162 xf.krFile Name : C:\Computer Station\Maths-IX\Chapter\Chap-6\Chap-6 (02-01-2006).PM65 GEETA vè;k;8 prqHk qZt8.1Hkwfedkvki vè;k;ks a 6 vkSj 7 esaf=kHkqtks a osQ vusd xq.kksaosQ ckjs esa vè;;u dj pqosQ gS aA vki;g Hkh tkurs gS a fd rhu vlajs [k fcanqvks a dks ;qXeksaes a tks M+ us ij tks vkÑfr izkIr gksrhgS ] f=kHkqt dgykrh gSA vc] vkb, pkj fcanq vafdr djsavkSj ns[ks a fd Øekuqlkj ;qXeksaesabudks tks M+ us ij D;k vkÑfr iz kIr gks rh gSA vkÑfr8.1 è;ku nhft, fd ;fn lHkh fcanq lajs [k gksa (,d gh js [kk es a gksa)] rks ges a ,d js [kk[kaMiz kIr gks rk gS [ns f[k, vkÑfr8.1 (i)] A ;fn pkj fcanqvks a esals rhu lajs[k gksa] rks ges a ,df=kHkqt iz kIr gksrk gS [ nsf[k, vkÑfr8.1 (ii)] vkSj ;fn pkj esals dks bZrhu fcanq lajs [k ugks a] rks ges a pkj Hkqtkvks a okyh ,d vkÑfr iz kIr gksrh gS [ nsf[k, vkÑfr8.1 (iii) vkSj(iv)]A pkjks afcUnq vks a dks ,d Øe esatksM+ us ls bl iz dkj iz kIr v kÑfr pr qHkZqt (quadrilateral) dgykrh gS A bl iqLrd es a ge osQoy vkÑfr 8.1 (iii) es a fn, x, tS ls prqHkqZtks a dk ghvè;;u djs axs vkS jvkÑfr8.1 (iv) es a fn, x, tS ls prqHkZqtks a dk ughaA
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8/13/2019 Ganit (8) -
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8/13/2019 Ganit (8) -
http://slidepdf.com/reader/full/ganit-8- 2/21
prqHkZ qt 163
File Name : C:\Computer Station\Maths-IX\Chapter\Chap-6\Chap-6 (02-01-2006).PM65 GEETA
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
File Name : C:\Computer Station\Maths-IX\Chapter\Chap-6\Chap-6 (02-01-2006).PM65 GEETA
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, :
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
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
File Name : C:\Computer Station\Maths-IX\Chapter\Chap-6\Chap-6 (02-01-2006).PM65 GEETA
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
File Name : C:\Computer Station\Maths-IX\Chapter\Chap-6\Chap-6 (02-01-2006).PM65 GEETA
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
File Name : C:\Computer Station\Maths-IX\Chapter\Chap-6\Chap-6 (02-01-2006).PM65 GEETA
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