD Kkr dhft,A

2. ;fn E,F,G vkS j H e'k% lekarj prq Hkq Zt ABCD dh Hkq tkvks a os Q e;&fcUnq gS a] rks n'kkZb, fd

ar (EFGH) =1

a r ( A B C2

gSA

3. P vkSjQ e'k% lekarj prq HkqZ t ABCD dh Hkq tkvksa DC vkS j AD ij fLFkr fcUnq gSa A n'kkZ b, fd ar (APB) = ar (BQC) gSA

4. vko` Qfr 9.16 es a] P lekarj prqHkq Z t ABCD os Q vH;arj esa fLFkr dksbZ fcUnq gSA n'kkZ b, fd

(i) ar (APB) + ar (PCD) =1

a r ( A B C2

(ii) ar (APD) + ar (PBC) = ar (APB) + ar (PCD)

[la osQr : P ls gks dj AB osQ lekarj ,d js[kk [khafp,A]

5. vko` Qfr 9.17 es a] PQRS vkS j ABRS lekarj prq HkqZ t gS a rFkkX Hkqtk BR ij fLFkr dks bZ fcUnq gSA n'kkZ b, fd

(i) ar (PQRS) = ar (ABRS)

(ii) ar (AX S) =1

a r ( P Q R 2

6. ,d fdlku osQ ikl lekarj prq Hkq Zt PQRS osQ :i dk ,d [ks r FkkA mlusRS ij fLFkr dksbZ fcUnq A fy;k vkSj mls P vkS j Q ls feyk fn;kA [ksr fdrus Hkkxksa es a foHkkftr gks x;k gS\ bu Hkkxksa osQ vkdkj D;k gSa \ og fdlku [ks r esa xsgw vkSj nkys a cjkcj&cjkcj Hkkxks a esa vyx&vyx cks uk pkgrh gSA og ,slk oS Qls djs\

vko` Qfr 9.15

vko` Qfr 9.16

vko` Qfr 9.17

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9.4 ,d gh vk/kj vkSj ,d gh leka rj js[kkvks a osQ chp fLFkr f=kHkqt vkb, vko` Qfr 9.18 dks ns[ksa A blesa vki nks f=kHkqtABCvkSj PBC ,s ls ns[ksa xs tks ,d gh vk/kj BC vkS j ,d

gh lekarj js [kkvks a BC vkS j AP osQ chp fLFkr gS a A ,s ls f=kHkq tksa osQ {ks=kiQyksa osQ ckjs esa vki D;k dg ldrs gS a\ bl iz'u dk mkj nsus osQ fy,] vki ,d vkys[k dkxt ij ,d gh vk/kj vkS j ,d gh leka rj js [kkvksa os Q chp f=kHkqtksa os Q dbZ ;qXe cukdj vkS j oxks dks fxudj muosQ {ks=kiQyksa dks Kkr djus dk

f;kdyki dj ldrs gS aA iz R;sd ckj vki ik,xs fd ,sls nks uks a f=kHkqtks a os Q {ks =kiQy (yxHkx) cjkcj gS aA bl f;kdyki dks ,d ft;kscksMZ ys dj Hkh fd;k tk ldrk gSA vki iq u% ik, xs fd nksuks {ks=kiQy (yxHkx) cjkcj gS aA

bl iz 'u dk ,d roZ Qla xr mkj iz kIr djus osQ fy,] vki fuEu izdkj vkxs c

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vc] eku yhft, ABCD ,d lekarj prq Hkq Zt gS ftldk ,d fod.kZ AC gS (nsf[k, vko` Qfr 9.20)A vkb, eku ysa fd AN ^ DC gSA ;ku nhft, fd

D ADC @ D CBA (D;ks a \)vr%] ar (ADC) = ar (CBA) (D;ks a \)

blfy,] ar (ADC) = 1 a r ( A B C2

=1

( D C A2

(D;ks a \)

vr%]D ADC dk {ks =kiQy = 12 vk/kj DC la xr 'kh"kZyEcAN nw ljs 'kCnksa es a]fdlh f=kHkqt dk {ks=kiQy mlosQ vk/kj(,d Hkqtk ) vkS j la xr 'kh"kZ yEc (;k pkbZ ) os Q xq .kuiQy os Q vk/s osQ cjkcj gksrk gS A D;k vkidks ;kn gS fd vki f=kHkq t

os Q {ks =kiQy os Q bl lw=k osQ ckjs es a d{kk VII esa i

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=1 B D A2

=1

C D A2

( pw fd BD = CD)

=12

vk/kj 'kh"kZyEc ( D ACD dk )= ar(ACD)

mnkgj.k 4 : vko` Qfr 9.22 esa] ABCD ,d prq Hkq Zt gSvkSjBE || AC bl izdkj gS fd BE c

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4. vko` Qfr 9.24 esa ] ABC vkSjABD ,d gh vk/kj AB ij cus nks f=kHkqt gSa A ;fn js [kk[ka M C D js[kk[kaM AB ls fcUnqO ij lef}Hkkftr gks rk gS] rks n'kkZ b,

fd ar (ABC) = ar (ABD) gSA

5. D, E vkS j F e'k% f=kHkq t ABC dh Hkq tkvks aBC, CAvkS j AB osQ e;&fcUnq gSa A n'kkZ b, fd

(i) BDEF ,d leka rj prq HkqZ t gS (ii) ar (DEF) = 14

ar (ABC)

(iii) ar (BDEF) =12

ar (ABC)

6. vko` Qfr 9.25 esa ] prqHkqZ t ABCD osQ fod.kZACvkS j BD ijLij fcUnq O ij bl iz dkj izfrPNs n

djrs gSa fd OB = OD gSA ;fn AB = CD gS] rks

n'kkZ b, fd (i) ar (DOC) = ar (AOB)(ii) ar (DCB) = ar (ACB)

(iii) DA || CB ;k ABCD ,d leka rj prqHkqZ t gSA[la osQr : D vkS j B ls AC ij yEc [kha fp, A]

7. fcUnq D vkS j E e'k% D ABC dh Hkqtkvksa AB vkSjAC ij bl iz dkj fLFkr gS a fd ar (DBC) = ar (EBC) gSA n'kkZ b, fd DE || BC gSA

8. XY f=kHkqt ABC dh Hkq tk BC os Q lekarj ,d js [kk gSA ;fnBE || ACvkS j CF || AB js[kk XY ls e'k% E vkS j F ij feyrh gS a] rks n'kkZ b, fd%ar (ABE) = ar (ACF)

vko` Qfr 9.24

vko`Qfr9.25

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9. lekarj prqHkqZ t ABCD dh ,d Hkqtk AB dks ,d fcUnq P rd c

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ar (AOD) = ar (BOC) gSA fl dhft, fd ABCD ,d leya c gSA

16. vko` Qfr9.29 esa ] ar (DRC) = ar (DPC) gS vkS j ar (BDP) = ar (ARC) gSA n'kkZ b, fd nks uksa

prq HkqZ t ABCD vkS j DCPR leyac gSa A

iz 'ukoyh 9.4 ( ,s fPNd)*1. lekarj prqHkq Zt ABCD vkSj vk;rABEF ,d gh vk/kj ij fLFkr gSa vkS j muos Q {ks=kiQy

cjkcj gS a A n'kkZb, fd leka rj prq Hkq Zt dk ifjeki vk;r os Q ifjeki ls vf/d gSA2. vko` Qfr 9.30 esa ] Hkq tk BC ij nks fcUnqD vkS j E

bl iz dkj fLFkr gSa fd BD = DE = EC gSA n'kkZ b, fd ar (ABD) = ar (ADE) = ar (AEC) gSA D;k vki vc ml iz 'u dk mkj ns ldrs gSa ] tksvkius bl v;k; dh ^Hkwfedk* es a Nks M+ fn;k Fkk

fd D;kcq f/;k dk [ksr okLro esa cjkcj {ks =kiQyks a

okys rhu Hkkxksa esa foHkkftr gks x;k gS\[ fVIi.kh : ;ku nhft, fd BD = DE = EC ys us ls D ABC rhu f=kHkq tks a ABD, ADEvkS j AEC es a foHkkftr gks tkrk gS ftuos Q {ks =kiQy cjkcj gSa aA blh iz dkj] BC dks n cjkcj Hkkxks a esa fo Hkkftr djos Q vkS j bl Hkq tk dks foHkkftr djus okys fcUnq vks a dks lEeq[k 'kh"kZ A ls

feyk dj vki bl f=kHkqt dks cjkcj {ks=kiQyksa okysn f=kHkqtks a es a foHkkftr dj ldrs gSa A3. vko` Qfr 9.31 esa ] ABCD, DCFE vkSj ABFE leka rj prq Hkq Z t gS aA n'kkZb, fd ar (ADE) =

ar (BCF) gSA4. vko` Qfr 9.32 esa ] ABCD ,d lekarj prqHkq Z t gS vkS j BC dks ,d fcUnq Q rd bl iz dkj

c

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vko` Qfr 9.31 vko` Qfr 9.325. vko` Qfr 9.33 esa ] ABC vkS j BDE nks leckgq f=kHkqt bl

iz dkj gS a fd D Hkq tk BC dk e;&fcUnq gSA ;fn A EHkq tk BC dks F ij iz frPNsn djrh gS] rks n'kkZ b, fd

(i) ar (BDE) =14

ar (ABC)

(ii) ar (BDE) =12

ar (BAE)

(iii) ar (ABC) = 2 ar (BEC)(iv) ar (BFE) = ar (AFD)

(v) ar (BFE) = 2 ar (FED)

(vi) ar (FED) =18 ar (AFC)

[la osQr : EC vkS j AD dks feykb,A n'kkZb, fdBE || AC vkS j DE || AB gS] bR;kfnA]6. prq HkqZ t ABCD os Q fod.kZ AC vkSjBD ijLij fcUnqP ij izfrPNs n djrs gS a A n'kkZ b, fd

ar (APB) ar (CPD) = ar (APD) ar (BPC) gSA[la osQr : A vkS j C ls BD ij yEc [khafp, A]

7. P vkS j Q e'k% f=kHkq t ABC dh Hkqtkvksa AB vkS j BC osQ e;&fcUnq gS a rFkkR js[kk[kaM

vko` Qfr 9.33

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AP dk e;&fcUnq gSA n'kkZ b, fd%

(i) ar (PRQ) =12

ar (ARC) (ii) ar (RQC) =38 ar (ABC)

(iii) ar (PBQ) = ar (ARC)

8. vko` Qfr 9.34 eas] ABC ,d ledks.k f=kHkqt gS ftldk dks.kA ledks .k gSA BCED, ACFGvkS j ABMN e'k% Hkq tkvks a BC, CAvkS j AB ij cus oxZ gS aA js [kk[ka M AX ^ DE Hkq tk BC

dks fcUnq Y ij feyrk gSA n'kkZ b, fd%

vko` Qfr 9.34

(i) D MBC @ D ABD (ii) ar (BYXD) = 2 ar (MBC)

(iii) ar (BYXD) = ar (ABMN) (iv) D FCB @ D ACE

(v) ar (CYXE) = 2 ar (FCB) (vi) ar (CYXE) = ar (ACFG)

(vii) ar (BCED) = ar (ABMN) + ar (ACFG)

fVIi.kh :

ifj.kke(vii) izfl (lqifjfpr) ikbFkkxks jl iz es; gSA bl iz es; dh ,d ljyre miifk vki d{kk X es a i

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9.5 lkjka 'k bl v;k; esa ] vkius fuEufyf[kr fcUnqvks a dk v;;u fd;k gS%

1. ,d vko`Qfr dk {ks=kiQy ml vko`Qfr }kjk ?ksjs x, ry os Q Hkkx ls lac (fdlh ek=kd es a) ,d la [;k gks rh gSA

2. nks lokxle vko` Qfr;ks a os Q {ks=kiQy cjkcj gks rs gS a] ijUrq bldk foykse vko';d :i ls lR; ugha gSA

3. ;fn ,d vko`Qfr T }kjk fuer dks bZ ryh; {ks=k fdUgha nks vko`Qfr;ksa P vkS j Q }kjk fuer nks vukfrO;kih ryh; {ks=kks a ls fey dj cuk gS] rks ar (T) = ar (P) + ar (Q) gS, tgkar (X)vko`Qfr X dk {ks =kiQy O;Dr djrk gSA

4. nks vko` Qfr;k ,d gh vk/kj vkSj ,d gh leka rj js [kkvksa osQ chp fLFkr dgh tkrh gSa ] ;fn mues a ,d mHk;fu"B vk/kj (,d Hkqtk) gks rFkk mHk;fu"B vk/kj os Q lEeq[k iz R;sd vko` Qfr osQ 'kh"kZ (dk 'kh"kZ ) ml vk/kj os Q lekarj fdlh js [kk ij fLFkr gks aA

5. ,d gh vk/kj (;k cjkcj vk/kjks a) okys vkSj ,d gh leka rj js[kkvksa os Q chp fLFkr lekarj prq Hkq Zt {ks =kiQy esa cjkcj gksrs gSa A

6. leka rj prqHkqZ t dk {ks =kiQy mlos Q vk/kj vkSj laxr 'kh"kZyEc dk xq.kuiQy gksrk gSA7. ,d gh vk/kj (;k cjkcj vk/kjksa ) okys vkS j cjkcj {ks =kiQyks a okys lekarj prqHkq Zt ,d gh

leka rj js [kkvksa os Q chp fLFkr gks rs gSa A8. ;fn ,d f=kHkq t vkS j ,d lekarj prq HkqZ t ,d gh vk/kj vkS j ,d gh leka rj js [kkvks a os Q chp

fLFkr gksa ] rks f=kHkq t dk {ks=kiQy lekarj prq Hkq Z t osQ {ks=kiQy dk vk/k gksrk gSA9. ,d gh vk/kj (;k cjkcj vk/kjks a) okys vkSj ,d gh leka rj js [kkvksa os Q chp fLFkr f=kHkq t

{ks=kiQy esa cjkcj gks rs gS aA10. f=kHkqt dk {ks =kiQy mlos Q vk/kj vkSj la xr 'kh"kZ yEc osQ xq .kuiQy dk vk/k gksrk gSA11. ,d gh vk/kj (;k cjkcj vk/kjksa ) okys vkS j cjkcj {ks =kiQyks a okys f=kHkq t ,d gh lekarj

js [kkvksa os Q chp fLFkr gksrs gS aA12. f=kHkqt dh ,d ekf;dk mls cjkcj {ks=kiQyksa okys nks f=kHkq tks a es a foHkkftr djrh gSA

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