ganit (8) - चतुर्भुज

8/13/2019 Ganit (8) -

http://slidepdf.com/reader/full/ganit-8- 1/21

8/13/2019 Ganit (8) -


prqHkZ qt 163

File Name : C:\Computer Station\Maths-IX\Chapter\Chap-6\Chap-6 (02-01-2006).PM65 GEETA

,d prqHkZ qt dh pkj Hkqtk,¡] pkj dks.k vkSj pkj 'kh"kZ gks rs gSa [ ns f[k, vkÑfr 8.2 (i)]A

vkÑfr 8.2

prq HkZ qt ABCD esa , AB, BC, CD vkS j DA pkj Hkqtk,¡ gSa ; A, B, C vkSj D pkj 'kh"kZ gSa

rFkk Ð A, Ð B, Ð C

vkS j Ð D

'kh"kks ± ij cus pkj dks .k gS aAvc lEeq [k 'kh"kks± A vkSj C rFkk B vkS j D dks tksfM+, [ nsf[k, vkÑfr 8.2 (ii)]A

AC vkSj BD prq HkZq t ABCD osQ nks fod.kZ (diagonals ) dgykrs gS aA

bl vè;k; esa ] ge fofHkUu iz dkj osQ prq HkqZ tks a vkS j muosQ xq.kks a os Q ckjs esa vè;;u djsa xsA fo'ks "k rkS j ij ge lekarj prqHkZ qtks a os Q ckjs esa i<+ sa xsA

vki lksp ldrs gSa fd ge prq HkqZ tks a (;k lekarj prq HkZq tksa ) dk D;ksa vè;;u djs aAvius ifjos'k es a nsf[k,A vki vius vkl&ikl prq HkqZ t osQ vkdkj dh vusd oLrq,¡ ns[k

ldrs gSa ] tSls& vkidh d{kk dk iQ'kZ ] nhokj] Nr] f[kM+ fd;k¡] ';keiV~V] MLVj (duster)

dk iz R;s d iQyd] vkidh iqLrd dk izR;sd i` "B] i<+us dh es t dk Åijh i` "B] bR;kfnA bues a ls oq QN dks uhps fn[kk;k x;k gS (nsf[k, vkÑfr 8.3)A

vkÑfr 8.3

;|fi gekjs vkl&ikl fn[kus okyh vf/dka 'k oLrq ,¡ vk;r os Q vkdkj dh gS a] fiQj Hkh ge prqHkZ qtks a vkS j fo'ks"kdj lekarj prq Hkq Ztksa os Q ckjs es a vkSj vf/d vè;;u djs axs] D;ks afd ,d vk;r ,d leka rj prq Hkq Zt gh gS vkSj lekarj prqHkZ qt os Q lHkh xq.k vk;r os Q fy, Hkh lR; gksrs gS aA

xf .kr

d{ kk

os Q fy,

IX

iq Lrd es t ' ; k e i V ~ V

http://schools.aglasem.com/

8/13/2019 Ganit (8) -


164 xf.kr


8.2 prqHkq Zt dk dks .k ;ks x xq.k

vc] vkb, ,d prqHkZ qt osQ dks .k ;k sx xq .k dk i q ufoZ yks du djsa A

prq HkZq t os Q dks .kks a dk ;ks x 3600 gksrk gSA ge bldh tk¡ p prq HkZq t dk ,d fod.kZ [kha p dj mls nks f=kHkqtks a esa foHkkftr djos Q dj ldrs gSa A

eku yhft, ABCD ,d prqHkqZ t gS vkS j AC mldk ,d fod.kZ gS ( nsf[k, vkÑfr 8.4)A

D ADC osQ dks .kksa dk D;k ; ksx gS\

ge tkurs gS a fd Ð DAC + Ð ACD + Ð D = 180° (1)

blh izdkj] D ABC esa] Ð CAB + Ð ACB + Ð B = 180° (2)

(1) vkSj (2) dks tksM+us ij] ges a iz kIr gks rk gS %Ð DAC + Ð ACD + Ð D + Ð CAB + Ð ACB + Ð B = 180° + 180° = 360°

lkFk gh] Ð DAC + Ð CAB = Ð A vkSj Ð ACD + Ð ACB = Ð C

vr%] Ð A + Ð D + Ð B + Ð C = 360° gSAvFkkZ r~ prqHk q Zt osQ dks .kk s a dk ;ksx 360° gk s rk gS A

8.3 prqHkZ qt os Q izdkj uhps fn, x, fofHkUu prqHkZ qtks a dks nsf[k, :

vkÑfr 8.5

vkÑfr 8.4


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prqHkZ qt 165


è;ku nhft, fd :

l vkÑfr 8.5 (i) esa ] prqHkZ q t ABCD dh lEeq[k Hkqtkvksa AB vkSj CD dk ,d ;q Xe lekarj gSA vki tkurs gSa fd ;g ,d leyac (trapezium ) dgykrk gSA

l vkÑfr;ksa 8.5 (ii), (iii) , (iv) vkSj (v) es a fn, lHkh prq HkqZ tksa es a lEeq[k Hkqtkvks a os Q nks uksa ;qXe lekarj gSa A ;s prq Hkq Zt leka rj prq Hk q Z t ( parallelograms ) dgykrs gS aAvr%] vkÑfr 8.5 (ii) dk prq HkZ qt PQRS ,d lekarj prq HkZ qt gSA blh izdkj] vkÑfr;ksa 8.5 (iii), (iv) vkSj (v) esa fn, lHkh prq HkZq t lekarj prq HkZ q t gS aA

l è;ku nhft, fd vkÑfr 8.5 (iii) osQ lekarj prqHkZq t MNRS es a ,d dks.k M

ledks .k gSA ;g fo'ks"k leka rj prqHkq Zt D;k dgykrk gS\ ;kn dhft,] ;g ,d vk;r (rectangle ) dgykrk gSA

l vkÑfr 8.5 (iv) esa fn, leka rj prq HkqZ t DEFG dh lHkh Hkq tk,¡ cjkcj gS a vkSj ge tkurs gSa fd ;g ,d leprq HkZq t (rhombus ) dgykrk gSA

l vkÑfr 8.5 (v) os Q lekarj prq HkZq t ABCD es a] Ð A = 90° vkS j lHkh Hkq tk,¡ cjkcj gS aA ;g ,d oxZ ( square ) dgykrk gSA

l vkÑfr 8.5 (vi) os Q prq HkZ qt ABCD esa ] AD = CD vkS j AB = CB gS] vFkkZ r~ vklUu Hkqtkvks a os Q nks ;q Xe cjkcj gSa A ;g ,d leka rj prqHkq Zt ugha gSA ;g ,d

ir a x (kite) dgykrk gSA

è;ku nhft, fd oxZ] vk;r vkSj leprq HkZq t es a ls iz R;s d ,d leka rj prq HkZq t gks rk gSA

l ,d oxZ ,d vk;r gS vkSj ,d leprq HkZq t Hkh gSA

l ,d leka rj prq Hkq Z t ,d leya c gSA

l irax ,d lekarj prq HkZq t ugha gSA

l leya c ,d lekarj prq HkZq t ugha gS (D;ksa fd bles a lEeq[k Hkqtkvksa dk ,d ;qXe

gh leka rj gS vkS j lekarj prq Hkq Zt osQ fy, lEeq[k Hkqtkvks a os Q nksuksa ;q Xe lekarj gksus pkfg,)A

l ,d vk;r vFkok ,d leprq HkZq t ,d oxZ ugha gSA


8/13/2019 Ganit (8) -


8/13/2019 Ganit (8) -


168 xf.kr


foykse esa fn;k gq vk ekuk tkrk gSA è;ku nhft, fd iz es ; 8.2 dks fuEu :i esa Hkh fy[kk tk ldrk gS %

;fn ,d prqHkZ qt ,d lekarj prqHkZ qt gS] rks mldh lEeq[k Hkqtkvks a dk iz R;sd ;q Xe cjkcj gksrk gSA blfy,] bldk foykse fuEu gks xk %

iz es ; 8.3 : ;fn ,d pr q Hk Z qt dh lEeq [k Hkqtkvks a dk izR; s d ; q Xe cjkcj gks ] rks og ,d lekarj pr qHk Zq t gksrk gS A

D;k vki blos Q dkj.k ns ldrs gSa \

eku yhft, prq HkZ qt ABCD dh Hkqtk,¡ AB

vkSj CD cjkcj gS a vkSj lkFk gh AD = BC gS( nsf[k, vkÑfr 8.9)A fod.kZ AC [kha fp,A

Li"Vr%] D ABC @ D CDA (D;ks a\)

vr%] Ð BAC = Ð DCA

vkSj Ð BCA = Ð DAC (D;ks a\)

D;k vc vki dg ldrs gSa fd ABCD ,d leka rj prq HkZq t gS\ (D;ks a\)

vkius vHkh ns [kk gS fd ,d leka rj prq HkZ qt es a lEeq [k Hkqtkvksa dk izR;s d ;qXe cjkcj

gks rk gS vkS j foyks er% ;fn fdlh prq HkZ qt eas lEeq [k Hkqtkvks a dk iz R;sd ;q Xe cjkcj gks] rks og ,d lekarj prq HkZq t gks rk gSA D;k ge ;gh ifj.kke lEeq[k dks .kksa osQ ;qXeksa osQ ckjs esa Hkh fudky ldrs gSa \

,d leka rj prqHkZ qt [kha fp, vkS j mlosQ dks .kks a dks ekfi,A vki D;k ns[krs gS a\

lEeq [k dks.kksa dk iz R;s d ;qXe cjkcj gSA

bls oqQN vkSj leka rj prqHkZ qt ys dj nks gjkb,A blls ge ,d vU; ifj.kke ij igq¡ prs gS a] tks fuEu gS %

iz es ; 8.4 : ,d lekarj prqHkZ q t es a lEeq [k dks.k cjkcj gks rs gS aA vc] D;k bl ifj.kke dk foykse Hkh lR; gS\ gk¡] ,slk gh gSA prq HkZq t os Q dks .k ;ks x

xq .k vkSj fr;Z d js[kk }kjk izfrPNs fnr leka rj js [kkvksa osQ xq.kks a dk iz;ksx djosQ] ge ns[k

vkÑfr 8.9


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prqHkZ qt 169


vkÑfr 8.10

ldrs gSa fd mijks Dr dk foykse Hkh lR; gSA bl izdkj] ges a fuEu izes; izkIr gks rh gS%

iz es ; 8.5 : ;fn ,d prqHkZ qt esa lEeq[k dks.kks a dk izR;s d ;qXe cjkcj gks ] rks og ,d

lekarj prqHk Z qt gk s rk gSA

leka rj prq HkZq t dk ,d xq.k vkS j Hkh gSA vkb, bldk vè;;u djsa A ,d leka rj prq HkZq t ABCD [khafp, vkS j mlosQ nks uksa fod.kZ AC vkSj BD [khafp,] tks ijLij O ij izfrPNsn djrs gS a ( nsf[k, vkÑfr 8.10)A

OA, OB, OC vkS j OD dh yEckb;k¡ ekfi,A

vki D;k ns[krs gS a\ vki ns[ksa xs fd OA = OC vkSj OB = OD

gSA vFkkZ r~ O nks uksa fod.kks± dk eè;&fca nq gSA

oq QN vkS j lekarj prq HkZq t ysdj bl fØ;kdyki dks nks gjkb,A

iz R;sd ckj] vki iz kIr djsa xs fd O nks uks a fod.kks± dk eè;&fcanq gSA

bl iz dkj] ge fuEu iz es ; izkIr djrs gS a %

iz es ; 8.6 : leka rj pr q Hk q Zt osQ fod.k Z ,d nwljs dks (ijLij ) lef}Hkkftr djrs gS aA vc] ;fn ,d prqHkZ qt os Q fod.kZ ,d nwljs dks lef}Hkkftr djsa ] rks D;k gks xk\ D;k

;g ,d leka rj prq HkZq t gks xk\ okLro esa ] ;g lR; gSA

;g iz es; 8.6 osQ ifj.kke dk foyks e gSA bls uhps fn;k tk jgk gS %

iz es ; 8.7 : ;fn ,d pr q Hk Z q t os Q fod. kZ ,d nwlj s dk s lef}Hkkftr djs a] rks og ,d leka rj prq HkZq t gksrk gS A

vki bl ifj.kke os Q fy, roZ Q fuEu iz dkj ns ldrs gS a %

è;ku nhft, fd vkÑfr 8.11 esa ] ;g fn;k gS fd OA = OC vkS j OB = OD gSA

vr%] D AOB @ D COD (D;ks a \)


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170 xf.kr


blfy,] Ð ABO = Ð CDO (D;ks a \) blls gesa AB || CD iz kIr gksrk gSA

blh izdkj] BC || AD gSA

vr%] ABCD ,d leka rj prqHkZ qt gSA

vkb, vc oq QN mnkgj.k ys aA

mnkgj.k 1 : n'kkZb, fd ,d vk;r dk izR;s d dks .k ,d ledks.k gks rk gSA

gy : ;kn dhft, fd ,d vk;r D;k gks rk gSA

,d vk;r og lekarj prqHkq Zt gksrk gS ftldk ,d dks .k ledks.k gksA

eku yhft, ABCD ,d vk;r gS] ftlesa Ð A = 90° gSA

gesa n'kkZ uk gS fd Ð B = Ð C = Ð D = 90° gSA

AD || BC vkSj AB ,d fr;Zd js [kk gS ( ns f[k, vkÑfr 8.12)A

blfy,] Ð A + Ð B = 180° (fr;Z d js[kk os Q ,d gh vks j os Q va r% dks.k) ijUrq] Ð A =90° gSA

blfy,] Ð B =180° – Ð A = 180° – 90° = 90°

vc Ð C = Ð A vkSj Ð D = Ð B (lekarj prq HkZq t os Q lEeq [k dks .k) blfy,] Ð C = 90° vkS j Ð D = 90°

vr%] vk;r dk izR;sd dks .k 900 gSA

mnkgj.k 2 : n'kkZb, fd ,d leprqHkZ qt osQ fod.kZ ijLij yEc gksrs gS aA

gy : leprqHkZ qt ABCD ij fopkj dhft, ( ns f[k, vkÑfr 8.13)A

vki tkurs gS a fd AB = BC = CD = DA (D;ks a\)

vc] D AOD vkSj D COD esa ] OA = OC (leka rj prq HkZq t osQ fod.kZ ijLij lef}Hkkftr djrs gS a)

OD = OD ( mHk;fu"B )

vkÑfr 8.11

vkÑfr 8.12


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prqHkZ qt 171


AD = CD (fn;k gS)vr%] D AOD @ D COD (SSS lok±xlerk fu;e )

blfy,] Ð AOD = Ð COD (CPCT)

ijUrq] Ð AOD + Ð COD = 180° (jS f[kd ;q Xe )

blfy,] 2Ð AOD = 180°

;k] Ð AOD = 90°

vr%] leprqZ Hkq t osQ fod.kZ ijLij yEc gSa A

mnkgj.k 3 : ABC ,d lef}ckg q f=kHkqt gS] ftles a AB = AC gSA AD cfg"dks .k PAC dks lef}Hkkftr djrk gS vkSj CD || BA gS ( nsf[k, vkÑfr 8.14)A n'kkZ b, fd

(i) Ð DAC = Ð BCA vkSj (ii) ABCD ,d leka rj prq HkZ q t gSA

gy : (i) ABC ,d lef}ckgq f=kHkqt gS] ftlesa AB = AC gSA ( fn;k gS)

blfy,] Ð ABC =Ð ACB (cjkcj Hkq tkvksa osQ lEeq [k dks.k)

lkFk gh] Ð PAC = Ð ABC + Ð ACB

(f=kHkqt dk cfg"dks.k)

;k] Ð PAC = 2ÐACB (1)

vc] AD dks .k PAC dks lef}Hkkftr djrh gSA

blfy,] Ð PAC = 2Ð DAC (2)

vr%]

2Ð DAC = 2ÐACB [(1) vkSj (2 ) ls]

;k] Ð DAC = Ð ACB

(ii) vc ;s nks uks a cjkcj dks.k os ,dka rj dks .k gSa tks js[kk[kaMks a BCvkS j AD dks fr;Zd js [kk AC }kjk iz frPNsn djus ls curs gSa A

blfy,] BC || AD

vkÑfr 8.13

vkÑfr 8.14


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172 xf.kr


lkFk gh] BA || CD gSA bl iz dkj] prqHkZ qt ABCD dh lEeq [k Hkqtkvks a osQ nks uksa ;qXe leka rj gS aA

vr%] ABCD ,d leka rj prqHkZ qt gSA

mnkgj.k 4 : nks lekarj js [kkvks a l vkSj m dks ,d fr;Zd js[kk p izfrPNsn djrh gS ( nsf[k, vkÑfr 8.15)A n'kkZb, fd va r% dks .kksa osQ lef}Hkktdks a ls cuk prqHkZ qt ,d vk;r gSA

gy : ;g fn;k gS fd l | | m gS vkS j fr;Zd js[kk p bUgsa Øe'k% fcanqvksa A vkSj C ij izfrPNsn djrh gSA

Ð PAC vkSj Ð ACQ os Q lef}Hkktd B ij iz frPNsn djrs gSa vkS j Ð ACR vkS j Ð SAC osQ lef}Hkktd D ij iz frPNsn djrs gS aA

gesa n'kkZuk gS fd prqHkZ qt ABCD ,d vk;r gSA

vc] Ð PAC = ÐACR

(l || m vkSj fr;Z d js [kk p ls cus ,dka rj dks.k )

blfy,] 1

2Ð PAC =

1

2 ÐACR

vFkkZr~ ] Ð BAC = ÐACD

;s cjkcj dks.k js[kkvks a AB vkSj DC osQ fr;Zd js [kk AC }kjk izfrPNs fnr djus ls curs gSa vkSj ;s ,dka rj dks.k gS aA

blfy,] AB || DC

blh izdkj] BC || AD (Ð ACB vkSj Ð CAD ysus ij )

vr%] ABCD ,d leka rj prqHkq Zt gSA

lkFk gh] Ð PAC + Ð CAS = 180° (jSf[kd ;qXe)

blfy,] 1

2

Ð PAC +1

2

Ð CAS =1

2

× 180° = 90°

;k] Ð BAC + Ð CAD = 90°

;k] Ð BAD = 90°

vkÑfr 8.15


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prqHkZ qt 173


vkÑfr 8.16

blfy,] ABCD ,d lekarj prqHkZ qt gS ftldk ,d dks.k ledks.k gSAvr% ABCD ,d vk;r gSA

mnkgj.k 5 : n'kkZb, fd ,d lekarj prqHkZ qt os Q dks .kks a osQ lef}Hkktd ,d vk;r cukrs gS aA gy : eku yhft, P, Q, R vkS j S Øe'k% leka rj prq HkZq t ABCD osQ Ð A vkS j Ð B, Ð B vkS j Ð C,

Ð C vkS j Ð D rFkk Ð D vkS j Ð A os Q lef}Hkktdks a os Q iz frPNsn fcanq gSa ( nsf[k, vkÑfr 8.16)A

D ASD esa vki D;k ns[k ldrs gSa \

pw¡ fd DS dks .k D dks vkS j AS dks .k A dks lef}Hkkftr djrs gS a] blfy,

Ð DAS + Ð ADS =1

2 Ð A +

1

2Ð D

=1

2 (Ð A + ÐD)

=1

2 × 180°

(Ð A vkSj Ð D fr;Zd js [kk osQ ,d gh vksj os Q va r% dks .k gS a )= 90°

lkFk gh] Ð DAS + Ð ADS + Ð DSA = 180° ( f=kHkq t dk dks.k ;ksx xq .k )

;k] 90° + Ð DSA = 180°

;k] Ð DSA = 90°

vr%] Ð PSR = 90° (Ð DSA dk 'kh"kkZfHkeq[k dks.k )

blh iz dkj] ;g n'kkZ ;k tk ldrk gS fd Ð APB = 90° ;k Ð SPQ = 90° ( tSlk fd Ð DSA os Q fy, fd;k Fkk )A blh i z dkj] Ð PQR = 90° vkSj Ð SRQ = 90° gSA

blfy,] PQRS ,d ,slk prq HkZq t gS ftlos Q lHkh dks .k ledks .k gS aA

D;k ge fu"d"kZ fudky ldrs gS a fd ;g ,d vk;r gS\ vkb, bldh tk¡p djs aA

ge n'kkZ pq os Q gSa fd Ð PSR = Ð PQR = 90° vkSj Ð SPQ = Ð SRQ = 90° gS] vFkkZ r~ lEeq[k dks.kks a os Q nksuks a ;qXe cjkcj gS aA


8/13/2019 Ganit (8) -


8/13/2019 Ganit (8) -


prqHkZ qt 175


gy : (i) prq HkZ qt APCQ esa ]

AP || QC ( pw ¡fd AB || CD) (1)

AP =1

2 AB, CQ =

1

2 CD (fn;k gS)

lkFk gh] AB = CD (D;ksa \)

blfy,] AP = QC (2)

vr%] APCQ ,d leka rj prq HkZq t gSA [ (1) vkS j (2) rFkk izes; 8.8 ls]

(ii) blh izdkj] DPBQ ,d leka rj prqHkq Zt gS] D;ksafd DQ || PB vkS j DQ = PB gSA

(iii) prq HkZ qt PSQR esa ]

SP || QR (SP, DP dk ,d Hkkx gS vkS j QR, QB dk ,d Hkkx gS)

blh izdkj] SQ || PR gSA

vr%] PSQR ,d lekarj prqHkZq t gSA

iz 'ukoyh 8.1

1. ,d prqHkZ q t os Q dks. k 3 : 5 : 9 : 13 osQ vuq ikr es a gSa A bl prqHkZ q t osQ lHkh dks.k Kkr dhft,A

2. ;fn ,d lekarj prqHkZq t os Q fod.kZ cjkcj gks a] rks n'kkZ b, fd og ,d vk;r gSA

3. n'kkZ b, fd ;fn ,d prqHkZq t os Q fod.kZ ijLij ledks .k ij lef}Hkkftr djsa ] rks og ,d leprqHkZ qt gks rk gSA

4. n'kkZ b, fd ,d oxZ osQ fod.kZ cjkcj gksrs gSa vkSj ijLij ledks.k ij lef}Hkkftr djrs gS a A

5. n'kkZ b, fd ;fn ,d prqHkZ qt osQ fod.kZ cjkcj gks a vkS j ijLij lef}Hkkftr djsa ] rks og

,d oxZ gksrk gSA


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176 xf.kr


6. leka rj prq HkZ q t ABCD dk fod.kZ AC dks .k A dks lef}Hkkftr djrk gS ( nsf[k, vkÑfr 8.19)A n'kkZ b, fd

(i) ;g Ð C dks Hkh lef}Hkkftr djrk gSA

(ii) ABCD ,d leprqHkZ qt gSA

7. ABCD ,d leprqHkZ qt gSA n'kkZ b, fd fod.kZ AC

dks.kks a A vkS j C nks uks a dks lef}Hkkftr djrk gS rFkk

fod.kZ BD dks.kksa B vkS j D nks uksa dks lef}Hkkftr djrk gSA

8. ABCD ,d vk;r gS ftles a fod.kZ AC nksuks a dks.kksa A vkS j C dks lef}Hkkftr djrk gSA

n'kkZ b, fd (i) ABCD ,d oxZ gS (ii) fod.kZ BD nks uksa dks .kks a BvkSj D dks lef}Hkkftr djrk gS

9. leka rj prqHkZ qt ABCD osQ fod.kZ BD ij nks fcanq

P vkSj Q bl izdkj fLFkr gSa fd DP = BQ gS( nsf[k, vkÑfr 8.20)A n'kkZb, fd

(i) D APD @ D CQB

(ii) AP = CQ

(iii) D AQB @ D CPD

(iv) AQ = CP

(v) APCQ ,d leka rj prq HkZq t gSA

10. ABCD ,d leka rj prqHkZ t gS rFkk AP vkSj CQ

'kh"kks± A vkSj C ls fod.kZ BD ij Øe'k% yEc gS a ( nsf[k, vkÑfr 8.21)A n'kkZb, fd

(i) D APB @ D CQD

(ii) AP = CQ

vkÑfr 8.20

vkÑfr 8.21

vkÑfr 8.19


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prqHkZ qt 177


vkÑfr 8.23

11. D ABC vkS j D DEF esa ] AB = DE, AB || DE, BC = EF

vkS j BC || EF gSA 'kh"kks ± A, B vkS j C dks Øe'k% 'kh"kks± D,

E vkS j F ls tksM+k tkrk gS (nsf[k, vkÑfr 8.22)A

n'kkZ b, fd

(i) prqHkZ qt ABED ,d leka rj prq HkZq t gSA

(ii) prqHkZ qt BEFC ,d lekarj prq HkZ qt gSA

(iii) AD || CF vkS j AD = CF gSA

(iv) prqHkZ qt ACFD ,d leka rj prq HkZq t gSA

(v ) AC = DF gSA(vi) D ABC @ D DEF gSA

12. ABCD ,d leya c gS] ftles a AB || DC vkSj AD = BC gS ( ns f[k, vkÑfr 8.23)A n'kkZb, fd

(i) Ð A = Ð B

(ii) Ð C = Ð D

(iii) D ABC @ D BAD

(iv) fod.kZ AC = fod.kZ BD gSA

[ la os Qr : AB dks c<+kb, vkS j C ls gks dj DA os Q leka rj ,d js[kk [khafp, tks c<+ h gq bZ Hkq tk

AB dks E ij iz frPNs n djsA]

8.6 eè;&fca nq iz es ;

vki ,d f=kHkq t vkSj ,d prqHkZ qt osQ vus d xq.kks a dk vè;;u dj pqos Q gS aA vkb, f=kHkqt os Q ,d vU; xq.k dk vè;;u djs a] tks ,d f=kHkq t dh Hkq tkvksa osQ eè;&fca nqvks a ls la caf/r gSA blos Q fy,] fuEufyf[kr fØ;kdyki dhft, %

,d f=kHkqt ABC [khafp, vkSj mldh nks Hkq tkvks a AB vkS j AC os Q eè;&¯cnq E vkS j F

va fdr dhft,A E vkSj F dks feykb, ( nsf[k, vkÑfr 8.24)AEF vkSj BC dks ekfi,A lkFk gh] Ð AEF vkSj Ð ABC dks Hkh ekfi,A vki D;k

ns[krs gS a\

vkÑfr 8.22


8/13/2019 Ganit (8) -


178 xf.kr


vki ik,¡ xs fd

EF =1

2 BC vkS j Ð AEF = ÐABC

gSA vr%] EF || BC gSA

oq QN vU; f=kHkq t ys dj] bl fØ;kdyki dks nksgjkb,A

bl iz dkj] vki ljyrk ls fuEu iz es ; ij igq¡ p ldrs gS a%

iz es ; 8.9 : fdlh f=kHk qt dh fdUgh nks Hkqtkvks a os Q eè;&fc anqvks a dks feykus okyk js [kk[ka M

rhljh Hkqtk os Q lek a rj gksrk gS A vki bl izes; dks fuEufyf[kr la os Qr dh lgk;rk ls fl¼ dj ldrs gS aA

vkÑfr 8.25 dks ns f[k,] ftlesa E vkS j F Øe'k%DABC dh Hkqtkvks a AB vkS j AC os Q eè;&fca nq gS a rFkk CD || BA gSA

D AEF @ D CDF (ASA fu;e )

blfy,] EF = DF vkSj BE = AE = DC (D;ksa \)

vr%] BCDE ,d leka rj prq HkZq t gSA (D;ks a\)

blls EF || BC izkIr gksrk gSA

è;ku nhft, fd EF =1

2 ED =

1

2BC gSA

D;k vki iz es ; 8.9 dk foykse fy[k ldrs gSa \ D;k ;g foyks e lR; gS\vki ns [ks axs fd Åij fn, x, iz es; dk foykse Hkh lR; gSA bls uhps fn;k tk jgk gS %

iz es ; 8.10 : fdlh f=kHk qt dh ,d H kq tk os Q eè;&fcanq l s nw ljh Hkq tk osQ lekarj [kha ph xbZ js [kk rhljh Hkqtk dks lef}Hkkftr djrh gS A

vkÑfr 8.26 es a nsf[k, fd Hkqtk AB dk eè;&fca nq

E gS vkS j E ls gksdj tkus okyh js[kk l Hkqtk BC os Q lekarj gSA lkFk gh] CM || BA gSAD AEF vkSj D CDF dh lok±xlerk dk iz ;ks x

djosQ] AF = CF fl¼ dhft,A

vkÑfr 8.24

vkÑfr 8.25

vkÑfr 8.26


8/13/2019 Ganit (8) -


prqHkZ qt 179


vkÑfr 8.28

mnkgj.k 7 : D ABC es a] D, E vkSj F Øe'k% Hkqtkvks a AB, BC vkSj CA os Q eè;&fcanq gS a ( ns f[k, vkÑfr 8.27)A n'kkZb, fd fcUnqvksa D, E vkS j F dks feykus ij D ABC pkj lok±xle f=kHkqtksa esa foHkkftr gks tkrk gSA

gy : pw¡ fd D vkS j E Øe'k% Hkq tkvksa AB vkSj BC osQ eè;&fca nq gS a] blfy, iz es; 8-9 }kjk

DE || AC

blh izdkj] DF || BC vkSj EF || AB gSA blfy,] ADEF, BDFE vkS j DFCE es a ls izR;sd ,d lekarj prq HkZq t gSA

vc] DE lekarj prq Hkq Z t BDFE dk ,d fod.kZ gSA

blfy,] D BDE @ D FED

blh izdkj] D DAF @ D FED

vkSj D EFC @ D FED

vr%] pkjks a f=kHkqt lok± xle gS aA

mnkgj.k 8 : l , m vkSj n rhu leka rj js[kk,¡ gSa ] tks fr;Zd js[kkvks a p vkSj q }kjk bl iz dkj iz frPNsfnr gSa fd l , m vkSj n js[kk p ij leku var% [kaM AB vkS j BC dkVrh gSa ( nsf[k, vkÑfr 8.28)A n'kkZb, fd l , m

vkSj n js [kk q ij Hkh leku var% [ka M DE vkSj EF

dkVrh gSa A

gy : ges a AB = BC fn;k gS vkSj ges a DE = EF fl¼ djuk gSA

vkb, A dks F ls feyk,¡ vkS j blls AF js [kk m dksG ij izfrPNsn djrh gSA leyac ACFD nks f=kHk q tks a ACF vkSj AFD es a foHkkftr gks tkrk gSA

vkÑfr 8.27


8/13/2019 Ganit (8) -


180 xf.kr


D ACF es a ;g fn;k gS fd B, Hkq tk AC dk eè;&fca nq gSA (AB = BC)

lkFk gh] BG || CF ( pw ¡ fd m || n g S )

vr%] G Hkq tk AF dk eè;&fca nq gSA ( izes ; 8.10 }kjk )

vc] D AFD es a Hkh ge blh roZ Q dk iz;ksx dj ldrs gSa A D;ks afd G Hkq tk AF dk eè;&fcanq gS vkS j GE || AD gS] blfy, izes; 8.10 ls E Hkqtk DF dk eè;&fca nq gSA

vFkkZ r~ DE = EF gSA

nw ljs 'kCnks a esa ] l , m vkSj n fr;Z d js[kk q ij Hkh cjkcj var% [ka M dkVrh gaSA

iz 'ukoyh 8.2

1. ABCD ,d prq Hkq Z t gS ftles a P, Q, R vkS j S Øe'k%Hkq tkvksa AB, BC, CD vkS j DA osQ eè;&fca nq gS a ( nsf[k, vko` Qfr 8.29)A AC mldk ,d fod.kZ gSA

n'kkZ b, fd

(i) SR || AC vkSj SR =1

2 AC gSA

(ii) PQ = SR gSA(iii) PQRS ,d lekarj prqHkZ qt gSA

2. ABCD ,d leprq HkZ qt gS vkS j P, Q, R vkSj S Øe'k% Hkq tkvks a AB, BC, CD vkS j DA osQ eè;&fca nq gSA n'kkZb, fd prq HkZ qt PQRS ,d vk;r gSA

3. ABCD ,d vk;r gS] ftles a P, Q, R vkS j S Øe'k% Hkqtkvksa AB, BC, CD vkS j DA osQ eè;&fcanq gSa A n'kkZ b, fd prq HkZq t PQRS ,d leprq HkZ qt gSA

4. ABCD ,d leya c gS] ftlesa AB || DC gSA lkFk gh] BD ,d fod.k Z gS vkS j E Hkqtk AD dk eè;&fcanq gSA E ls gks dj ,d js [kk AB os Q leka rj [kha ph xbZ gS] tks BC dks F ij izfrPNs n djrh gS ( nsf[k, vkÑfr 8.30)A n'kkZb, fd F Hkqtk BC dk eè;&fcanq gSA

vkÑfr 8.30

vkÑfr 8.29


8/13/2019 Ganit (8) -


prqHkZ qt 181


5. ,d leka rj prq HkZq t ABCD es a E vkS j F Øe'k% Hkqtkvksa AB vkS j CD os Q eè;&fca nq gSa ( ns f[k, vkÑfr 8.31)A n'kkZ b, fd js [kk[ka M AF vkS j EC fod.kZ BD dks lef=kHkkftr djrs gSa A

6. n'kkZ b, fd fdlh prq HkZ qt dh lEeq[k Hkq tkvks a osQ eè;&fca nq vks a dks feykus okys js[kk[ka M

ijLij lef}Hkkftr djrs gSa A

7. ABC ,d f=kHkq t gS ftldk dks.k C ledk s .k gSA d.kZ AB osQ eè;&fcanq M ls gks dj BC

osQ leka rj [khaph xbZ js [kk AC dks D ij iz frPNs n djrh gSA n'kkZb, fd

(i) D Hkq tk AC dk eè;&fca nq gSA (ii) MD ^ AC gSA

(iii) CM = MA =1

2AB gSA

8.7 lkjka 'k

bl vè;k; esa ] vkius fuEufyf[kr fca nq vks a dk vè;;u fd;k gS %1. fdlh prq HkZq t osQ dks.kks a dk ;ksx 360° gksrk gSA

2. leka rj prqHkZ qt dk ,d fod.kZ mls nks lok±xle f=kHkqtksa es a foHkkftr djrk gSA

3. ,d leka rj prq HkZ qt es a]

(i) lEeq [k Hkqtk,¡ cjkcj gksrh gS aA (ii) lEeq [k dks .k cjkcj gks rs gS aA

(iii) fod.kZ ijLij lef}Hkkftr djrs gSa A

4. ,d prq HkZ qt lekarj prq HkZ q t gks rk gS] ;fn

(i) lEeq [k Hkqtk,¡ cjkcj gksa _ ;k (ii) lEeq [k dks .k cjkcj gks a_ ;k (iii) fod.kZ ijLij lef}Hkkftr djrs gksa _

;k (iv) lEeq[k Hkq tkvks a dk ,d ;q Xe cjkcj gks vkSj leka rj gksA

vkÑfr 8.31


8/13/2019 Ganit (8) -


ganit (8) - चतुर्भुज

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