Extra R¨vwgwZi mgvavbAwZwi³ Ask
gMwYZj (Avewk¨K)
†U÷ Q†ccvim-Gi
493
-
Extra-R¨vwgwZi mgvavb
. Dccv`¨ .
Abykxjbx-2 : †iLv, †KvY, wÎfzR I PZzfz©R m¤žwK©Z
Aby: 2.4. i¤^‡mi KY©Øq ci¯ži‡K mg‡Kv‡Y mgwØLwÊZ K‡i|
[P: †ev: 11, 09; h: †ev: 07]
DËi : mvaviY wbe©Pb : cÖgvY Ki‡Z n‡e †h, i¤^‡mi KY©Øq
ci¯ži‡K mg‡Kv‡Y mgwØLwÊZ K‡i|
A
B
×
cÖgvY Ki‡Z n‡e †h, AB + AC > BD + DC.
A¼b : CD †K ewa©Z Kwi †hb Zv AB evûi mv‡_ E we›`y‡Z wgwjZ nq|
cÖgvY : Rvbv Av‡Q, wÎfz‡Ri †h‡Kv‡bv `yB evûi mgwó Z…Zxq evû
A‡c¶v e„nËi|
∴ ∆ ACE-G AC + AE > CE
ev, AC + AE > CD + DE ... ... ... (i) [Q CE = CD + DE ]
Avevi, ∆ BED-G,
DE + BE > BD
O
... ... ... (ii)
GLb (i) bs I (ii) bs †hvM K‡i cvB,
×
D
C
we‡kl wbe©Pb : g‡b Kwi, ABCD i¤^‡mi AC Ges BD KY©Øq
ci¯ži‡K O we›`y‡Z †Q` K‡i‡Q| cÖgvY Ki‡Z n‡e †h, AO = CO
Ges BO = DO Ges ∠AOB = ∠AOD = ∠COD = ∠BOC
= GK mg‡KvY|
cÖgvY : AB I DC †iLvØq mgvšivj Ges AC I BD Zv‡`i †Q`K|
AZGe, ∠BAC = ∠ACD [GKvši †KvY]
Ges ∠BDC = ∠ABD [GKvši †KvY]
∆AOB Ges ∆COD-G,
∠OAB = ∠OCD;
∠OBA = ∠ODC
Ges AB = Abyicƒ DC
myZivs ∆ AOB ≅ ∆COD
myZivs, AO = CO Ges BO = DO
(cÖgvwYZ)
AO mvaviY evû|
∴ ∆AOB ≅ ∆AOD
myZivs ∠AOB = ∠AOD.
∴ ∠AOB + ∠AOD = GK mij‡KvY
∴ ∠AOB = ∠AOD = GK mg‡KvY|
Aevi, ∠COD = wecÖZxc ∠AOB = GK mg‡KvY|
AZGe ∠AOB = ∠AOD = ∠COD = ∠BOC =
AB + AC > 2AD.
[Kz: †ev: 08; w`: †ev: 11; P: †ev: 09, 01; wm: †ev: 09; h: †ev: 08]
A
DËi :
D
C
we‡kl wbe©Pb : ∆ABC-Gi BC evûi ga¨we›`y D| A, D †hvM
Kwi| cÖgvY Ki‡Z n‡e †h, AB + AC > 2AD.
A¼b : AD †K E ch© š Ggbfv‡e ewa© Z Kwi †hb,
AD = DE nq| E, C †hvM Kwi|
cÖgvY : ∆ABD Ges ∆ECD-G
BD = CD
[Q D, BC Gi ga¨we›`y ]
AD = DE
[ A¼b Abymv‡i ]
Ges Ašfz³
© ∠ADB = Ašfz³
© ∠EDC [ wecÖZxc †KvY e‡j ]
GK
mg‡KvY (cªgvwYZ)
Aby: 5. ∆ABC Gi Af¨š‡i D GKwU we›`y| cÖgvY Ki †h,
[wm: †ev: 01]
A
E
D
B
Aby: 6. ∆ABC Gi BC evûi ga¨we›`y D n‡j, cÖgvY Ki †h,
E
AB = AD
BO = DO
DËi :
(cÖgvwYZ)
B
GLb, ∆AOB Ges ∆AOD-G
AB + AC > BD + DC.
AC + AE + DE + BE > CD + DE + BD
ev, AC + AB + DE > CD + DE + BD
[Q AE + BE = AB]
∴ AB + AC > BD + CD.
[Dfq c¶ n‡Z DE ev` w`‡q]
C
we‡kl wbe©Pb : ∆ABC Gi Af¨š‡i D †h‡Kv‡bv GKwU we›`y| B, D
Ges C, D †hvM Ki‡j `yBwU †iLv BD Ges CD cvIqv hvq|
∴
∆ABD ≅ ∆ECD
[Q wÎfzR؇qi `yBwU evû
myZivs AB = CE ... ... ... (i)
Ges Zv‡`i Ašfz©³ †KvY mgvb]
GLb, ∆ AEC-G, AC + CE > AE
[Q wÎfz‡Ri `yB evûi mgwó Z…Zxq evû A‡c¶v e„nËi ]
[Q (i) bs †_‡K AB = CE ]
ev, AC + AB > AD + DE
ev, AB + AC > AD + AD
[Q A¼bvbymv‡i, DE = AD]
∴ AB + AC > 2AD. (cÖgvwYZ)
Aby: 7. cÖgvY Ki †h, wÎfz‡Ri ga¨gv·qi mgwó Zvi cwimxgv
A‡c¶v ¶z`ªZi|
[wm: †ev: 03]
DËi : mvaviY wbe©Pb : cÖgvY Ki‡Z n‡e †h, wÎfz‡Ri ga¨gv·qi
mgwó Zvi cwimxgv A‡c¶v ¶z`ªZi|
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
494
1
A
F
B
C
D
G
we‡kl wbe©Pb : g‡b Kwi, ∆ABC-Gi BC, CA Ges AB evûi
ga¨gv h_vµ‡g AD, BE, Ges CF; †hLv‡b D, E I F h_vµ‡g BC,
CA Ges AB evûi ga¨we›`y|
cÖgvY Ki‡Z n‡e †h, AB + BC + CA > AD + BE + CF.
A¼b : AD iwk¥‡K G ch©š ewa©Z Kwi †hb DG = AD nq| G, C
†hvM Kwi|
cªgvY : ∆ABD Ges ∆GCD-G
BD = CD
[Q D, BC Gi ga¨we›`y Kíbvbymv‡i ]
AD = GD
[ A¼b Abymv‡i ]
Ges Ašfz³
© ∠ADB = Ašfz³
© ∠GDC
[ wecÖZxc †KvY e‡j ]
[we: `ª: ewikvj †evW©-2002 cÖkœc‡Î A, B, C Gi ¯’‡j
Kzwgj−v †evW©-2009 G 'O' Gi ¯’‡j 'P' †`Iqv n‡q‡Q|]
AB + CA > 2AD ... ... ... (ii)
Abyicƒ fv‡e, cÖgvY Kiv hvq †h,
BC + AB > 2BE ... ... ... (iii)
Ges CA + BC > 2CF ... ... (iv)
(ii) bs, (iii) bs I (iv) bs †hvM K‡i cvB,
AB + CA + BC + AB + CA + BC > 2AD + 2BE + 2CF
ev, 2 (AB + BC + CA) > 2 (AD + BE + CF)
∴ AB + BC + CA > AD + BE + CF
[Dfq c¶‡K 2 Øviv fvM K‡i] (cÖgvwYZ)
Aby: 9. ∆ABC Gi ∠B I ∠C Gi mgwØLÊKØq O we›`y‡Z wgwjZ
1
[Xv: †ev: 09; w`: †ev: 10; P: †ev: 08; h: †ev 12, 10, 06, 02; wm: †ev: 10,
02,06; iv: †ev:, e: †ev: 02, Kz: †ev: 12, 06, 09]
A
P, Q, R
Ges
1
n‡j, cÖgvY Ki †h, ∠BOC = 90° – 2 ∠A. [e: †ev: 09, 04; Kz: †ev: 01]
DËi :
A
C
D
E
O
we‡kl wbe©Pb : g‡b Kwi, ∆ABC-Gi AB evû‡K D ch©š Ges AC
evû‡K E ch©š ewa©Z Kivq B Ges C we›`y‡Z `yBwU ewnt¯’ †KvY
h_vµ‡g ∠DBC Ges ∠ECB Drcbœ n‡q‡Q| GLb ∠DBC Gi
mgwØLÊK BO Ges ∠ECB Gi mgwØLÊK CO ci¯ži O we›`y‡Z
wgwjZ n‡q‡Q|
1
cÖgvY Ki‡Z n‡e †h, ∠BOC = 90° – 2 ∠A.
cÖgvY : Avgiv Rvwb, wÎfz‡Ri wZb †Kv‡Yi mgwó 180°
∴ ∆ABC-G, ∠A + ∠B + ∠C = 180°
Avevi, ∆BOC-G, ∠BOC + ∠OBC + ∠OCB = 180°
1
nq| cÖgvY Ki †h, ∠BOC = 90° + 2 ∠A.
mgwØLÊKØq]
Aby: 10. ∆ABC-Gi AB I AC evû‡K ewa©Z Ki‡j B I C we›`y‡Z †h
ewnt¯’‡KvY `yBwU Drcbœ nq, Zv‡`i mgwØLÊK `yBwU O we›`y‡Z wgwjZ
B
[Q †Kv‡bv wÎfz‡Ri `yB evûi mgwó Z…Zxq evû A‡c¶v e„nËi]
DËi :
1
1
1
∠B + ∠C = 90° – ∠A ... ... (i)
2
2
2
GLb, ∆BOC-G
∠BOC + ∠OBC + ∠OCB = 180°
1
1
ev, ∠BOC + 2 ∠B + 2 ∠C = 180°
[Q BO Ges CO †iLv h_vµ‡g ∠B I ∠C-Gi
1
ev, ∠BOC + 90° – 2 ∠A = 180° [(i) bs n‡Z]
1
ev, ∠BOC = 180° – 90° + 2 ∠A
1
∴ ∠BOC = 90° + ∠A (cÖgvwYZ)
2
Zv‡`i Ašfz©³ †KvY mgvb ]
ev, CA + AB > AD + GD [Q (i) bs †_‡K AB = CG ]
ev, CA + AB > AD + AD [A¼b Abymv‡i GD = AD ]
∴
1
∴
E
∴ ∆ ABD ≅ ∆ GCD
[Q wÎfzR؇qi `yBwU evû Ges
... ... ... (i)
myZivs AB = CG
GLb, ∆ ACG-G CA + CG > AG
1
ev, 2 ∠A + 2 ∠B + 2 ∠C = 90° [ Dfq c¶‡K 2 Øviv fvM K‡i ]
1
ev, ∠BOC + 2 ∠DBC + 2 ∠ECB = 180°
[Q BO Ges CO h_vµ‡g ∠DBC I ∠ECB Gi mgwØLÊK]
1
1
ev, ∠BOC + 2 (∠A + ∠C) + 2 (∠A + ∠B) = 180°
[Q ∆ABC-Gi ewn:¯’ ∠CBD = ∠A + ∠C Ges ewn:¯’ ∠BCE = ∠A+ ∠B]
1
ev, ∠BOC + 2 (∠A + ∠C + ∠A + ∠B) = 180°
O
1
B
C
we‡kl wbe©Pb : ∆ABC Gi ∠B Ges ∠C Gi mgwØLÊKØq O
we›`y‡Z wgwjZ n‡q‡Q| A_©vr BO Ges CO h_vµ‡g ∠ABC Ges
∠ACB Gi mgwØLÊK|
cÖgvY Ki‡Z n‡e †h,
1
∠BOC = 90° + ∠A.
2
cÖgvY : ∆ABC-G,
∠A + ∠B + ∠C = 180° [Q
wÎfz‡Ri wZb †Kv‡Yi mgwó 180°]
ev, ∠BOC + 2 (∠A + ∠B + ∠C + ∠A) = 180°
1
ev, ∠BOC + 2 (180° + ∠A) = 180° [Q ∠A + ∠B + ∠C = 180°]
1
ev, ∠BOC + 90° + 2 ∠A = 180°
1
ev, ∠BOC = 180° – 90° – 2 ∠A
∴ ∠BOC = 90° –
1
∠A
2
(cÖgvwYZ)
[we: `ª: Kzwgj−v †evW©-2001 cÖkœc‡Î O Gi ¯’‡j P †`Iqv n‡q‡Q|]
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
495
Aby: 12. cÖgvY Ki †h, wÎfz‡Ri †h‡Kv‡bv `yB evûi ga¨we›`yi
ms‡hvRK †iLvsk Z…Zxq evûi mgvšivj Ges ˆ`‡N©¨ Zvi A‡a©K|
[iv: †ev:, w`: †ev: 09; h: †ev:, 05; wm: †ev: 11, 08, 05; P: †ev: 12, 06,
03; Xv: †ev: 12, 07, 02; e: †ev: 11, 07; Kz: †ev: 11]
DËi : mvaviY wbe©Pb : cÖgvY Ki‡Z n‡e †h, wÎfz‡Ri †h‡Kv‡bv
`yB evûi ga¨we›`yi ms‡hvRK †iLvsk Z…Zxq evûi mgvšivj I
ˆ`‡N©¨ Zvi A‡a©K|
A
C
B
∴∆ABE ≅ ∆ACF [`yB †KvY I GK evû mgvb e‡j]
∴ AB = AC (cÖgvwYZ)
Aby: 15. cÖ gvY Ki †h, wÎfz ‡ Ri †h‡Kv‡bv `y B evûi Aši Zvi
Z…Z xq evû A‡c¶v ¶z` ªZ i|
[iv: †ev: 00; Kz: †ev: 01]
DËi : mvaviY wbe©Pb : cÖgvY Ki‡Z n‡e †h, wÎfz‡Ri †h‡Kv‡bv
`yB evûi Aši Z…Zxq evû A‡c¶v ¶z`ªZi|
D
F
E
cÖgvY Ki‡Z n‡e †h, AB = AC
cÖgvY : ∆ABE Ges ∆ACF -Gi g‡a¨
∠BAE = ∠CAF,
[mvaviY †KvY]
∠AEB = ∠AFC
[mg‡KvY e‡j]
Ges BE = CF
A
we‡kl wbe©Pb : g‡b Kwi, ∆ABC-Gi AB Ges AC evûi ga¨we›`y
h_vµ‡g E Ges F| E, F †hvM Kivq EF †iLvsk cvIqv hv‡e| cÖgvY
1
Ki‡Z n‡e †h, EF | | BC Ges EF = 2 BC.
A¼b : EF †K D ch©š Ggbfv‡e ewa©Z Kwi †hb EF = FD nq| D,
C †hvM Kwi|
cÖgvY : ∆AEF Ges ∆CDF-G
AF = CF
[Q Kíbvbymv‡i, F, AC Gi ga¨we›`y]
EF = FD
[ A¼b Abymv‡i ]
Ges Ašfz³
© ∠AFE = Ašfz³
© ∠CFD [wecÖZxc †KvY e‡j ]
∴ ∆ AEF ≅ ∆ CDF
myZivs AE = CD Ges ∠EAF = ∠DCF.
GLb, ∠EAF Ges ∠DCF GKvši †KvY Ges G‡`i †Q`K AC.
∴ AE | | DC.
A_©vr BE | | CD
[Q AE Ges BE GKB mij‡iLv ]
Avevi, AE = CD
ev, BE = CD
[Q E, AB Gi ga¨we›`y ]
BCDE
Gi
BE
Ges
CD
wecixZ evûØq ci¯ži
Kv‡RB PZzfR
z©
mgvb Ges mgvšivj|
Avgiv Rvwb, PZzfz©‡Ri `yBwU wecixZ evû mgvb I mgvšivj n‡j,
Zvi Aci evû `yBwUI mgvb I mgvšivj nq|
∴ ED | | BC A_©vr EF | | BC
Ges ED = BC
ev, EF + FD = BC
[Q ED = EF + FD ]
ev, EF + EF = BC
[Q A¼b Abymv‡i EF = FD ]
ev, 2 EF = BC
∴
EF =
1
BC
2
1
myZivs EF | | BC Ges EF = 2 BC. (cÖgvwYZ)
[we: `ª: XvKv †evW©-2002 G ∆BCD Gi BC I BD evû؇qi ga¨we›`yi
ms‡hvRK †iLv Z…Zxq evûi mgvšivj I ˆ`‡N©¨i A‡a©K †`Iqv n‡q‡Q|]
Aby: 14. ∆ABC-Gi B I C kxl© †_‡K wecixZ evûi Dci Aw¼Z
j¤^ h_vµ‡g BE I CF. hw` BE = CF nq, Z‡e †`LvI †h,
AB = AC.
[e: †ev: 10]
A
DËi :
F
B
D
C
B
we‡kl wbe©Pb : g‡b Kwi, ∆ ABC-Gi wZbwU evû n‡jv AB, BC
Ges CA| †hLv‡b AB > BC > AC. A_©vr AC ¶z`ªZi evû|
cÖgvY Ki‡Z n‡e †h, wÎfzRwUi †h †Kv‡bv `yB evûi Aši Z…Zxq evû
A‡c¶v ¶z`ªZi|
A¼b : AB evû †_‡K AC Gi mgvb K‡i AD Ask KvwU|
C, D †hvM Kwi|
[ A¼b Abymv‡i ]
cÖgvY : ∆ ADC-G, AC = AD
∴
∠ADC = ∠ACD
[Q wÎfz‡Ri mgvb mgvb evûi wecixZ †KvYØq ci¯ži mgvb ]
GLb, ∠ADB > ∠ACB [Q ∠ADB GK mij †KvY ev 180°]
ev, ∠ADC + ∠BDC > ∠ACD + ∠BCD
ev, ∠ACD + ∠BDC > ∠ACD + ∠BCD [Q ∠ADC = ∠ACD]
∴ ∠BDC > ∠BCD [ Dfqc¶ †_‡K ∠ACD ev` w`‡q]
A_©vr ∆ BCD-G, ∠BDC > ∠BCD
Rvbv Av‡Q, wÎfz‡Ri e„nËi †Kv‡Yi wecixZ evû ¶z`ªZi †Kv‡Yi
wecixZ evû A‡c¶v e„nËi|
∴ BC > BD
ev, BD < BC
ev, AB – AD < BC
[Q BD = AB – AD]
ev, AB – AC < BC
[Q AD = AC]
ev, AB – BC < AC
[ c¶vši K‡i]
∴ AB – BC < AC
A_©vr wÎfz‡Ri `yB evûi Aši Z…Zxq evû A‡c¶v ¶z`ªZi| (cÖgvwYZ)
Aby: 17. ∆ ABC G AB > AC Ges ∠A Gi mgwØLÊK AD, BC
evû‡K D we›`y‡Z †Q` K‡i| cÖgvY Ki †h, ∠ADB ¯’‚j‡KvY|
[Xv: †ev: 11; iv: †ev: 12, 10, 06, Kz: †ev: 10, 07, 05, e †ev: 05; wm: †ev:
07, 04; h: †ev: 11, 09; P:† ev: 10; w`: †ev: 12]
A_ev, ∆PQR G PR > PQ Ges ∠QPR Gi mgwØLÊK PC, QR
evû‡K C we›`y‡Z †Q` K‡i| cÖgvY Ki †h, ∠PCR ¯’‚j‡KvY|
[Xv: †ev: 01; P: †ev: 07, 04]
DËi :
E
A
C
we‡kl wbe©Pb : †`Iqv Av‡Q, ∆ABC-Gi B Ges C kxl©we›`y
†_‡K wecixZ evûØq (AC I AB) Gi Ici j¤^ h_vµ‡g BE
Ges CF. Avevi BE = CF †`Iqv Av‡Q|
B
D
C
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
496
we‡kl wbe©Pb : †`Iqv Av‡Q, ∆ABC-G AB > AC Ges ∠A Gi
mgwØLÊK AD, BC evûi mv‡_ D we›`y‡Z wgwjZ n‡q‡Q|
cÖgvY Ki‡Z n‡e †h, ∠ADB GKwU ¯’‚j‡KvY|
cÖgvY : ∆ABC-G AB > AC
[wÎfz‡Ri `yBwU evû Amgvb n‡j, e„nËi evûi
∴ ∠C > ∠B
wecixZ †KvY ¶z`ªZi evûi wecixZ †KvY A‡c¶v e„nËi]
1
1
Abykxjbx-4 : †¶Îdj msµvš
Aby: 1. cÖgvY Ki †h, mvgvšwi‡Ki KY©Øq mvgvšwiK †¶ÎwU‡K
[wm: †ev: 03]
PviwU mgvb wÎfzR‡¶‡Î wef³ K‡i|
DËi : mvaviY wbe©Pb : cÖgvY Ki‡Z n‡e †h, mvgvšwi‡Ki KY©Øq
mvgvšwiK †¶ÎwU‡K PviwU mgvb wÎfzR †¶‡Î wef³ K‡i|
A
B
1
ev, ∠C + 2 ∠A > ∠B + 2 ∠A [ Dfq c‡¶ 2 ∠A †hvM K‡i ]
O
1
Avevi, ∆ACD-G ewnt¯’ ∠ADB = Ašt¯’ (∠C + 2 ∠A)
Avevi, ∆ABD-G ewnt¯’ ∠ADC = Ašt¯’
1
2
1
(∠B + ∠A)
2
1
2
∴ ∠ADB > ∠ADC [Q ∠C + ∠A > ∠B + ∠A]
†h‡nZz †KvY `yBwU mwbœwnZ †KvY Ges Amgvb|
∴ 90° < ∠ADB ∠180°
∴∠ADB ¯’‚j‡KvY| (cÖgvwYZ)
Aby: 21. †`LvI †h, PZzf‡z© Ri evû¸‡jvi ga¨we›`y ch©vqµ‡g †hvM
Ki‡j GKwU mvgvšwiK Drcbœ nq|
[h: †ev: 03, 88]
DËi : mvaviY wbe©Pb : †`Lv‡Z n‡e †h, PZzfz©‡Ri evû¸‡jvi
ga¨we›`y ch©vqµ‡g †hvM Ki‡j GKwU mvgvšwiK Drcbœ nq|
A
P
B
S
Q
D
R
C
z© | AB, BC, CD
we‡kl wbe©Pb : g‡b Kwi, ABCD GKwU PZzfR
I DA evû PviwUi ga¨we›`y h_vµ‡g P, Q, R I S; ga¨we›`y¸‡jv
ch©vqµ‡g †hvM Ki‡j PQRS GKwU †¶Î cvIqv hv‡e| †`Lv‡Z
n‡e †h, PQRS GKwU mvgvšwiK|
A¼b : B, D †hvM Kwi|
cÖgvY : Rvbv Av‡Q, wÎfz‡Ri †h‡Kv‡bv `yB evûi ga¨we›`yi
ms‡hvRK mij‡iLv Z…Zxq evûi mgvšivj I ˆ`‡N©¨ Zvi A‡a©K|
∆ABD Gi AD I AB evûi ga¨we›`y h_vµ‡g S I P Ges SP
Zv‡`i ms‡hvRK mij‡iLv|
1
BD
2
CB I CD evûi
D
C
we‡kl wbe©Pb : g‡b Kwi, ABCD GKwU mvgvšwiK| Zvi AC I
BD KY©Øq ci¯ži O we›`y‡Z †Q` K‡i‡Q|
cÖgvY Ki‡Z n‡e †h, ∆ †¶Î AOB = ∆ †¶Î BOC = ∆ †¶Î
COD = ∆ †¶Î AOD.
cÖgvY : ABCD mvgvšwi‡Ki KY©Øq AC I BD ci¯ži‡K O we›`y‡Z
†Q` K‡i|
myZivs AO = OC Ges BO = OD.
[Q mvgvšwi‡Ki KY©Øq ci¯ži‡K mgwØLwÊZ K‡i]
Avgiv Rvwb, wÎfz‡Ri ga¨gv wÎfzRwU‡K mgvb †¶Îdjwewkó
`yBwU wÎfz‡R wef³ K‡i|
GLb, ∆ ABC-Gi ga¨gv BO [Q AO = OC ]
∴ ∆ †¶Î AOB = ∆ †¶Î BOC ... ... ... (i)
Ges ∆ BCD-Gi ga¨gv CO [Q BO = OD]
∴ ∆ †¶Î BOC = ∆ †¶Î COD... ... ... (ii)
Avevi, ∆ CAD Gi ga¨gv DO [Q AO = OC]
∴ ∆ †¶Î COD = ∆ †¶Î AOD ... ... ... (iii)
(i) bs ,(ii) bs Ges (iii) bs n‡Z cvB,
∆ †¶Î AOB = ∆ †¶Î BOC = ∆ †¶Î COD = ∆ †¶Î AOD.
A_©vr, mvgvšwi‡Ki KY©Øq mvgvšwiK †¶ÎwU‡K PviwU mgvb
wÎfzR‡¶‡Î wef³ K‡i| (cÖgvwYZ)
Aby: 3. ∆ABC-Gi AB I AC evû؇qi ga¨we›`y h_vµ‡g X I
1
cÖgvY Ki †h, ∆†¶Î AXY Gi †¶Îdj = 4 (∆†¶Î ABC
Gi †¶Îdj). [iv: †ev: 11; P: †ev: 12, 09, 04; h: †ev: Kz: †ev: 02, 04, 08;
Y.
wm: †ev: 12, 09; e: †ev: 11]
X
∴ PS | | BD Ges PS =
Avevi, ∆CBD Gi
QR Zv‡`i ms‡hvRK mij‡iLv|
∴ RQ | | DB Ges RQ =
A
DËi :
ga¨we›`y h_vµ‡g Q I R Ges
1
DB
2
1
AZGe, PS = RQ [Q PS I RQ cÖ‡Z¨‡K 2 BD Gi mgvb]
Ges PS | | RQ [Q PS I RQ cÖ‡Z¨‡K BD Gi mgvšivj]
Abyicƒ fv‡e, A, C †hvM K‡i cÖgvY Kiv hv‡e †h,
PQ = SR Ges PQ | | SR
Zvn‡j, PQRS PZzfz©‡Ri `yB wecixZ evû ci¯ži mgvb I
mgvšivj|
∴ PQRS GKwU mvgvšwiK| (†`Lv‡bv n‡jv)
Y
B
C
we‡kl wbe©Pb : g‡b Kwi ∆ABC-Gi AB Ges AC evû؇qi
ga¨we›`y h_vµ‡g X Ges Y. X,Y †hvM Kwi|
1
cÖgvY Ki‡Z n‡e †h, ∆-†¶Î AXY Gi †¶Îdj = 4 (∆-†¶Î
ABC Gi †¶Îdj)
A¼b : B, Y †hvM Kwi|
cÖgvY : ∆ ABY-G XY, AB Gi Ici ga¨gv|
1
∆-†¶Î AXY = 2 (∆-†¶Î ABY)
[ Q XY ga¨gv
∆-†¶Î ABY-†K mgwØLwÊZ K‡i]
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
497
Avevi, ∆ABC- G BY, AC-Gi Ici ga¨gv|
Aby: 7. ABC GKwU mgevû wÎfzR Ges AD, BC-Gi Ici j¤^|
†`LvI †h, 4AD 2 = 3AB 2. [Xv. †ev. 10; wm: †ev: 10, 08; P: †ev: 11, 05,
1
∴ ∆-†¶Î ABY = 2 (∆-†¶Î ABC) [GKB Kvi‡Y]
03; e: †ev: 12, 09, 03; Kz: †ev: 11, 07, 04; iv: †ev: 05; h: †ev: 11, 08, 07, 03]
1 1
∴ ∆-†¶Î AXY = 2{2 (∆-†¶Î ABC)}
1
= (∆-†¶Î ABC)
4
1
A_©vr ∆-†¶Î AXY Gi †¶Îdj = 4 (∆-†¶Î ABC Gi †¶Îdj)
A_ev, cÖgvY Ki †h, mgevû wÎfz‡Ri †h‡Kvb kxl© n‡Z wecixZ
evûi Ici Aw¼Z j‡¤^i Pvi¸Y Dnvi †h‡Kvb evûi e‡M©i
[Xv: †ev: 2000]
wZb¸‡Yi mgvb|
A
DËi :
(cÖgvwYZ)
Abykxjbx-5 : cx_v‡Mviv‡mi Dccv`¨ I Zvi e¨envi welqK
D`v-1. ABC mg‡KvYx wÎfz‡R ∠A = GK mg‡KvY| BE I CF
ga¨gv| cÖgvY Ki †h, 4(BE2 + CF2) = 5BC2.
[iv: †ev: 11; Kz: †ev: 10; P: †ev: 07; wm: †ev: 01]
DËi :
C
E
∴
A
F
B
we‡kl wbe©Pb : ∆ABC-Gi ∠A = GK mg‡KvY Ges BE I CF
ga¨gv| cÖgvY Ki‡Z n‡e †h, 4(BE2 + CF2) = 5BC2.
cÖgvY : ∆ABE G BE2 = AB2 + AE2 ... ... ... (i)
Ges ∆AFC-G CF2 = AC2 + AF2 ... ... ... (ii)
∴ 4(BE2 + CF2) = 4(AB2 + AE2 + AC2 + AF2)
[(i) bs I (ii)bs n‡Z]
= 4(AB2 + AC2) + 4AE2 + 4AF2
= 4BC2 + (2AE)2 + (2AF)2
[cx_v‡Mviv‡mi Dccv`¨ Abymv‡i]
= 4BC2 + AC2 + AB2 [Q BE I CF ga¨gv]
= 4BC2 + BC2 [cx_v‡Mviv‡mi Dccv`¨ Abymv‡i]
∴ 4(BE2 + CF2) = 5BC2 (cÖgvwYZ)
Aby: 6. ABC wÎfz‡Ri ∠A = GK mg‡KvY| D, AC-Gi Dci¯’
GKwU we›`y| cÖgvY Ki †h, BC2 + AD2 = BD2 + AC2.
DËi :
B
[wm: †ev: 04; iv: †ev: 99]
B
D
C
we‡kl wbe©Pb : ∆ ABC mgevû A_©vr AB = BC = CA Ges AD, BC
Gi Ici j¤^|
†`Lv‡Z n‡e †h, 4AD2 = 3AB2.
cÖgvY : AD ⊥ BC [†`Iqv Av‡Q ]
∠ADB = ∠ADC = 90°.
GLb, mg‡KvYx ∆ ABD Ges mg‡KvYx ∆ ACD-G,
AwZfzR AB = AwZfzR AC [Q ABC mgevû wÎfzR]
Ges AD mvaviY evû|
∴ ∆ABD ≅ ∆ACD
[Q mg‡KvYx wÎfzR؇qi AwZfzR Ges Aci GKwU evû mgvb ]
myZivs BD = CD
∴ BC = 2BD
Avevi, mg‡KvYx ∆ ABD-G, ∠ADB=90° Ges AwZfzR = AB.
∴ cx_v‡Mviv‡mi Dccv`¨ Abymv‡i,
AB2 = AD2 + BD2
ev, AD2 = AB2 – BD2
ev, 4AD2 = 4AB2 – 4BD2 [Dfq c¶‡K 4 Øviv ¸Y K‡i]
ev, 4AD2 = 4AB2 – (2BD)2
[Q BC = 2BD ]
ev, 4AD2 = 4AB2 – BC2
ev, 4AD2 = 4AB2 – AB2
[Q AB = BC]
2
2
∴ 4AD = 3AB . (†`Lv‡bv n‡jv)
[XvKv †evW©-2010 Gi cÖ‡kœ ÔDÕ Gi ¯’‡j ÔEÕ †`Iqv n‡q‡Q|]
Aby: 8. ABC GKwU mgwØevû mg‡KvYx wÎfzR| BC-Gi
AwZfzR Ges P, BC-Gi Ici †h‡Kv‡bv we›`y| cÖgvY Ki †h,
PB2 + PC2 = 2PA2. [Kz: †ev: 05, 02; Xv: †ev: 08, 06; P: †ev: 10, 08;
e: †ev: 10, 08, 01; iv: †ev: 07, 03; wm: †ev: 07; w`: †ev: 10; h: †ev: 12]
DËi :
A
D
C
we‡kl wbe©Pb : ∆ABC-Gi ∠A = GK mg‡KvY Ges D, AC Gi
Dci¯’ GKwU we›`y| B, D †hvM Kwi| cÖgvY Ki‡Z n‡e †h, BC2 +
AD2 = BD2 + AC2.
cªgvY : ∆ABC mg‡KvYx wÎfz‡R AwZfzR = BC
∴ BC2 = AB2 + AC2 ... ... (i) [cx_v‡Mviv‡mi Dccv`¨ Abymv‡i]
Avevi, ∆ABD mg‡KvYx wÎfz‡R AwZfzR = BD
∴ BD2 = AB2 + AD2
[cx_v‡Mviv‡mi Dccv`¨ Abymv‡i]
ev, AD2 = BD2 – AB2 ... ... (ii)
(i) bs I (ii) bs †hvM K‡i cvB,
∴ BC2 + AD2 = AB2 + AC2 + BD2 – AB2
ev, BC2 + AD2 = AC2 + BD2
∴ BC2 + AD2 = BD2 + AC2. (cÖgvwYZ)
B
E
A
P
D
C
we‡kl wbe©Pb : g‡b Kwi, mgwØevû mg‡KvYx ∆ABC-Gi AB = AC
Ges AwZfzR BC| P, BC Gi Ici †h‡Kv‡bv we›`y| P, A †hvM Kwi|
cÖgvY Ki‡Z n‡e †h, PB2 + PC2 = 2PA2.
A¼b : P we›`y †_‡K AB Ges AC evûi Ici h_vµ‡g PE Ges
PD j¤^ Uvwb|
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
498
cÖgvY : ∆ABC- Gi, ∠A = 90°
Ges AC = AB nIqvq, ∠B = ∠C = 45°
GLb, ∆PDC-Gi, ∠D = 90°
[Q PD ⊥ AC]
myZivs, ∠ DPC = ∠DCP = 45°
∴
Aby: 11. ∆ABC Gi AD GKwU ga¨gv| †`LvI †h,
AB2 + AC2 = 2(BD2 + AD2).
[P: †ev: 02]
A
DËi :
CD = PD
Abyiƒcfv‡e cÖgvY Kiv hvq, PBE mg‡KvYx wÎfz‡R, PE = BE
GLb, PDC mg‡KvYx wÎfz‡R PC AwZfzR nIqvq
PC2 = PD2 + CD2 = PD2 + PD2 [Q PD = CD]
∴ PC2 = 2PD2 ... ... ... (i)
Avevi, PBE mg‡KvYx wÎfz‡R PB AwZfzR nIqvq,
PB2 = BE2 + PE2
= PE2 + PE2 [Q BE = PE]
∴ PB2 = 2PE2 ... ... ... (ii)
GLb, (i) Ges (ii) bs †hvM K‡i cvB,
PC2 + PB2 = 2PD2 + 2PE2 = 2(PD2 + PE2)
Avevi, ∠E = ∠A = ∠D = GK mg‡KvY nIqvq ADPE GKwU AvqZ|
∴ PE = AD
[Q AvqZ‡¶‡Îi wecixZ evûØh ci¯ži mgvb]
∴ PC2 + PB2 = 2(PD2 + AD2) ... ... ... (iii)
ADP mg‡KvYx wÎfz‡R PA AwZfzR nIqvq,
PA2 = PD2 + AD2
Zvn‡j, (iii) bs n‡Z cvB,
PC2 + PB2 = 2PA2
∴ PB2 + PC2 = 2PA2 (cÖgvwYZ)
Aby: 10. ∆ABC Gi ∠C m²‡KvY; AD, BC Gi Ici j¤^|
[wm: †ev: 06]
†`LvI †h, AB2 = AC2 + BC2 – 2BC·CD.
DËi :
A
A
C
D
B
C
wPÎ-1
B
D
wPÎ-2
we‡kl wbe©Pb : †`Iqv Av‡Q, ∆ABC Gi ∠C m²‡KvY| AD, BC
Gi (ev CB Gi ewa©Zvs‡ki) Ici j¤^|
A_©vr ∠ADB = ∠ADC = 90°.
†`Lv‡Z n‡e †h, AB2 = AC2 + BC2 – 2BC·CD.
cÖgvY : ∆ACD G ∠ADC GK mg‡KvY Ges AwZfzR = AC.
myZivs cx_v‡Mviv‡mi Dccv`¨ Abymv‡i,
AC2 = AD2 + CD2 ... ... (i)
Avevi, ∆ABD G ∠ADB = GK
Ges AwZfzR = AB.
B
D
E
C
we‡kl wbe©Pb : ∆ABC Gi AD GKwU ga¨gv|
†`Lv‡Z n‡e †h, AB2 + AC2 = 2(BD2 + AD2)
A¼b : A we›`y n‡Z BC Gi Ici GKwU j¤^ AE AuvwK|
cÖgvY : cx_v‡Mviv‡mi Dccv`¨ Abymv‡i mg‡KvYx wÎfzR ADE G
AD2 = AE2 + DE2 ... ... ... (i)
Avevi, cx_v‡Mviv‡mi Dccv`¨ Abymv‡i mg‡KvYx ∆ABE G
AB2 = AE2 + BE2
= AE2 + (BD + DE)2
= AE2 + BD2 + DE2 + 2BD.DE
= (AE2 + DE2) + BD2 + 2BD.DE
= AD2 + BD2 + 2BD.DE ... ... ... (ii) [(i) bs n‡Z]
Ges ∆ACE G [GKB Kvi‡Y]
AC2 = AE2 + EC2
= AE2 + (DC – DE)2
= AE2 + DC2 + DE2 – 2DC.DE
= (AE2 + DE2) + BD2 – 2BD.DE
[ †h‡nZz, AD, BC evûi ga¨gv ∴ DC = BD]
= AD2 + BD2 – 2BD.DE ... ... ... (iii) [(i) bs n‡Z]
GLb, (ii) bs I (iii) bs †hvM K‡i,
AB2 + AC2 = AD2 + AD2 + BD2 + BD2
+ 2BD.DE – 2BD.DE
∴ AB2 + AC2 = 2(AD2 + BD2) (†`Lv‡bv n‡jv)
Aby: 12. ABCD AvqZ‡¶‡Îi Af¨š‡i O †h‡Kv‡bv we›`y|
cÖgvY Ki †h, OA2 + OC2 = OB2 + OD2.
[wm: †ev:06; h: †ev: 10; e: †ev: 01, 04; Kz: †ev: 03; w`: †ev: 12]
A_ev, ABCD AvqZ‡¶‡Îi Af¨š‡i P †h‡Kv‡bv we›`y| cÖgvY
[h: †ev: 06, 01]
Ki †h, PA2 + PC2 = PB2 + PD2.
A
F
B
DËi :
O
mg‡KvY
myZivs cx_v‡Mviv‡mi Dccv`¨ Abymv‡i, AB2 = AD2 + BD2
wPÎ-1 G
AB2 = AD2 + BD2
[Q BD = BC – CD ]
= AD2 +(BC – CD)2
= AD2 + BC2 + CD2 – 2BC·CD [mÎ e¨envi K‡i ]
= (AD2 + CD2) + BC2 – 2BC· CD
= AC2 + BC2 – 2BC·CD
[Q (i) bs †_‡K, AD2 + CD2 = AC2]
wPÎ-2 G
AB2 = AD2 + BD2
= AD2 + (CD – BC)2 [Q BD = CD – BC ]
= AD2 + CD2 + BC2 – 2BC·CD [mÎ e¨envi K‡i]
= AC2 + BC2 – 2BC·CD
[Q (i) bs †_‡K AD2 + CD2 = AC2 ]
2
2
2
myZivs, AB = AC + BC – 2BC·CD. (†`Lv‡bv n‡jv)
D
E
C
we‡kl wbe©Pb : g‡b Kwi, ABCD AvqZ‡¶‡Îi Af¨š‡i O
†h‡Kv‡bv we›`y| O, A; O, B; O, C Ges O, D †hvM Kwi| cÖgvY
Ki‡Z n‡e †h, OA2 + OC2 = OB2 + OD2.
A¼b :O we›`y †_‡K CD Ges AB Gi Ici h_vµ‡g OE Ges OF
j¤^ Uvwb|
cÖgvY : FE, AB A_ev CD Gi Ici j¤^ [A¼bvbymv‡i]|
myZivs FE †iLv ABCD AvqZ‡¶ÎwU‡K AFED Ges FBCE `yBwU
AvqZ‡¶‡Î wef³ K‡i|
myZivs AF = DE Ges CE = BF [Q AvqZ‡¶‡Îi wecixZ evû]
GLb, cx_v‡Mviv‡mi Dccv`¨ Abymv‡i,
mg‡KvYx ∆ OAF-G, OA2 = OF2 + AF2
... ... ... (i)
mg‡KvYx ∆ OCE-G, OC2 = OE2 + CE2
... ... ... (ii)
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
mg‡KvYx ∆ OBF-G, OB2 = OF2 + BF2
... ... ... (iii)
2
2
2
Ges mg‡KvYx ∆ ODE-G, OD = OE + DE ... ... ... (iv)
(i) bs I (ii) bs †hvM K‡i cvB,
OA2 + OC2 = OF2 + AF2 + OE2 + CE2
= OF2 + CE2 + OE2 + AF2
= (OF2 + BF2) + (OE2 + DE2)
[Q CE = BF Ges AF = DE ]
2
2
= OB + OD [(iii) bs Ges (iv) bs n‡Z ]
∴
OA2 + OC2 = OB2 + OD2.
Aby: 4. cªgvY Ki †h, wÎfz‡Ri †h‡Kv‡bv †Kv‡Yi AšwØ©LÊK I
Aci `yBwU †Kv‡Yi ewnwØ©LÊKØq mgwe›`y|
[Kz: †ev: 04; h: †ev: 01, 04; e: †ev: 02]
DËi : mvaviY wbe©Pb : cÖgvY Ki‡Z n‡e †h, wÎfz‡Ri †h‡Kv‡bv
†Kv‡Yi AšwØ©LÊK I Aci `yBwU †Kv‡Yi ewnwØ©LÊKØq mgwe›`y|
E
Y
O
X
A
Q
B
C
P
F
we‡kl wbe©Pb : g‡b Kwi, ∆ABC Gi BA evû‡K E ch©š Ges
BC evû‡K F ch©š ewa©Z Kivq `yBwU ewn:¯’ †KvY h_vµ‡g
∠EAC Ges ∠FCA Drcbœ n‡q‡Q|
cÖgvY Ki‡Z n‡e †h, Aš:¯’ ∠ABC-Gi mgwØLÊK Ges ewn:¯’
∠EAC I ∠FCA-Gi mgwØLÊKØq mgwe›`y A_©vr GKB we›`yMvgx|
A¼b : ∠EAC Ges ∠FCA-Gi mgwØLÊK evûØq h_vµ‡g AX
Ges CY Uvwb †hb Zviv O we›`y‡Z †Q` K‡i| B, O †hvM Kwi| OP,
OQ, OR h_vµ‡g BCF, AC Ges BAE evû·qi Ici j¤^ Uvwb|
cÖgvY : Rvbv Av‡Q, ci¯•i‡”Q`x `yBwU mij‡iLv †_‡K mg`ieZ©x
†Kv‡bv we›`yi mÂvic_ D³ wbw`©ó †iLv `yBwUi Ašf©³
y †Kv‡Yi
mgwØLÊK nq|
GLb, CY ci¯•i‡”Q`x AC Ges CF †iLvs‡ki Ašf©³
y ∠FCA
Gi mgwØLÊK|
myZivs CY-Gi Ici †h‡Kv‡bv we›`y AC Ges CF †_‡K mg`ieZ©x|
∴ OQ = OP ... ... ... (i)
Avevi AX ci¯•i‡”Q`x AC
Aby: 6. cÖgvY Ki †h, wÎfz‡Ri †KvY¸‡jvi mgwØLÊKÎq mgwe›`y|
[Xv: †ev: 07, 04, wm: †ev:, P: †ev: 06, 02; Kz: †ev: 12, 09, 00, P: †ev:, h:
12, †ev: 09, 00; iv: †ev: 08, 01; wm: †ev: 11; e: †ev: 12]
DËi : mvaviY wbe©Pb : cÖgvY Ki‡Z n‡e †h, wÎfz‡Ri †KvY¸‡jvi
A
mgwØLÊKÎq mgwe›`y|
Q
P
I
(cÖgvwYZ)
Abykxjbx-9 : mÂvic_ welqK
R
499
Ges AE †iLvs‡ki Ašf©³
y ∠EAC
Gi mgwØLÊK|
myZivs AX Gi Ici †h‡Kv‡bv we›`y AC Ges AE †_‡K mg`ieZ©x|
∴ OQ = OR ... ... ... (ii)
(i) bs Ges (ii) bs Zzjbv K‡i cvB,
∴ OP = OR
A_©vr, O we›`y †_‡K BCF Ges BAE evû `yBwUi `iZ¡ mgvb|
∴ O we›`y, ∠ABC Gi mgwØLʇKi Ici Aew¯’Z|
myZivs BO, ∠ABC Gi mgwØLÊK|
AZGe, ∠ABC Gi mgwØLÊK Ges ∠BAC I ∠ACB
Gi
ewnwØ©LÊK mgwe›`y|
A_©vr, wÎfz‡Ri †h‡Kv‡bv †Kv‡Yi AšwØ©LÊK I Aci `yBwU †Kv‡Yi
ewnwØ©LwÊKØq mgwe›`y| (cÖgvwYZ)
B
O
C
we‡kl wbe©Pb : g‡b Kwi, ∆ABC Gi wZbwU †KvY n‡jv ∠A, ∠B
Ges ∠C. cÖgvY Ki‡Z n‡e †h, ∠A, ∠B Ges ∠C Gi
mgwØLÊKÎq mgwe›`y A_©vr GK we›`yMvgx n‡e|
A¼b : ∠B Ges ∠C Gi mgwØLÊK h_vµ‡g BI Ges CI AuvwK
†hb Zviv I we›`y‡Z †Q` K‡i| A, I †hvM Kwi| I we›`y n‡Z BC, CA
I AB evûi Ici h_vµ‡g IO, IP Ges IQ j¤^ Uvwb|
cÖgvY : Avgiv Rvwb, ci¯•i‡”Q`x `yBwU mij‡iLv †_‡K
mg`ieZ©x †Kv‡bv we›`yi mÂvic_ D³ wbw`©ó †iLv `yBwUi Ašf©³
y
†Kv‡Yi mgwØLÊK nq|
BI, ci¯•i‡”Q`x AB Ges BC †iLvs‡ki Ašf©³
y ∠ABC Gi
mgwØLÊK|
myZivs BI-Gi Ici †h †Kv‡bv we›`y, AB Ges BC †_‡K mg`ieZ©x|
∴
IQ = IO ... ... ... (i)
∴
(i)
IO = IP ... ... ... (ii)
Avevi, CI, ci¯•i‡”Q`x BC Ges AC †iLvs‡ki Ašf©³
y ∠ACB
Gi mgwØLÊK|
myZivs CI-Gi Ici †h‡Kv‡bv we›`y BC Ges AC †_‡K mg`ieZ©x|
bs Ges (ii) bs Zzjbv K‡i cvB,
IQ = IP
[Q DfqwU PO Gi mgvb]
A_©vr, I we›`y †_‡K AB Ges AC †iLvs‡ki `iZ¡ mgvb|
∴ I we›`y AB Ges AC Gi Ašf©³
y ∠BAC ev, ∠A Gi
mgwØLʇKi Ici Aew¯’Z|
myZivs, AI, ∠A Gi mgwØLÊK|
∴ ∠A, ∠B Ges ∠C Gi mgwØLÊKÎq GKwe›`y I-†Z wgwjZ n‡q‡Q
A_©vr, mgwØLÊKÎq mgwe›`y| (cÖgvwYZ)
Abykxjbx-10.2 : e„Ë m¤žK©xq
D`v: 10.1. cÖgvY Ki †h, e„‡Ëi `yBwU R¨v Gi g‡a¨ †K‡›`ªi
wbKUZi R¨vwU Aci R¨v A‡c¶v e„nËi|
[wm: †ev: 11; Xv: †ev: 10; P: †ev: 10]
DËi : mvaviY wbe©Pb : cÖgvY Ki‡Z n‡e †h, e„‡Ëi `yBwU R¨v
Gi g‡a¨ †K‡›`ªi wbKUZi R¨vwU Aci R¨v A‡c¶v e„nËi|
A
C
O
E
F
B
D
we‡kl wbe©Pb : g‡b Kwi, O †K›`ªwewkó e„‡Ë AB Ges CD `yBwU
R¨v| O †_‡K AB I CD ch©š j¤^Øq h_vµ‡g OE I OF Ges OE
<OF| cÖgvY Ki‡Z n‡e †h, AB > CD.
A¼b : O I A Ges O I C †hvM Kwi|
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
500
cÖgvY : †h‡nZz OE ⊥ AB Ges OF ⊥ CD
1
AE = AB
2
Ges
1
CF = CD
2
GLb ∆ OAE Ges ∆ OCF mg‡KvYx wÎfzR؇qi g‡a¨
Avgiv cvB,
OA2 = AE2 + OE2 Ges
OC2 = CF2 + OF2
wKšy OA = OC [ GKB e„‡Ëi e¨vmva© e‡j
ev, OA2 = OC2 [ Dfqc¶‡K eM© K‡i ]
ev, AE2 + OE2 = CF2 + OF2
]
†h‡nZz OE2 < OF2 [Q OE < OF]
∴ AE2 > CF2
1
ev, 2 AB > 2 CD
∴ AB > CD
1
AB
2
O e„‡Ëi
∴ BP =
Avevi,
∴
†K›`ª Ges OQ ⊥ R¨v CD.
CQ = DQ
∴ DQ =
1
CD
2
wKš‘, †`Iqv Av‡Q, AB = CD
1
1
ev, 2 AB = 2 CD
∴
BP = DQ ... ... ... (i)
∴
∆ OPE ≅ ∆ OQE
GLb, mg‡KvYx ∆ OPE Ges mg‡KvYx ∆ OQE-G,
AwZfzR OE Dfq wÎfz‡Ri mvaviY evû|
Ges OP = OQ [Q mgvb mgvb R¨v †K›`ª †_‡K mg`ieZ©x ]
ev, AE > CF
1
cÖgvY : O e„‡Ëi †K›`ª Ges OP ⊥ R¨v AB.
∴ AP = BP [Q Rvbv Av‡Q, e„‡Ëi †K›`ª n‡Z e¨vm wfbœ Ab¨ R¨vGi Ici Aw¼Z j¤^ H R¨v-†K mgwØLwÊZ K‡i|]
(cÖgvwYZ)
D`v: 10.2. cÖgvY Ki †h, e„‡Ëi e¨vmB e„nËg R¨v|
[Kz: †ev: 05; wm: †ev: 09; h: †ev: 10; e: †ev: 06]
DËi : mvaviY wbe©Pb : cÖgvY Ki †h, e„‡Ëi e¨vmB e„nËg R¨v|
O
A
B
D
C
myZivs PE = QE ... ... ... (ii)
GLb (i) bs I (ii) bs †hvM K‡i cvB,
BP + PE = DQ + QE
BE = DE ... ... ... (iii)
Avevi, AB = CD ... ... ...(iv)
(iv) bs n‡Z (iii) bs we‡qvM K‡i cvB,
AB – BE = CD – DE
∴ AE = CE
... ... ... (v)
myZivs (v) bs Ges (iii) bs †_‡K †`Lv
∴
BE = DE
we‡kl wbe©Pb : g‡b Kwi, O †K›`ªwewkó ABDC GKwU e„Ë| AB
e¨vm Ges CD e¨vm wfbœ Ab¨ GKwU R¨v| cÖgvY Ki‡Z n‡e †h,
AB > CD
A¼b : O, C I O, D
†hvM Kwi|
cÖgvY : OA = OB = OC = OD [GKB e„‡Ëi e¨vmva©]
GLb ∆ COD-G
(†`Lv‡bv n‡jv)
Aby: 2. cÖgvY Ki †h, e„‡Ëi mgvb R¨v-Gi ga¨we›`y¸‡jv mge„Ë|
[wm: †ev: 07; Xv: †ev: 12]
DËi : mvaviY wbe©Pb : cÖgvY Ki‡Z n‡e †h, e„‡Ëi mgvb R¨v-Gi
ga¨we›`y¸‡jv mge„Ë|
A
OC + OD > CD
[Xv: †ev: 05; Kz: †ev: 06, 00; h: †ev: 00, 04]
B
D
O
P
Q
E
A
C
we‡kl wbe©Pb : g‡b Kwi, O †K›`ªwewkó ACBD e„‡Ë AB I CD `yBwU
mgvb R¨v Ges Zviv ci¯ži‡K E we›`y‡Z †Q` K‡i‡Q|
†`Lv‡Z n‡e †h, AE = CE Ges BE = DE.
A¼b: †K›`ª O †_‡K AB Ges CD Gi Ici h_vµ‡g OP Ges OQ
j¤^ AvuwK Ges O, E †hvM Kwi|
N
O
ev, OA + OB > CD [Q OA = OB = OC = OD]
∴ AB > CD (cÖgvwYZ)
DËi : mvaviY wbe©Pb : †`Iqv Av‡Q, e„‡Ëi `ywU mgvb R¨v ci¯ži‡K
†Q` Ki‡j, †`Lv‡Z n‡e †h, Zv‡`i GKwUi AskØq AciwUi
Ask؇qi mgvb|
C
M
[wÎfy‡Ri †h‡Kvb `yB evûi †hvMdj Z…Zxq evû A‡c¶v e„nËi]
Aby: 1. e„‡Ëi `yBwU mgvb R¨v ci¯ži‡K †Q` Ki‡j, †`LvI †h,
Zv‡`i GKwUi AskØq AciwUi Ask؇qi mgvb|
hvq †h, AE = CE Ges
B
E
D
P
F
we‡kl wbe©Pb : g‡b Kwi, ABDC e„‡Ëi †K›`ª O. AB, CD, I
e„‡Ëi mgvb mgvb wZbwU R¨v hv‡`i ga¨we›`y¸‡jv nj
h_vµ‡g M, N I P | cÖgvY Ki‡Z n‡e †h, M, N I P mge„Ë|
A¼b : O, M ; O, N Ges O, P †hvM Kwi|
cÖgvY : Avgiv Rvwb, e„‡Ëi R¨v-Gi ga¨we›`y I †K‡›`ªi ms‡hvRK
†iLvsk H R¨v Gi Ici j¤^|
M, AB R¨v-Gi ga¨we›`y|
EF
∴ OM ⊥ AB
Z`ªc“ ON ⊥ CD
Ges OP ⊥ EF
Avgiv Rvwb, e„‡Ëi mgvb mgvb R¨v †K›`ª n‡Z mg`ieZ©x|
†K›`ª O n‡Z AB, CD I EF R¨v-Gi j¤^`iZ¡ h_vµ‡g OM, ON I OP.
†h‡nZz AB = EF = CD [†`Iqv Av‡Q ]
∴ OM = ON = OP [Q mgvb mgvb R¨v †K›`ª †_‡K mg`ieZ©x ]
myZivs O †K †K›`ª K‡i OM ev ON ev OP Gi mgvb e¨vmva©
e„Ë A¼b Ki‡j e„ËwU M, N I P we›`y w`‡q hv‡e|
AZGe, M, N I P mge„Ë| (cÖgvwYZ)
wb‡q
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
501
Aby: 5. †`LvI †h, e„‡Ëi `yBwU R¨v-Gi g‡a¨ e„nËi R¨v-wU
¶z`ªZi R¨v A‡c¶v †K‡›`ªi wbKUZi|
[Xv: †ev: 10, 02; P: †ev: 10, 06, iv: †ev: 05; e: †ev: 11, 04]
DËi : mvavib wbe©Pb : †`Lv‡Z n‡e †h, e„‡Ëi `yBwU R¨v Gi g‡a¨
e„nËi R¨v-wU ¶z`ªZi R¨v A‡c¶v †K‡›`ªi wbKUZi|
we‡kl wbe©Pb : g‡b Kwi, O
C
A
†K›`ªwewkó ABDC GKwU e„Ë| AB
I CD Gi `yBwU R¨v Ges AB > CD.
O
F
OE Ges OF †K›`ª O n‡Z h_vµ‡g
E
AB I CD Gi Ici j¤^| †`Lv‡Z
D
n‡e †h, OE < OF.
B
A¼b : O, A Ges O, C †hvM Kwi|
cÖgvY : Rvbv Av‡Q, e„‡Ëi †K›`ª †_‡K e¨vm wfbœ Ab¨ R¨v-Gi Ici
Aw¼Z j¤^ R¨v-†K mgwØLwÊZ K‡i|
1
AB. [O e„‡Ëi †K›`ª Ges OE ⊥ AB]
2
1
∴ CF = DF = CD [O e„‡Ëi †K›`ª Ges OF ⊥ CD ]
2
GLb, Kíbv Abymv‡i, AB > CD
1
1
1
ev, 2 AB > 2 CD [Dfq c¶‡K 2 Øviv ¸Y K‡i ]
1
1
[Q AE = AB Ges CF = CD]
ev, AE > CF
2
2
∴ AE2 > CF2
[ Dfqc¶‡K eM© K‡i]
GLb, mg‡KvYx ∆ OAE Ges mg‡KvYx ∆ OCF-Gi,
AwZfzR h_vµ‡g OA Ges OC.
∴ AE = BE =
Zvn‡j, cx_v‡Mviv‡mi Dccv`¨ Abymv‡i,
OA2 = OE2 + AE2
Ges
OC2 = OF2 + CF2
wKšy, OA = OC
[Q GKB e„‡Ëi e¨vmva©
Zvn‡j, OA2 = OC2
[Dfq c¶‡K eM© K‡i
A_©vr, OE2 + AE2 = OF2 + CF2
ev,
AE2 – CF2 = OF2 – OE2 ... ... ... (i)
†h‡nZz AE2 > CF2
ev, AE2 – CF2 > 0
ev, OF2 – OE2 > 0
[ (i) bs †_‡K
ev, OF2 > OE2
ev, OF > OE
∴
OE< OF.
]
]
cªgvY : Rvbv Av‡Q, e„‡Ëi GKB Pv‡ci Ici `Êvqgvb e„˯’ †KvY
†K›`ª¯’ †Kv‡Yi A‡a©K|
GLb, GKB Pvc AB Gi Ici `Êvqgvb e„˯’ ∠ACB
Ges †K›`ª¯’ ∠AOB.
1
∠AOB
2
ev, ∠AOB = 2 ∠ACB ... ... (i)
GKB Pvc CD Gi Ici `Êvqgvb e„˯’ ∠CBD Ges
†K›`ª¯’ ∠COD.
1
∴ ∠CBD = ∠COD
2
ev, ∠COD = 2 ∠CBD ... ... (ii)
(i) bs I (ii) bs †hvM K‡i cvB,
∠AOB + ∠COD = 2∠ACB + 2∠CBD
ev, ∠AOB + ∠COD = 2(∠ACB + ∠CBD)
ev, ∠AOB + ∠COD = 2(∠ECB + ∠EBC) ... ... (iii)
GLb, ∆ CBE-Gi,
∠ECB + ∠EBC + ∠BEC = `yB mg‡KvY
[Q wÎfz‡Ri wZb †Kv‡Yi mgwó `yB mg‡KvY]
Avevi, BE iwk¥i cªvšwe›`y E-†Z AC mij‡iLv wgwjZ
∴ ∠ACB =
Avevi,
n‡q‡Q|
d‡j ∠BEC Ges ∠AEB mwbœwnZ †KvYØq Drcbœ n‡q‡Q|
∴ ∠BEC + ∠AEB = `yB mg‡KvY
myZivs ∠ECB + ∠EBC + ∠BEC = ∠BEC + ∠AEB
[Q Dfq c¶B `yB mg‡Kv‡Yi mgvb]
ev, ∠ECB + ∠EBC = ∠AEB ... ... (iv)
(iv) bs Ges (iii) bs n‡Z cvB
∠AOB + ∠COD = 2 ∠AEB. (cªgvwYZ)
Aby: 3. wP‡Î, R¨v AB ⊥ R¨v CD. AC I BD PvcØq †K‡›`ª
h_vµ‡g ∠AOC I ∠BOD Drcbœ K‡i‡Q|
]
Abykxjbx-10.3 : e„Ë m¤žK©xq
Aby: 1. O †K›`ªwewkó †Kv‡bv e„‡Ë ABCD GKwU Ašwj©wLZ
PZzfR
z© | AC, BD KY©Øq E we›`y‡Z †Q` Ki‡j cÖgvY Ki †h,
DËi :
Kiv n‡jv| cªgvY Ki‡Z n‡e †h,
A
A_©vr, e„‡Ëi `yBwU R¨v Gi g‡a¨ e„nËi R¨vwU ¶z`ªZg R¨v A‡c¶v
†K‡›`ªi wbKUZi| (†`Lv‡bv n‡jv)
∠AOB + ∠ COD = 2∠AEB
B, O; C, O Ges D, O †hvM
∠AOB + ∠COD = 2∠AEB.
O
E
B
cªgvY Ki †h, ∠AOC + ∠BOD = `yB mg‡KvY|
[P: †ev: 04]
A_ev, O †K›`ªwewkó e„‡Ëi AB I CD R¨vØq ci¯ži j¤^fv‡e P
we›`y‡Z †Q` Ki‡j cÖgvY Ki †h, ∠AOC + ∠BOD = 180°.
A
DËi :
[Xv: †ev: 08]
O
A
E
B
O
D
C
D
D
B
E
C
we‡kl wbe©Pb : O †K›`ªwewkó †Kv‡bv e„‡Ë ABCD GKwU Ašwj©wLZ
PZzfR
z© | Bnvi AC, BD KY©Øq E we›`y‡Z †Q` K‡i‡Q| Bnvi A, O;
C
we‡kl wbe©Pb : g‡b Kwi, O †K›`ªwewkó ACBD e„‡Ëi R¨v
AB ⊥ CD Ges R¨vØq P we›`y‡Z j¤^fv‡e †Q` K‡i‡Q| A, O; B,
O; C, O Ges D, O †hvM Kwi, Zvn‡j AC Ges BD PvcØq †K‡›`ª
h_vµ‡g ∠AOC Ges ∠BOD Drcbœ K‡i| cªgvY Ki‡Z n‡e †h,
∠AOC + ∠BOD = `yB mg‡KvY|
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
502
A¼b : A, D †hvM Kwi|
cªgvY : ∆ADE-G, ewnt¯’ ∠BEC = Ašt¯’ (∠ADE + ∠EAD)
A_©vr ∠BEC = ∠ADC + ∠BAD
AC Pv‡ci Ici Aew¯’Z ∠ADC e„˯’ †KvY Ges ∠AOC
†K›`ª¯’ †KvY|
∴ ∠AOC = 2∠ADC
Abyicƒ fv‡e, ∠BOD = 2∠BAD
∴ ∠AOC + ∠BOD = 2∠ADC + 2∠BAD
= 2(∠ADC + ∠BAD)
= 2∠BEC
= 2 × 1 mg‡KvY
[QAB Ges CD ci¯•i j¤^fv‡e †Q` K‡i,
∴ ∠BEC = 1 mg‡KvY]
= 2 mg‡KvY|
∴ ∠AOC + ∠BOD = `yB mg‡KvY| ( cÖgvwYZ )
Aby: 4. O †K›`ªwewkó ABCD e„‡Ë, ∠ADB + ∠BDC = GK
mg‡KvY| cªgvY Ki †h, A, O Ges C GK mij‡iLvq Aew¯’Z|
A¼b : A, D †hvM Kwi|
cÖgvY : ∆ADE-G, ewnt¯’ ∠AEC = Ašt¯’ (∠ADE + ∠EAD)
A_©vr ∠AEC = ∠ADC + ∠BAD
GLb, AC Pv‡ci Ici Aew¯’Z ∠ADC e„˯’ †KvY Ges ∠AOC
†K›`ª¯’ †KvY|
∴∠AOC = 2∠ADC
Avevi, BD Pv‡ci Ici Aew¯’Z ∠BAD
∠BOD †K›`ª¯’ †KvY|
∴ ∠BOD = 2∠BAD
∴ ∠AOC + ∠BOD = 2∠ADC + 2∠BAD
= 2(∠ADC + ∠BAD)
= 2∠AEC
∴ ∠AOC + ∠BOD = 2∠AEC ( cÖgvwYZ )
Aby: 6. AB I CD `yBwU R¨v e„‡Ëi evB‡i E we›`y‡Z †Q` K‡i‡Q|
cÖgvY Ki †h, AC I BD PvcØq †K‡›`ª †h `yBwU †KvY Drcbœ K‡i,
[e: †ev: 06, Xv: †ev: 03; Kz: †ev: 01]
Zv‡`i Aši ∠AEC Gi wظY|
DËi :
A
[w`: †ev: 11; iv: †ev: 10, 08; Kz: †ev: 10, 07; Xv: †ev:, wm: †ev: 10, 06,
e: †ev: 10, 08, 05; h: †ev: 08, 03; P: †ev: 12]
DËi :
e„˯’ †KvY Ges
B
E
D
O
D
O
A
C
C
B
we‡kl wbe©Pb : †`Iqv Av‡Q, O †K›`ªwewkó ABCD e„‡Ë,
∠ADB + ∠BDC = GK mg‡KvY|
cªgvY Ki‡Z n‡e †h, A, O Ges C GKB mij‡iLvq Aew¯’Z|
A¼b : A, C †hvM Kwi|
cÖgvY : ∠ADB + ∠BDC = GK mg‡KvY| [ †`Iqv Av‡Q ]
ev, ∠ADC = GK mg‡KvY|
wKš‘ Rvbv Av‡Q, Aa©e„˯’ †KvY GK mg‡KvY|
∴ ∠ADC = Aa©e„˯’ †KvY
myZivs AC = e„‡Ëi e¨vm
A_©vr e„‡Ëi †K›`ª O e¨vm AC Gi Ici Aew¯’Z|
∴ A, O Ges C GK mij‡iLvq Aew¯’Z| ( cÖgvwYZ )
Aby: 5. AB I CD `yBwU R¨v e„‡Ëi Af¨š‡i E we›`y‡Z †Q`
K‡i‡Q| cÖgvY Ki †h, AC I BD PvcØq †K‡›`ª †h `yBwU †KvY
Drcbœ K‡i, Zv‡`i mgwó ∠AEC-Gi wظY|
[Xv: †ev: 11; P: †ev: 09; e: †ev: 06; Kz: †ev: 04; wm: †ev: 05]
DËi :
A
E
∠AOC – ∠BOD = 2 ∠AEC.
A¼b : A, D †hvM Kwi|
cÖgvY : ∆ADE-G,
ewnt¯’ ∠ADC = Ašt¯’ (∠AED + ∠EAD)
A_©vr ∠ADC = ∠AEC + ∠BAD
ev, ∠ADC − ∠BAD = ∠AEC
∴ ∠AEC = ∠ADC − ∠BAD
GLb, AC Pv‡ci Ici Aew¯’Z ∠ADC
∠AOC †K›`ª¯’ †KvY|
∴ ∠AOC = 2∠ADC
Avevi, BD Pv‡ci Ici Aew¯’Z ∠BAD
∠BOD †K›`ª¯’ †KvY|
∴ ∠BOD = 2∠BAD
∴ ∠AOC − ∠BOD = 2∠ADC − 2∠BAD
= 2 (∠ADC − ∠BAD)
= 2∠AEC
∴ ∠AOC − ∠BOD = 2∠AEC (cÖgvwYZ)
e„˯’ †KvY Ges
e„˯’ †KvY Ges
Aby: 7. †`LvI †h, e„˯’ UªvwcwRqv‡gi wZh©K evûØq ci¯•i mgvb|
O
D
we‡kl wbe©Pb : g‡b Kwi, O †K›`ªwewkó †Kv‡bv e„‡Ëi AB I CD
R¨vØq‡K ewa©Z Kivq Bnviv e„‡Ëi evB‡i E we›`y‡Z †Q` K‡i‡Q|
A, O; B, O; C, O Ges D, O †hvM Kivq AC I BD PvcØq †K‡›`ª
h_vµ‡g ∠AOC Ges ∠BOD Drcbœ K‡i‡Q| cÖgvY Ki‡Z n‡e †h,
[iv: †ev: 01; wm: †ev: 03]
C
DËi : mvaviY wbe©Pb : †`Lv‡Z n‡e †h, e„˯’ UªvwcwRqv‡gi
wZh©K evûØq ci¯ži mgvb|
B
we‡kl wbe©Pb : g‡b Kwi, AB I CD R¨v `yBwU O †K›`ªwewkó
e„‡Ëi Af¨š‡i E we›`y‡Z †Q` K‡i‡Q| O, A; O, B; O, C Ges O,
D †hvM Kwi| AC I BD PvcØq †K‡›`ª h_vµ‡g ∠AOC Ges
∠BOD Drcbœ K‡i‡Q|
cÖgvY Ki‡Z n‡e †h, ∠AOC + ∠BOD = 2 ∠AEC.
A
D
B
C
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
503
we‡kl wbe©Pb : g‡b Kwi, ABCD e„˯’ UªvwcwRqv‡gi mgvšivj
evûØq h_vµ‡g AB I CD. myZivs, Bnvi wZh©K evûØq n‡jv AD
I BC. cÖgvY Ki‡Z n‡e †h, AD = BC.
A¼b : B, D †hvM Kwi|
cÖgvY : ABCD UªvwcwRqv‡g,
AB ⎢⎢CD Ges BD †Q`K
[ Kíbv Abymv‡i ]
∴ ∠ABD = ∠BDC
[ GKvši †KvY ]
A_©vr, AD Pv‡ci Ici e„˯’ †KvY = BC Pv‡ci Ici
e„˯’ †KvY
ev, Pvc AD = Pvc BC. [Q e„‡Ë mgvb mgvb Pv‡ci
Ici `Êvqgvb e„˯’ †KvY¸‡jv ci¯ži mgvb]
ev, AD R¨v = BC R¨v
[Q e„‡Ë mgvb mgvb Pvc mgvb R¨v wQbœ K‡i ]
∴ AD = BC. ( cÖgvwYZ )
Aby: 8. AB I AC †Kv‡bv e„‡Ëi `yBwU R¨v Ges P I Q h_vµ‡g
Zv‡`i Øviv wQbœ DcPvc `yBwUi ga¨we›`y| PQ R¨v AB I AC R¨v
†K h_vµ‡g D I E we›`y‡Z †Q` K‡i| †`LvI †h, AD = AE.
[P: †ev: 00, 03]
DËi :
A
P
D
Abykxjbx-10.5 : e„Ë m¤žK©xq
Aby: 1. ∆ABC G ∠B I ∠C Gi mgwØLÊKØq P we›`y‡Z Ges
ewn©wØLÊKØq Q we›`y‡Z wgwjZ n‡j cÖgvY Ki †h, B, P, C, Q we›`y
[h: †ev: 02; iv: †ev: 03; e: †ev: 09]
PviwU mge„Ë|
DËi :
A
P
B
C
E
F
Q
we‡kl wbe©Pb : ∆ ABC-G ∠B I ∠C-Gi mgwØLÊKØq h_vµ‡g
BP I CP, P we›`y‡Z wgwjZ n‡q‡Q| AB evû‡K E ch©š Ges AC
evû‡K F ch©š ewa©Z Kivq h_vµ‡g ∠EBC Ges ∠FCB ewnt¯’
†KvYØq Drcbœ n‡q‡Q| ∠EBC Ges ∠FCB Gi mgwØLÊKØq A_©vr
∠B Ges ∠C Gi ewntwØLÊKØq h_vµ‡g BQ Ges CQ, Q we›`y‡Z
wgwjZ n‡q‡Q| cÖgvY Ki‡Z n‡e †h, B, P, C, Q we›`y PviwU mge„Ë|
cÖgvY : ∠ABC + ∠EBC = 2 mg‡KvY [Q ˆiwLK hyMj †KvY]
1
1
ev, 2 ∠ABC + 2 ∠EBC = 1 mg‡KvY
Q
E
[ Dfqc¶‡K 2 Øviv fvM K‡i ]
ev, ∠PBC + ∠QBC = 1 mg‡KvY
B
[Q BP, ∠ABC- Gi
O
C
we‡kl wbe©Pb : g‡b Kwi, O †K›`ªwewkó e„‡Ëi AB I AC `ywU
R¨v| P, AB DcPv‡ci Ges Q, AC DcPv‡ci ga¨we›`y| PQ R¨v
AB I AC R¨vØq‡K h_vµ‡g D I E we›`y‡Z †Q` K‡i‡Q|
†`Lv‡Z n‡e †h, AD = AE.
A¼b : A, P Ges A, Q †hvM Kwi|
cÖgvY : P, APB Pv‡ci ga¨we›`y|
∴ Pvc AP = Pvc BP
Zvn‡j, e„˯’ ∠AQP = e„˯’ ∠BAP
[Q e„‡Ëi mgvb mgvb Pv‡ci Ici `Êvqgvb e„˯’ †KvY¸‡jv
ci¯ži mgvb ]
A_©vr, ∠AQE = ∠PAD
... ... ... (i)
Z`ªc“ ∠EAQ = ∠APD
... ... ... (ii)
(i) bs I (ii) bs †hvM K‡i cvB,
∠AQE + ∠EAQ = ∠PAD + ∠APD ... ... ... (iii)
Avevi, ∆ APD-Gi, ewnt¯’ ∠ADE, Bnvi Ašt¯’ `yB wecixZ
∠APD I ∠PAD Gi mgwói mgvb|
A_©vr, ∠ADE = ∠PAD + ∠APD
... ... ... (iv)
Abyi‡ƒ c, ∆ AQE-Gi ewnt¯’ ∠AED, Bnvi Aš:¯’ `yB wecixZ
∠AQE I ∠EAQ Gi mgwói mgvb|
A_©vr, ∠AED = ∠AQE + ∠EAQ = ∠PAD + ∠APD [(iii) bs †_‡K]
= ∠ADE [ (iv) bs †_‡K ]
∴
∠AED = ∠ADE.
A_©vr ∆ADE-Gi, ∠AED = ∠ADE
∴ AD = AE
[Q wÎfz‡Ri mgvb mgvb †Kv‡Yi
wecixZ evûØq ci¯ži mgvb]
(†`Lv‡bv n‡jv)
mgwØLÊK]
1
∴ ∠ABC = ∠PBC Ges BQ, ∠EBC Gi mgwØLÊK
2
1
∴ ∠EBC = ∠QBC ]
2
ev, ∠PBQ = 1 mg‡KvY
... ... ... (i)
Abyicƒ fv‡e, ∠PCQ = 1 mg‡KvY ... ... ... (ii)
(i) bs Ges (ii) bs †hvM K‡i cvB,
∠PBQ + ∠PCQ = 2 mg‡KvY
∴ PZzfR
z© BPCQ-G ∠PBQ + ∠PCQ = 2 mg‡KvY
A_©vr PZzfR
z© BPCQ Gi `ywU wecixZ †KvY ∠PBQ Ges
∠PCQ ci¯ži m¤žiK|
myZivs B, P, C, Q we›`y PviwU mge„Ë| (cÖgvwYZ)
Aby: 4. O †K›`ªwewkó e„‡Ëi AB I CD R¨v `yBwU e„‡Ëi Af¨š‡i
Aew¯’Z †Kv‡bv we›`y‡Z mg‡Kv‡Y wgwjZ n‡q‡Q| cÖgvY Ki †h,
∠AOD + ∠BOC = `yB mg‡KvY| [Xv: †ev:, w`: †ev: 09; h: †ev: 11,
09, 06, iv: †ev: 10, 07, 04; Kz: †ev: 09, 05, 02; P: †ev: 02, 07, 08, 11;
wm: †ev: 02, 08; e: †ev: 12]
DËi :
C
O
E
B
A
D
we‡kl wbe©Pb : g‡b Kwi, O †K›`ªwewkó e„‡Ëi AB I CD R¨v
`yBwU e„‡Ëi Af¨š‡i Aew¯’Z E we›`y‡Z mg‡Kv‡Y wgwjZ
n‡q‡Q| A, O Ges D, O †hvM Kivq ∠AOD Drcbœ nq| Avevi,
O, C Ges O, B †hvM Kivq ∠BOC Drcbœ nq| cÖgvY Ki‡Z n‡e
†h, ∠AOD + ∠BOC = `yB mg‡KvY|
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
504
A¼b : B, D †hvM Kwi|
cÖgvY :GKB Pvc AD-Gi Ici `Êvqgvb †K›`ª¯’ ∠AOD Ges
e„˯’ ∠ABD.
∴
∠ABD =
1
∠AOD
2
∠AOD + ∠BOC = 2 ∠ABD + 2 ∠BDC
ev, ∠AOD + ∠BOC = 2 (∠ABD + ∠BDC)
A_©vr, ∠AOD + ∠BOC = 2 (∠EBD + ∠EDB) ... ... ... (iii)
GLb, DE ⊥ AB. Zvn‡j, ∠DEB = GK mg‡KvY
∴ EBD mg‡KvYx wÎfz‡Ri,
∠EBD + ∠EDB = 1 mg‡KvY|
Zvn‡j, (iii) bs n‡Z cvB,
∠AOD + ∠BOC = 2 × 1 mg‡KvY = 2 mg‡KvY
∴ ∠AOD + ∠BOC = `yB
mg‡KvY| (cÖgvwYZ)
Aby: 6. ABCD PZzfz©‡Ri wecixZ †KvYØq ci¯ži m¤žiK|
AC †iLv hw` ∠BAD Gi mgwØLÊK nq, Z‡e cÖgvY Ki †h,
DËi :
Aby: 4. O †K›`ª w ewkó †Kv‡bv e„ ‡ Ëi AB I CD R¨v `y BwU
e„ ‡ Ëi Af¨ši¯’ E we›`y ‡ Z †Q` Ki‡j cª gvY Ki †h,
1
(∠BOD + ∠AOC).
2
[iv: †ev: 11; P: †ev: 05; Kz: †ev: 04, 01; iv: †ev: 09, 02; h: †ev: 12]
∠AEC =
[Q e„‡Ëi GKB Pv‡ci Ici `Êvqgvb e„˯’ †KvY †K›`ª¯’
†Kv‡Yi A‡a©K]
A_©vr, ∠AOD = 2∠ABD
... ... ... (i)
Abyi‡ƒ c †`Lv‡bv hvq †h,
∴ ∠BOC = 2 ∠BDC
... ... ... (ii)
(i) bs I (ii) bs †hvM K‡i cvB,
[Kz: †ev: 11; wm: †ev: 04; e: †ev: 07]
BC = CD.
Abykxjbx-10.6 : e„Ë m¤žK©xq
A
B
C
D
we‡kl wbe©Pb : ABCD PZzfz©‡Ri wecixZ †KvYØq ci¯ži
m¤žiK| AC †iLv, ∠BAD-Gi mgwØLÛK| cÖgvY Ki‡Z n‡e †h,
BC = CD.
A¼b : B, D †hvM
Kwi|
cÖgvY : ABCD PZzfz©‡Ri wecixZ †KvYØq ci¯ži m¤žiK|
∴ A, B, C, D we›`y PviwU mge„Ë|
[Q PZzf‡z© Ri `yB wecixZ †KvY m¤žiK n‡j Bnvi kxl©we›`y PviwU mge„Ë]
AC, ∠BAD Gi mgwØLÊK
[†`Iqv Av‡Q ]
∴ ∠BAC = ∠DAC ... ... ... (i)
GLb, GKB Pvc CD-Gi Ici e„˯’ ∠DAC Ges e„˯’ ∠DBC.
∴ ∠DAC = ∠DBC ... ... ... (ii)
DËi :
A
D
O
E
C
B
we‡kl wbe©Pb : O †K›`ªwewkó ACBD e„‡Ëi AB I CD R¨v `yBwU
e„‡Ëi Af¨ši¯’ E we›`y‡Z †Q` K‡i‡Q| O, A ; O, D; O, B I O,
C †hvM Kiv n‡jv| AC I BD PvcØq †K‡›`ª h_vµ‡g ∠AOC Ges
∠BOD Drcbœ K‡i| cªgvY Ki‡Z n‡e †h,
1
(∠BOD + ∠AOC).
2
A, D †hvM Kwi|
∠AEC =
A¼b :
cªgvY : Rvbv Av‡Q, e„‡Ëi GKB Pv‡ci Ici `Êvqgvb e„˯’ †KvY
†K›`ª¯’ †Kv‡Yi A‡a©K|
e„‡Ëi AC Pv‡ci Ici `Êvqgvb †K›`ª¯’ ∠AOC Ges e„˯’ ∠ADC
1
∠AOC ... ... ... (i)
2
Avevi, e„‡Ëi BD Pv‡ci Ici `Êvqgvb
e„˯’ ∠BAD.
1
∴ ∠BAD = ∠BOD ... ... ... (ii)
2
∴ ∠ADC =
†K›`ª¯’ ∠BOD Ges
Avevi, Rvbv Av‡Q, wÎfz‡Ri †h‡Kv‡bv ewnt¯’ †KvY Bnvi Aš:¯’
wecixZ †KvY؇qi mgwói mgvb|
GLb, ∆ADE Gi ewn:¯’ ∠AEC Ges Bnvi Aš:¯’ `yB wecixZ
∠ADE I ∠DAE
∴
∠ADE +∠DAE = ∠AEC
ev, ∠AEC = ∠ADC + ∠BAD
1
1
ev, ∠AEC = 2 ∠AOC + 2 ∠BOD [(i) Ges (ii) bs †_‡K]
∴
∠AEC =
1
(∠BOD + ∠AOC).
2
(cªgvwYZ)
[we: `ª: Kzwgj−v †evW©-2001 cÖkœc‡Î 'E'Gi ¯’‡j 'K' †`Iqv n‡q‡Q|]
Aby: 5. O †K›`ªwewkó †Kv‡bv e„‡Ëi AB I CD R¨v `yBwU e„‡Ëi
ewnt¯’ E we›`y‡Z †Q` Ki‡j cªgvY Ki †h,
∠AEC =
1
(∠BOD ∼ ∠AOC).
2
[iv: †ev: 06, Kz: †ev: 02]
GKB Pv‡ci Ici `Êvqgvb e„˯’ †KvY¸‡jv mgvb]
Avevi, GKB Pvc BC-Gi Dci e„˯’ ∠BAC Ges ∠BDC.
∴ ∠BAC = ∠BDC
[H GKB Kvi‡Y]
ev, ∠DAC = ∠BDC
[ (i) bs †_‡K ]
[ (ii) bs †_‡K ]
ev, ∠DBC = ∠BDC
A_©vr ∆ BCD-Gi,
DËi :
∠BDC = ∠DBC
∴ BC = CD
[Q mgvb mgvb †Kv‡Yi wecixZ evûØq ci¯ži mgvb ]
we‡kl wbe©Pb : O †K›`ªwewkó ACDB e„‡Ëi AB I CD R¨v `yBwU
e„‡Ëi ewnt¯’ †Kv‡bv we›`y E-†Z †Q` Kij| O, A; O, B; O, D I
O, C †hvM Kiv n‡jv|
[Q e„‡Ëi
(cÖgvwYZ)
A
O
B
E
D
C
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
505
Zvn‡j, AC I BD PvcØq †K‡›`ª h_vµ‡g ∠AOC Ges ∠BOD
Drcbœ Kij| cªgvY Ki‡Z n‡e,
A_© v r
1
∠AEC = (∠BOD ∼ ∠AOC)
2
1
1
∠AEC = (∠BOD – ∠AOC) A_ev ∠AEC = (∠AOC – ∠BOD)
2
2
A¼b : B, C †hvM Kwi|
cªgvY : Rvbv Av‡Q, e„‡Ëi GKB Pv‡ci Ici `Êvqgvb e„Ët¯’
†KvY †K›`ª¯’ †Kv‡Yi A‡a©K|
e„‡Ëi AC Pv‡ci Ici `Êvqgvb †K›`ª¯’ ∠AOC I e„˯’ ∠ABC .
∴
∠BCD.
†K›`ª¯’ ∠BOD Ges e„˯’
1
∠BOD
2
1
∠BCE = ∠BOD ... ... ... (ii)
2
∠BCD =
ev,
Avevi, Rvbv Av‡Q, wÎfz‡Ri †h‡Kvb ewnt¯’ †KvY Bnvi Ašt¯’
`yB wecixZ †Kv‡Yi mgwói mgvb|
GLb, ∆BCE Gi ewnt¯’ ∠ABC Ges Ašt¯’ `y B wecixZ
∠BCE I ∠BEC .
∴
∴
∠APB = ∠AQB
A_©vr, ∠APQ = ∠AQP
GLb, ∆ APQ-G ∠APQ = ∠AQP
∴ AQ = AP
[ Q wÎfz‡Ri mgvb mgvb
myZivs ∆ PAQ mgwØevû|
†Kv‡Yi wecixZ evûØq ci¯ži mgvb ]
(cªgvwYZ)
Abykxjbx-10.7 : e„Ë m¤žK©xq
∠ABC =
Avevi, e„‡Ëi
Q
1
∠AOC ... ... ... (i)
2
BD Pv‡ci Ici `Êvqgvb
Avevi, Rvbv Av‡Q, mgvb mgvb e¨vmva©wewkó e„‡Ë mgvb mgvb
Pv‡ci Ici `Êvqgvb e„˯’ †KvYMy‡jv ci¯ži mgvb|
mgvb mgvb Pvc AFB Ges AEB Gi Ici `Êvqgvb h_vµ‡g,
e„˯’ ∠APB Ges e„˯’ ∠AQB
Aby: 1. O †K›`ªwewkó GKwU e„‡Ëi ewnt¯’ †Kv‡bv we›`y P
†_‡K e„‡Ë `yBwU ¯žk©K Uvbv n‡jv| cªgvY Ki †h, OP mij‡iLv
¯žk© R¨v-Gi j¤^wØLÊK| [Kz: †ev: 11; wm: †ev: 12, 10, 05; Xv: †ev: 04;
h: †ev: 07, 04; e: †ev: 05, 02; iv: †ev: 12, 09; P: †ev: 09, 07; w`: †ev: 12]
A_ev, O †K›`ªwewkó GKwU e„‡Ëi ewnt¯’ †Kv‡bv we›`y D †_‡K
e„‡Ë `yBwU ¯žk©K Uvbv n‡jv| cªgvY Ki †h, OD mij‡iLv ¯žk©
[Xv: †ev: 11]
R¨v Gi j¤^wØLÊK|
B
DËi :
∠ABC = ∠BCE + ∠BEC
ev, ∠ABC – ∠BCE = ∠BEC
O
1
1
ev, 2 ∠AOC – 2 ∠BOD = ∠AEC [(i) bs Ges (ii) bs †_‡K ]
1
1
∠AOC – ∠BOD
2
2
wPÎvby h vqx Pvc AC > Pvc BD A_© v r
1
∠AOC > ∠BOD e‡j, ∠AEC = (∠AOC – ∠BOD)
2
wKšy hw` Pvc BD > Pvc AC, A_© v r ∠BOD > ∠AOC nq
1
Z‡e, ∠AEC = 2 (∠BOD – ∠AOC) n‡e|
1
A_©vr Dc‡ii kZ©mv‡c‡¶, ∠AEC = 2 (∠BOD ∼ ∠AOC) (cªgvwYZ)
∴
∠AEC =
Aby: 6. `yBwU mgvb e¨vmva©wewkó e„‡Ëi mvaviY R¨v AB| B we›`y
w`‡q Aw¼Z †Kv‡bv mij‡iLv hw` e„Ë `yBwUi mv‡_ P I Q
we›`y‡Z wgwjZ nq, Z‡e cªgvY Ki †h, ∆ PAQ mgwØevû|
[h: †ev: 05; wm: †ev: 01]
A
DËi :
F
E
P
B
Q
we‡kl wbe©Pb : g‡b Kwi, APBF Ges AQBE mgvb e¨vmva© wewkó
`yBwU e„Ë| AB Zv‡`i mvaviY R¨v| B we›`y w`‡q Aw¼Z
mij‡iLv APBF e„ˇK P we›`y‡Z I AQBE e„ˇK Q we›`y‡Z †Q`
K‡i| A, P Ges A, Q †hvM Kiv n‡jv|
cªgvY Ki‡Z n‡e †h, ∆PAQ mgwØevû|
cªgvY : Rvbv Av‡Q, mgvb mgvb e¨vmva© wewkó e„‡Ë mgvb mgvb
R¨v mgvb mgvb Pvc wQbœ K‡i|
APBF Ges AQBE e„Ë `yBwUi mvaviY R¨v AB.
∴ Pvc AFB = Pvc AEB
C
P
A
we‡kl wbe©Pb : O †K›`ªwewkó e„‡Ëi ewnt¯’ †Kv‡bv we›`y P †_‡K
Aw¼Z `yBwU ¯žk©K PA I PB e„ˇK A I B we›`y‡Z ¯žk©
K‡i‡Q| A, B †hvM Kivq AB ¯žk©-R¨v cvIqv †Mj| P, O †hvM
Kiv nj| OP mij‡iLv ¯žk©-R¨v AB ‡K C we›`y‡Z †Q` K‡i‡Q|
cªgvY Ki‡Z n‡e †h, OP ¯žk©-R¨v AB Gi j¤^wØLÊK|
A¼b : O, A Ges O, B †hvM Kwi|
cªgvY : e„‡Ëi ewnt¯’ we›`y P †_‡K PA Ges PB `ywU ¯žk©K|
∴ PA = PB
[Q e„‡Ëi
ewnt¯’ †Kv‡bv ewnt¯’ we›`y †_‡K ¯žk© we›`y؇qi `iZ¡ mgvb]
GLb, ∆OAP Ges ∆OBP-G,
PA = PB
OA = OB
[GKB e„‡Ëi e¨vmva© ]
Ges OP Dfq wÎfz‡Ri mvaviY evû|
†hLv‡b AwZfzR OP = AwZfzR OP
∴ ∆ OAP ≅ ∆ OBP [wÎfzR؇qi wZbwU Abyiƒc evû ci¯ži mgvb]
myZivs ∠AOP = ∠BOP
A_©vr ∠AOC = ∠BOC ... ... ... (i)
GLb, ∆ OAC Ges ∆ OBC-G,
OA = OB,
[ GKB e„‡Ëi e¨vmva© ]
OC Dfq wÎfz‡Ri mvaviY evû
© ∠BOC [ (i) bs †_‡K cvB ]
Ges Ašfz³
© ∠AOC = Ašfz³
∴ ∆ OAC ≅ ∆ OBC
[Dfq wÎfz‡Ri `ywU Abyiƒc evû Ges Zv‡`i Ašfy³
© †KvYØq mgvb]
∴ AC = BC ... ... ... (ii)
Ges ∠OCA = ∠OCB
wKš‘ Giv ˆiwLK hyMj †KvY e‡j cÖ‡Z¨‡KB mg‡KvY|
∴ ∠OCA = ∠ OCB = 1 mg‡KvY
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
506
A_©vr OP ⊥ AB ... ... ... (iii)
(ii) bs Ges (iii) bs †_‡K cvB,
AC = BC A_©vr C ¯žk©-R¨v AB Gi ga¨we›`y Ges OP ⊥ ¯žk©-R¨v
cªgvY : Avgiv Rvwb, e„‡Ëi ewnt¯’ †Kv‡bv we›`y †_‡K e„‡Ë `yBwU ¯žk©K
Uvb‡j, H we›`y †_‡K ¯žk© we›`y؇qi `iZ¡ mgvb|
∴ AE = AD;
[Q O †Kw›`ªK e„‡Ëi ewnt¯’ we›`y A n‡Z `yBwU ¯•k©K]
GLb, ∆ OAD Ges ∆ OAE -G
AD = AE,
OD = OE [Q GKB e„‡Ëi e¨vmva© ]
Ges OA Dfq wÎfz‡Ri mvaviY evû|
∴
∆ OAD ≅ ∆ OAE
[ Q wÎfzR؇qi Abyiƒc evûÎq ci¯ži mgvb ]
∴ ∠OAD = ∠OAE
∴
OA, ∠DAE Z_v ∠BAC-Gi mgwØLÊK|
A_©vr, O we›`ywU ∠BAC-Gi mgwØLÛ‡Ki Dci Aew¯’Z|
Abyicƒ fv‡e, †`Lv‡bv hvq †h, M I N we›`y ∠ABC-Gi
AB.
∴ OP
¯žk©-R¨v AB- Gi j¤^wØLÊK| (cªgvwYZ)
[we: `ª: ewikvj †evW©-2005 cÖkœc‡Î 'P' Gi ¯’‡j 'T' Ges 'OP' Gi
¯’‡j 'OT' †`Iqv n‡q‡Q |]
Aby: 2. †`Iqv Av‡Q, O e„‡Ëi †K›`ª Ges PA I PB ¯žk©KØq
e„ˇK h_vµ‡g A I B we›`y‡Z ¯žk© K‡i‡Q| cªgvY Ki †h, PO,
∠APB †K mgwØLwÊZ K‡i| [wm: †ev:, e: †ev: 12, 03; Kz: †ev:, h: †ev: 00]
DËi :
A
O
mgwØLÛ‡Ki Dci Aew¯’Z|
wKšy ∠BAC- Gi AšwØ©LÊK G‡Ki AwaK mij‡iLv
n‡Z cv‡i bv|
myZivs OA, MA I NA GKB mij‡iLv|
∴ O, M, N †K›`ªÎq mg‡iL| (cªgvwYZ)
P
B
we‡kl wbe©Pb : g‡b Kwi, O †K›`ªwewkó e„‡Ëi ewn:¯’ we›`y P
†_‡K Aw¼Z `yBwU ¯žk©K PA I PB e„ˇK h_vµ‡g A I B
we›`y‡Z ¯žk© K‡i‡Q| e„‡Ëi †K›`ª O Ges ewnt¯’ P we›`yi
ms‡hvRK mij‡iLv PO|
cªgvY Ki‡Z n‡e †h, PO, ∠APB †K mgwØLwÊZ K‡i|
A_©vr ∠APO = ∠BPO.
A¼b : O, A Ges O, B †hvM Kwi|
cªgvY : e„‡Ëi ewnt¯’ we›`y P †_‡K PA Ges PB `yBwU ¯žk©K|
∴ PA = PB [ Q e„‡Ëi ewnt¯’ †Kv†bv we›`y †_‡K ¯žk© we›`y؇qi `iZ¡ mgvb ]
GLb, ∆ OAP Ges ∆ OBP-G
PA = PB;
OA = OB [ Q GKB e„‡Ëi e¨vmva© ]
Ges OP, Dfq wÎfz‡Ri mvaviY evû|
∴ ∆ OAP ≅ ∆ OBP [Q wÎfzR؇qi wZbwU Abyic
ƒ evû ci¯ži mgvb]
myZivs ∠APO = ∠BPO
A_©vr OP, ∠APB †K mgwØLwÊZ K‡i| (cªgvwYZ)
Aby: 3. cªgvY Ki †h, †hme e„Ë `yBwU ci¯ži‡ŽQ`x mij‡iLv؇qi
[P: †ev: 10]
cª‡Z¨KwU‡K ¯žk© K‡i, Zv‡`i †K›`ªmgn mg‡iL|
DËi : mvaviY wbe©Pb : cªgvY Ki‡Z n‡e †h, †hme e„Ë `yBwU
ci¯ži‡ŽQ`x mij‡iLv؇qi cª‡Z¨KwU‡K ¯žk© K‡i, Zv‡`i
†K›`ªmgn mg‡iL|
C
I
E
O
M
A
O
B
D
C
we‡kl wbe©Pb : g‡b Kwi, AB GKwU e„‡Ëi e¨vm Ges BC e„‡Ëi
e¨vmv‡a©i mgvb GKwU R¨v| e„ËwUi †K›`ª O. e„‡Ëi A I C we›`y‡Z
Aw¼Z ¯žk©KØq D we›`y‡Z wgwjZ n‡q‡Q| A, C †hvM Kivq ACD
wÎfzR Drcbœ n‡q‡Q|
cªgvY Ki‡Z n‡e †h, ∆ACD mgevû|
A¼b : O, C †hvM Kwi|
cªgvY : ∆ OBC-G, OB = OC [ GKB e„‡Ëi e¨vmva© ]
Ges BC = OB
[†`Iqv Av‡Q, R¨v BC e„ËwUi e¨vmv‡a©i mgvb]
∴
OB = BC = OC
∠OBC = ∠OCB = ∠BOC = 60°
[ mgevû wÎfz‡Ri
N
D
F
H
[h: †ev: 11, 02; e: †ev: 07]
DËi :
A_©vr ∆ OBC mgevû|
G
A
Aby: 6. AB †Kv‡bv e„‡Ëi e¨vm Ges BC e¨vmv‡a©i mgvb GKwU
R¨v| hw` A I C we›`y‡Z Aw¼Z ¯žk©KØq ci¯ži D we›`y‡Z
wgwjZ nq, Z‡e cªgvY Ki †h, ACD GKwU mgevû wÎfzR|
B
we‡kl wbe©Pb : g‡b Kwi, AB Ges AC `yBwU ci¯ži‡ŽQ`x mij‡iLv
ci¯ži‡K A we›`y‡Z †Q` K‡i‡Q| O, M, N †K›`ªwewkó wZbwU e„Ë
AB †iLvsk‡K h_vµ‡g D, F, H we›`y‡Z Ges AC †iLvsk‡K h_vµ‡g
E, G, I we›`y‡Z ¯žk© K‡i‡Q| cªgvY Ki‡Z n‡e †h, O, M, N
†K›`ªÎq mg‡iL|
A¼b : O, A; M, A; N, A †hvM Kwi| Avevi, O, E ; O, D †hvM Kwi|
cªwZwU †KvY 60°]
∴ ∠ABC = 60°
Avevi, A we›`y‡Z AD GKwU ¯•k©K Ges AC ¯•k©
we›`yMvgx R¨v|
[GKvši e„Ëvsk †KvY e‡j]
∴ ∠DAC = ∠ABC
∴ ∠DAC = 60°
Avevi, D we›`y n‡Z AD Ges DC e„‡Ëi `ywU ¯•k©K
e‡j AD = DC.
∴ ∆ADC-G AD = DC,
∴ ∠ACD = ∠DAC = 60°
[ wÎfz‡Ri
mgvb mgvb evûi wecixZ †KvYØq ci¯•i mgvb]
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
507
GLb, ∆ ACD-Gi,
∠ACD + ∠CAD + ∠ADC = 180°
[ wÎfz‡Ri
wZb‡Kv‡Yi mgwó `yB mg‡KvY ev 180° Gi mgvb]
ev, 60° + 60° + ∠ADC = 180°
[ (ii) Ges (iii) bs
†_‡K ]
ev, ∠ADC = 180° – 120°
∴
∠ADC = 60° ... ... ... (iv)
myZivs (ii) bs , (iii) bs Ges (iv) bs †_‡K †`Lv hv‡ŽQ †h,
∆ ACD Gi cªwZwU †KvY 60°.
Avevi †h‡nZz cÖwZwU †KvY mgvb †m‡nZz cÖwZwU †Kv‡Yi
wecixZ evûI mgvb A_©vr CD = AD = AC.
∴ ∆ ACD GKwU mgevû wÎfzR| (cªgvwYZ)
myZivs, ∠PAQ = GK mg‡KvY| (cÖgvwYZ)
[we. `ª. h‡kvi †evW©-2009 cÖkœc‡Î ÔAÕ I ÔTT′Õ Gi ¯’‡j h_vµ‡g ÔOÕ I
ÔRRÕ †`Iqv n‡q‡Q|]
Aby: 4. †`Iqv Av‡Q, A I B e„Ë؇qi †K›`ª Ges C e„Ë؇qi
¯žk©we›`y| C we›`y w`‡q Aw¼Z †iLvsk e„ËØq‡K P I Q we›`y‡Z
†Q` K‡i‡Q| cÖgvY Ki †h, AP | | BQ.
[Xv: †ev: 06, wm: †ev: 11, 04, 06, e: †ev: 09, 04; Kz: †ev: 12, 05, 03; P: †ev:
06, 03; h: †ev: 08, 05, 01; iv: †ev: 10, 08; w`: †ev: 11]
DËi :
P
A
B
B
C
C
A
Abykxjbx-10.8 : e„Ë m¤žK©xq
Q
Aby: 2. †`Iqv Av‡Q, `yBwU e„Ë A we›`y‡Z ewnt¯žk© K‡i‡Q|
TT´ ¯žk©K e„Ë `yBwU‡K P I Q we›`y‡Z ¯žk© K‡i‡Q| cÖgvY Ki
†h, ∠PAQ = GK mg‡KvY| [Xv: †ev:, h: †ev: 12, 09; wm: †ev: 08; Kz:
†ev: 08, 07; P: †ev: 04; w`: †ev: 10; e: †ev: 10]
DËi :
O
A
O´
T
P
R
Q
T
©
we‡kl wbe©Pb : g‡b Kwi, O I O © †K›`ªwewkó `yBwU e„Ë A we›`y‡Z
ewnt¯žk© K‡i‡Q| GKwU mij mvaviY ¯žk©K TT´ H e„Ë `yBwU‡K P I
Q we›`y‡Z ¯žk© K‡i‡Q| P, A; Q, A †hvM Kwi|
cÖgvY Ki‡Z n‡e †h, ∠PAQ = GK mg‡KvY|
A¼b : e„ËØq A we›`y‡Z ewnt¯žk© K‡i‡Q| myZivs A we›`y‡Z Dfq
e„‡Ëi Rb¨ GKwU mvaviY ¯žk©K _vK‡e| A we›`y w`‡q AR mvaviY
¯žk©KwU Uvwb †hb Zv TT © †K R we›`y‡Z †Q` K‡i|
cÖgvY : PR Ges AR, O †Kw›`ªK e„‡Ëi ewnt¯’ R we›`y n‡Z Aw¼Z ¤žk©K|
∴
PR = AR [Q e„‡Ëi ewnt¯’ we›`y n‡Z ¯žk© we›`y؇qi `iZ¡ mgvb]
A_©vr ∆ PAR-G PR = AR
Q
P
wPÎ-1
wPÎ-2
we‡kl wbe©Pb : g‡b Kwi, A I B †K›`ªwewkó `yBwU e„Ë C we›`y‡Z ¯žk©
K‡i‡Q| C we›`y w`‡q Aw¼Z PQ †iLvsk e„Ë `yBwU‡K P I Q we›`y‡Z
†Q` K‡i‡Q| A, P Ges B, Q †hvM Kiv n‡jv|
cÖgvY Ki‡Z n‡e †h, AP | | BQ.
A¼b : A, C I B, C †hvM Kwi|
cÖgvY : A †Kw›`ªK I B †Kw›`ªK e„ËØq ci¯ži‡K C we›`y‡Z ¯žk© K‡i‡Q|
∴ A, B, C mg‡iL|
[Q `yBwU e„Ë ci¯ži ¯žk© Ki‡j,Zv‡`i †K›`ªØq I ¯žk© we›`y
mg‡iL nq ]
GLb ∆ PAC-G, AP = AC [GKB e„‡Ëi e¨vmva© ]
∴ ∠ACP = ∠APC
[Q wÎfz‡Ri mgvb mgvb
Z`ªc~ , ∠BCQ = ∠BQC
evû؇qi wecixZ †KvYØq ci¯ži mgvb ]
e„ Ë Ø‡qi Aš:¯ž‡k© i (wPÎ-1) †¶‡Î mvaviY †KvY Ges
ewn:¯ž‡k© i (wPÎ-2) †¶‡Î wecÖ Z xc †KvY nIqvq,
∠ ACP = ∠BCQ
ev, ∠APC = ∠BQC [ Q ∠ACP = ∠APC Ges ∠BCQ = ∠BQC ]
wKšy Giv e„Ë؇qi Aš:¯ž‡k©i ((wPÎ-1) †¶‡Î Abyiƒc Ges
ewnt¯ž‡k©i (wPÎ-2) †¶‡Î GKvši †KvY hv‡`i †Q`K PQ.
A_©vr, AP | | BQ (cÖgvwYZ)
∴
∠PAR = ∠APR ... ... (i)
[ Q wÎfz‡Ri mgvb mgvb evûi wecixZ †KvYØq mgvb]
Avevi, QR Ges AR,O‡Kw›`ªK e„‡Ëi ewnt¯’ we›`y n‡Z Aw¼Z
Aby: 5. AB, AC †Kv‡bv e„‡Ëi `yBwU mgvb R¨v| cÖgvY Ki †h,
BC †iLvsk A we›`y‡Z e„‡Ëi ¯žk©‡Ki mgvšivj|
¯žk©K|
DËi :
∴
QR = AR
A_©vr ∆ ARQ-G, QR = AR
∴
∠QAR = ∠AQR ... ... (ii)
[Q wÎfz‡Ri mgvb mgvb evûi wecixZ †KvYØq mgvb ]
(i) bs Ges (ii) bs †hvM K‡i cvB,
∠APR + ∠AQR = ∠PAR + ∠QAR
ev, ∠APR + ∠AQR = ∠PAQ [Q ∠PAQ = ∠PAR + ∠QAR]
ev, ∠APQ + ∠AQP = ∠PAQ
ev, ∠APQ + ∠AQP + ∠PAQ = 2 ∠PAQ
[Dfqc‡¶ ∠PAQ †hvM K‡i ]
ev, 2 mg‡KvY = 2 ∠PAQ
ev, 2 ∠PAQ = 2 mg‡KvY
[Q ∆ APQ-Gi wZb †Kv‡Yi mgwó 2 mg‡KvY Gi mgvb ]
[P: †ev: 11; Kz: †ev: 06, 04; iv: †ev: 00; 02; wm: †ev: 09; e: †ev: 01]
A
T
O
B
C
we‡kl wbe©Pb : g‡b Kwi, O †K›`ªwewkó ABC e„‡Ëi `yBwU R¨v
AB I AC ci¯ži mgvb A_©vr AB = AC. B, C †hvM Kwi| AT
†iLv A we›`y‡Z Aw¼Z ¯žk©K|
cÖgvY Ki‡Z n‡e †h, BC †iLv A we›`y‡Z AT ¯žk©‡Ki mgvšivj
A_©vr BC | | AT.
cÖgvY : ∆ABC-G
AB = AC
[†`Iqv Av‡Q ]
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
508
∴
∠ACB = ∠ABC
[Q wÎfz„‡Ri mgvb mgvb evûi wecixZ †KvYØq ci¯ži mgvb]
GLb, ABC e„‡Ëi A we›`y‡Z AT ¯žk©K Ges AC ¯žk© we›`yMvgx R¨v|
∴ ∠TAC = GKvši e„Ëvskw¯’Z ∠ABC
ev, ∠TAC = ∠ACB [ Q ∠ABC = ∠ACB ]
wKšy Giv GKvši †KvY; hv‡`i †Q`K AC.
∴ BC | | AT. (cÖgvwYZ)
Aby: 6. `yBwU e„Ë O we›`y‡Z Ašt¯žk© K‡i‡Q Ges Zv‡`i
GKwUi e¨vmva© AciwUi e¨v‡mi mgvb| OPQ †iLvsk ¶z`ªZi
e„ËwU‡K P we›`y‡Z Ges e„nËi e„ËwU‡K Q we›`y‡Z †Q` K‡i‡Q|
cÖgvY Ki †h, OP = PQ. [e: †ev: 11; Xv: †ev: 03, 12; h: †ev: 03; P: †ev:
12, 05, 01; Kz: †ev: 10; w`: †ev: 09; wm: †ev: 07, 02]
DËi :
Q
P
M
N
A¼b : P we›`y‡Z e„Ë `yBwUi mvaviY ¯žk©K TPE Uvwb|
TPE ewa©Z AB †K E we›`y‡Z †Q` K‡i|
cÖgvY : ¶z`ªZi e„‡Ë EP I EC ewnt¯’ E we›`y n‡Z e„‡Ëi Ici `yBwU
¯žk©K|
∴ EP = EC [Q e„‡Ëi ewnt¯’
GLb, ∆ PCE-G, EP = EC
∴ ∠PCE = ∠CPE
we›`y †_‡K ¯žk©we›`y؇qi `iZ¡ mgvb]
[ wÎfz‡Ri mgvb mgvb evûi wecixZ †KvYØq ci¯ži mgvb]
... ... ... (i)
ev, ∠PCE = ∠BPC + ∠BPE
Avevi, ∆ ACP-Gi
∠APC + ∠CAP + ∠ACP = `yB mg‡KvY ... ... ... (ii)
wKš‘, CP iwk¥i cÖvš we›`y C-†Z AE mij‡iLv wgwjZ n‡q‡Q d‡j
∠PCE Ges ∠ACP mwbœwnZ †KvYØq Drcbœ n‡q‡Q|
∴ ∠PCE + ∠ACP = `yB mg‡KvY
... ... ... (iii)
(ii) bs Ges (iii) bs Zzjbv K‡i cvB,
∠APC + ∠CAP + ∠ACP = ∠PCE + ∠ACP
... ... ... (iv)
ev, ∠APC + ∠CAP = ∠PCE
(i) I (iv) bs Zzjbv K‡i cvB,
O
∠BPC + ∠BPE = ∠APC + ∠CAP
we‡kl wbe©Pb : g‡b Kwi, M I N †K›`ªwewkó `yBwU e„Ë O we›`y‡Z
Ašt¯žk© K‡i‡Q Ges M †Kw›`ªK e„nËi e„‡Ëi e¨vmva© OM, N
†Kw›`ªK ¶z`ªZi e„‡Ëi e¨v‡mi mgvb| OPQ †iLvsk ¶z`ªZi e„ˇK
P we›`y‡Z Ges e„nËi e„ˇK Q we›`y‡Z †Q` K‡i‡Q|
cªgvY Ki‡Z n‡e †h, OP = PQ.
A¼b : M, P †hvM Kwi|
cÖgvY : M I N e„Ë؇qi †K›`ª Ges O ¯žk©we›`y|
∴ M, N, O we›`y wZbwU mg‡iL|
[Q `yBwU
... ... ... (v)
wKš‘, ∠BPE = GKvši e„Ëvsk¯’ ∠CAP
AZGe, (v) bs n‡Z cvB, ∠BPC = ∠APC
myZivs PC, ∠APB Gi mgwØLÊK| (†`Lv‡bv n‡jv)
[we: `ª: XvKv †evW©-2002 cÖkœc‡Î P we›`yi ¯’‡j A. AB Gi ¯’‡j PQ ;
PL Gi ¯’‡j AC I ∠APB Gi ¯’‡j ∠PAQ †`Iqv n‡q‡Q|]
R¨vwgwZi mgvavb
m¤•v`¨
e„Ë ci¯ži ¯žk© Ki‡j, †K›`ªØq I ¯žk©we›`y mg‡iL]
N †Kw›`ªK e„‡Ë OM e¨vm|
∴ ∠MPO = GK mg‡KvY
[Q Aa©e„˯’ †KvY]
myZivs MP ⊥ OP A_©vr MP ⊥ OQ
M †Kw›`ªK e„‡Ë M †K›`ª Ges MP, OQ R¨v-Gi Ici j¤^|
∴ P, OQ Gi ga¨we›`y|
[Q e„‡Ëi †K›`ª †_‡K e¨vm wfbœ R¨v-Gi Ici Aw¼Z j¤^ R¨v †K
mgwØLwÊZ K‡i ]
Abykxjbx-3 : wÎfzR I PZzfz©R msµvš
Abywm. 3.1(6). wÎfz‡Ri `yBwU evû Ges Zv‡`i GKwUi wecixZ
m¶¥‡KvY †`Iqv Av‡Q| wÎfzRwU A¼b Ki| [Xv: †ev: 04; e: †ev: 06]
DËi : mvaviY wbe©Pb : wÎfz‡Ri `yBwU evû Ges Zv‡`i GKwUi
wecixZ m¶¥‡KvY †`Iqv Av‡Q| wÎfzRwU AuvK‡Z n‡e|
A_©vr OP = PQ. (cÖgvwYZ)
a
Aby: 9. `yBwU e„Ë P we›`y‡Z Ašt¯žk© K‡i‡Q| e„nËi e„‡Ëi AB R¨v
¶z`ªZi e„ËwU‡K C we›`y‡Z ¯žk© K‡i‡Q| †`LvI †h, PC †iLvsk
∠APB †K mgwØLwÊZ K‡i| [Xv: †ev: 10, 07, 05, 02; e: †ev: 08; h:
b
†ev: 10, 06; P: †ev: 08, 02; iv: †ev: 12, 05, 01; P: †ev: 00; Kz: †ev: 09]
DËi :
T
P
A
C
B
E
we‡kl wbe©Pb : g‡b Kwi, `yBwU e„Ë ci¯ži‡K P we›`y‡Z Ašt¯žk©
K‡i‡Q| e„nËi e„‡Ëi AB R¨v ¶z`ªZi e„ËwU‡K C we›`y‡Z ¯žk©
K‡i‡Q| A, P; B, P Ges C, P †hvM Kiv nj| †`Lv‡Z n‡e †h, PC
†iLv ∠APB †K mgwØLwÊZ K‡i|
M
A
a
b
x
b
x
B
C
C′
X
we‡kl wbe©Pb : g‡b Kwi, wÎfz‡Ri `yBwU evû h_vµ‡g a I b Ges
Zv‡`i GKwU wecixZ m¶¥‡KvY ∠x †`Iqv Av‡Q| wÎfzRwU AvuK‡Z
n‡e|
A¼b : †h †Kv‡bv iwk¥ BX Gi B we›`y‡Z ∠XBM = ∠x AvuwK|
BM iwk¥ †_‡K BA = a KvwU| A we›`y‡K †K›`ª K‡i b Gi mgvb
e¨vmva© wb‡q BX iwk¥i Ici GKwU e„ËPvc AvuwK| e„ËPvcwU BX
iwk¥‡K h_vµ‡g C I C′ we›`y‡Z †Q` K‡i| A, C I A, C′ †hvM Kwi|
Zvn‡j ABC Ges ABC′ DfqB DwÏó wÎfzR|
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
509
cÖgvY : A¼bvbymv‡i, ∆ABC-G BA = a, AC = b Ges ∠ABC = ∠x
Ges ∆ABC′-G BA = a, AC′ = b Ges ∠ABC′ = ∠x
∴ ∆ABC Ges ∆ABC′ DfqB wb‡Y©q wÎfzR| (cÖgvwYZ)
F
p
4
D
A
p
4
Aby: 1. GKwU wÎfz‡Ri f‚wg msjMœ `yBwU †KvY Ges kxl© †_‡K
f‚wgi Dci Aw¼Z j‡¤^i ˆ`N©¨ †`Iqv Av‡Q| wÎfzRwU AvuK|
p
4
[wm: †ev: 11; Xv: †ev 10, 06, wm: †ev: 07, 01; Kz: †ev: 10, 05, 02, h: †ev:, e: †ev: 02]
DËi : mvaviY wbe©Pb : GKwU wÎfy‡Ri f‚wgmsjMœ `yBwU †KvY Ges
kxl© n‡Z f‚wgi Ici Aw¼Z j‡¤^i ˆ`N©¨ †`Iqv Av‡Q| wÎfzRwU
AvuK‡Z n‡e|
h
x
y
p
4
B
E
C
we‡kl wbe©Pb : g‡b Kwi, †Kv‡bv eM©‡¶‡Îi cwimxgv p †`Iqv
Av‡Q| eM©‡¶ÎwU AvuK‡Z n‡e|
A¼b : cwimxgv p †K cÖ_‡g mgvb `yBfv‡M fvM K‡i Zvi A‡a©K
p
P
A
Ask‡K cybivq mgvb `yBfv‡M wef³ Kwi| d‡j ¶z`ªZg Ask 4 Gi
Q
x
p
mgvb n‡e| Gevi †h‡Kv‡bv iwk¥ BE n‡Z BC = 4 †K‡U †bB| BE
y
p
iwk¥i B we›`y‡Z ∠EBF = 90° AuvwK| BF n‡Z BA =4 wbB| AZtci
h
M
B
p
I C we›`y‡K †K›`ª K‡i 4 Gi mgvb e¨vmva© wb‡q ∠EBF Gi
Af¨š‡i `yBwU e„ËPvc AuvwK Ges g‡b Kwi e„ËPvc `yBwU ci¯ži‡K
D we›`y‡Z †Q` K‡i| A, D Ges C, D †hvM Kwi|
Zvn‡j, ABCD-B DwÏó eM©‡¶Î n‡e|
A
D
R
C
N
p
we‡kl wbe©Pb : g‡b Kwi, GKwU wÎfz‡Ri f‚wg msjMœ `yBwU †KvY ∠x I
∠y Ges kxl© we›`y †_‡K f‚wgi Ici Aw¼Z j‡¤^i ˆ`N©¨ h †`Iqv
Av‡Q| wÎfzRwU AvuK‡Z n‡e|
A¼b : †h‡Kv‡bv iwk¥ AR n‡Z AD = h †K‡U wbB| AD
†iLvs‡ki Ici A I D we›`y‡Z h_vµ‡g PAQ I MDN, AD
†iLvs‡ki Ici j¤^ A¼b Kwi| PQ †iLvi A we›`y‡Z PA Gi mv‡_
∠PAB = ∠x Ges QA Gi mv‡_ ∠QAC = ∠y AvuwK| AB I
AC iwk¥ `yBwU MN †iLv‡K h_vµ‡g B I C we›`y‡Z †Q` K‡i|
Zvn‡j, ∆ABC-B DwÏó wÎfzR|
cÖgvY : PQ Ges MN †iLvØq AD †iLvi Ici j¤^ e‡j Zviv mgvšivj|
∴ ∠ABC = GKvši ∠PAB = ∠x
Ges ∠ACB = GKvši ∠QAC = ∠y
AZGe, ∆ ABC-G
∠ABC = ∠x, ∠ACB = ∠y Ges DŽPZv AD = h
myZivs, ∆ ABC-B wb‡Y©q wÎfzR| (cªgvwYZ)
[we: `ª: Kzwgj−v †evW©-2002 cÖkœc‡Î j‡¤^i ˆ`N©¨ a †`Iqv n‡q‡Q]
cÖgvY : ABCD PZzf‡©y R AB = BC = CD = DA = 4 Ges
∠ABC = 90° [AKbvbymv‡i ]
∴ ABCD GKwU eM©‡¶Î
p
p
p
2
p
p
09, 08, 04; e: †ev: 09; Kz: †ev: 11, 08, 01; iv: †ev:, w`: †ev: 09; Xv: †ev: 08]
DËi : mvaviY wbe©Pb : †Kv‡bv mg‡KvYx wÎfz‡Ri AwZfzR I Aci
`yB evûi mgwó †`Iqv Av‡Q| wÎfzRwU AvuK‡Z n‡e|
a
s
H
A
D
A′
[iv: †ev: 10, 05; h: †ev: 12, 10, 08, 06,04; wm: †ev: 08, 03; Xv: †ev: 01; P:
†ev: 09, 02; e: †ev: 12, 07; Kz: †ev: 09]
p
4
p
Aby: 3. mg‡KvYx wÎfz‡Ri AwZfzR I Aci `yB evûi mgwó †`Iqv
Av‡Q| wÎfz RwU AvuK| [Xv: †ev: 11; Kz: †ev: 11; P: †ev: 11; wm: †ev:, P: †ev: 11,
Aby: 2. eM©‡¶‡Îi cwimxgv †`Iqv Av‡Q| eM©‡¶ÎwU AvuK|
DËi : mvaviY wbe©Pb : †Kv‡bv eM©‡¶‡Îi cwimxgv †`Iqv Av‡Q|
eM©‡¶ÎwU AvuK‡Z n‡e|
p
Ges AB + BC + CD + AB = 4 + 4 + 4 + 4 = 4 × 4 = p
AZGe, ABCD-B DwÏó eM©‡¶Î| (cªgvwYZ)
45°
E
B′
s
B
C
F
we‡kl wbe©Pb : g‡b Kwi, GKwU mg‡KvYx wÎfz‡Ri AwZfzR a Ges
Aci `yB evûi mgwó s †`Iqv Av‡Q| wÎfzRwU AvuK‡Z n‡e|
A¼b : †h‡Kv‡bv iwk¥ EF n‡Z EC = s †K‡U wbB| EC Gi E
we›`y‡Z ∠CEH = 45° AvuwK| C †K †K›`ª K‡i a Gi mgvb e¨vmva©
wb‡q GKwU EH Gi Ici e„ËPvc AvuwK Ges g‡b Kwi, e„ËPvcwU
EH †K A Ges A′ we›`y‡Z †Q` K‡i| A, C Ges A′, C †hvM Kwi|
GLb A Ges A′ n‡Z EC Gi Ici AB Ges A′B′ j¤^ Uvwb|
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
510
Zvn‡j, ABC Ges A′B′C DfqB DwÏó wÎfzR|
cÖgvY : AB, EC Gi Ici j¤^|
b
a
∴∠ABE = 90°
GLb, ∆ABE G ∠ABE = 90°
∠AEB = 45° [ A¼b Abymv‡i ]
∴ Aewkó ∠BAE = 45°
myZivs, ∆ AEB Gi ∠EAB = ∠AEB = 45°
∴ BE = AB
GLb mg‡KvYx ∆ ABC G,
AB + BC = BE + BC = EC = s Ges AwZfzR AC = a
myZivs, ∆ ABC-B DwÏó mg‡KvYx wÎfzR|
Abyi‡ƒ c †`Lv‡bv hvq †h, ∆A′B′C I DwÏó wÎfzR| (cªgvwYZ)
Aby: 4. wÎfz‡Ri f‚wg msjMœ GKwU †KvY, DŽPZv I Aci `yB evûi
mgwó †`Iqv Av‡Q| wÎfzRwU AvuK| [Kz: †ev:, iv: †ev: 12, 04; Xv: †ev:
09, 07, 03; iv: †ev:, h: †ev: 01; wm: †ev: 02; w`: †ev: 10; P: †ev: 12]
DËi : mvaviY wbe©Pb : †Kvb wÎfz‡Ri f‚wg msjMœ GKwU †KvY, DŽPZv
I Aci `yB evûi mgwó †`Iqv Av‡Q| wÎfzRwU AvuK‡Z n‡e|
s
h
R
N
M
x
A
P
h
Q
s
C′
F
G
H
D
C
z
y
b
z
x
y
a
A
B
E
we‡kl wbe©Pb : g‡b Kwi, PZzfz©‡Ri `yBwU evû a I b Ges wZbwU
†KvY ∠x, ∠y I ∠z †`Iqv Av‡Q| PZzfz©RwU AuvK‡Z n‡e|
A¼b : †h‡Kv‡bv iwk¥ AE n‡Z a = AB KvwU| B we›`y‡Z ∠ABF
= ∠y AuvwK| Gevi A we›`y‡Z ∠BAG = ∠x AuvwK| iwk¥ BF n‡Z
b-Gi mgvb K‡i BC Ask KvwU| Avevi BC †iLvi †h cv‡k¦© A
we›`y Aew¯’Z †m cv‡k¦© C we›`y‡Z ∠z = ∠BCH AuvwK| CH iwk¥,
AG iwk¥‡K D we›`y‡Z †Q` K‡i|
myZivs ABCD-B DwÏó PZzfR
z© |
ABCD
PZzf‡z© R
cªgvY : A¼b Abymv‡i
AB = a I BC = b.
∠DAB = ∠x, ∠ABC = ∠y
∠BCD = ∠z.
Ges
myZivs ABCD-B wb‡Y©q PZzfR
z© | (cÖgvwYZ)
h
B
x
C
E
we‡kl wbe©Pb : g‡b Kwi, †Kvb wÎfz‡Ri fwgmsjMœ GKwU †KvY
∠x, DŽPZv h I Aci `yB evûi mgwó s †`Iqv Av‡Q| wÎfzRwU
AvuK‡Z n‡e|
A¼b : †h‡Kv‡bv iwk¥ BE-Gi B we›`y‡Z ∠RBE = ∠x AvuwK| BR
iwk¥ n‡Z s-Gi mgvb K‡i BN Ask †K‡U wbB| Gevi B we›`y‡Z
BE Gi Ici BM j¤^ AvuwK| BM n‡Z DŽPZv h Gi mgvb K‡i BP
Ask KvwU| AZ:ci P we›`y‡Z PQ j¤^ AvuwK Ges g‡b Kwi, BR
iwk¥‡K PQ iwk¥ A we›`y‡Z †Q` Kij| Gevi AN Gi mgvb e¨vmva©
wb‡q BE Gi Ici GKwU e„ËPvc AvuwK| e„ËPvcwU BE †K C© I C
we›`y‡Z †Q` K‡i| A, C I A, C © †hvM Kwi|
myZivs ABC © I ABC DfqB DwÏó wÎfzR|
[ A¼b Abymv‡i ]
cÖgvY : ∆ ABC Gi ∠ABC = ∠x
Ges AB + AC = AB + AN = BN = s
[ A¼b Abymv‡i ]
Avevi, BE Ges PQ Df‡qB BM Gi Dci j¤^ e‡j BE | | PQ
Ges mgvšivj hyM‡ji DŽPZv BP = h
[ A¼b Abymv‡i ]
∆ ABC Gi DŽPZv h
AZGe, ∆ ABC DwÏó wÎfzR|
Abyi‡ƒ c cÖgvY Kiv hvq †h, ∆ABC′ I DwÏó wÎfzR| (cªgvwYZ)
Aby: 6. GKwU PZzf
© z‡Ri `yBwU evû I wZbwU †KvY †`Iqv Av‡Q|
PZzf
© zRwU AuvK| [w`: †ev: 11; h: †ev:, wm: †ev: 12, 09; Kz: †ev: 07, 03, P:
†ev: 03; iv: †ev: 02; e: †ev: 10]
DËi : mvaviY wbe©Pb : GKwU PZzf‡z© Ri `yBwU evû Ges wZbwU
†KvY †`qv Av‡Q| PZzfz©RwU AuvK‡Z n‡e|
Aby: 7. mgevû wÎfz‡Ri cwimxgv †`Iqv Av‡Q| wÎfzRwU AuvK|
[iv: †ev: 11, 08, 06; Xv: †ev: 12, 10, 05; P: †ev: 10, 05; wm: †ev: 10, 05;
e: †ev: 11, 05; Kz: †ev: 12, 11, 07; h: †ev: 11, 07; w`: †ev: 12]
DËi : mvaviY wbe©Pb : mgevû wÎfz‡Ri cwimxgv †`Iqv Av‡Q
wÎfzRwU A¼b Ki‡Z n‡e|
A
s
B
C
E
we‡kl wbe©Pb : g‡b Kwi, GKwU mgevû wÎfz‡Ri cwimxgv s †`Iqv
Av‡Q| wÎfzRwU A¼b Ki‡Z n‡e|
A¼b : BE †h‡Kv‡bv iwk¥ wbB| Gici cwimxgv s †K mgvb wZb
1
As‡k wef³ Kwi| Gici BE n‡Z BC = 3 s Ask †K‡U wbB| Gevi
BC evûi B I C we›`y‡K
1
†K›`ª K‡i BC Gi †h‡Kv‡bv GK cv‡k 3 s
Gi mgvb e¨vmva© wb‡q `yBwU e„ËPvc A¼b Kwi| g‡b Kwi,
e„ËPvcØq ci¯ži‡K A we›`y‡Z †Q` K‡i| cwi‡k‡l A, B; A, C
†hvM Kwi|
Zvn‡j ABC-B DwÏó wÎfzR|
1
1
1
cÖgvY : ∆ABC-Gi, cwimxgv = AB + BC + CA = 3s + 3s + 3s = s
∴ ABC-B
DwÏó wÎfzR| (cÖgvwYZ)
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
511
Abykxjbx-11 : e„Ë m¤žK©xq
Aby: 3. Giƒc GKwU e„Ë AvuK †hb Zv `yBwU mgvšivj mij‡iLv‡K
Ges Zv‡`i GKwU wbw`©ó †Q`K‡K ¯žk© K‡i|
[iv: †ev: 11; e: †ev: 08; Xv: †ev 06, Kz: †ev:, P: †ev: 12, 03; h: †ev: 10, 01; wm: †ev: 02]
DËi : mvaviY wbe©Pb : Giƒc GKwU e„Ë AvuK‡Z n‡e †hb Zv `yBwU
mgvšivj mij‡iLv‡K Ges Zv‡`i GKwU wbw`©ó †Q`K‡K ¯žk©
E
K‡i|
R
M
C
D
P
O
A
N
B
Q
F
we‡kl wbe©Pb : g‡b Kwi, AB I CD `yBwU mgvšivj mij‡iLv Ges
EF Zv‡`i GKwU wbw`©ó †Q`K| EF, CD †K M we›`y‡Z Ges AB †K N
we›`y‡Z †Q` K‡i| Ggb GKwU e„Ë AvuK‡Z n‡e, hv AB I CD
mgvšivj †iLvØq‡K Ges EF †Q`K‡K ¯žk© K‡i|
A¼b : ∠NMD Ges ∠MNB Gi mgwØLÊK `ywU iwk¥ h_vµ‡g MO
I NO AvuwK Ges g‡b Kwi, mgwØLÊKØq ci¯ži‡K O we›`y‡Z †Q`
K‡i| AZ:ci O †_‡K MN Gi Ici OP j¤^ Uvwb Ges g‡b Kwi,
OP, MN †K P we›`y‡Z †Q` K‡i| O †K †K›`ª K‡i OP Gi mgvb
e¨vmva© wb‡q GKwU e„Ë AvuwK|
Zvn‡j, GB e„ËwUB wb‡Y©q e„Ë n‡e|
cÖgvY : O n‡Z AB I CD †iLvs‡ki Ici h_vµ‡g OQ Ges OR
j¤^ Uvwb| g‡b Kwi, j¤^Øq AB I CD †K h_vµ‡g Q I R we›`y‡Z
†Q` K‡i|
OM, ∠NMD Gi mgwØLÊK|
∴ OP = OR
Avevi, ON, ∠MNB Gi mgwØLÊK|
∴ OP = OQ
Zvn‡j, OP = OQ = OR
myZivs, O †K †K›`ª K‡i OP Gi mgvb e¨vmva© wb‡q Aw¼Z e„Ë P,
Q Ges R we›`y w`‡q hv‡e|
Avevi, OP, OQ Ges OR Gi cÖvšwe›`y‡Z MN, NB Ges MD
†iLvsk wZbwU j¤^|
myyZivs e„ËwU †iLvsk wZbwU‡K h_vµ‡g P, Q I R we›`y‡Z ¯žk© K‡i|
AZGe, PQR-B wb‡Y©q e„Ë| (cÖgvwYZ)
Aby: 4. Ggb GKwU e„Ë AvuK hv GKwU wbw`©ó e„ˇK ¯žk© K‡i Ges
GKwU wbw`©ó mij‡iLv‡K GKwU wbw`©ó we›`y‡Z ¯žk© K‡i| [Xv: †ev: 10,
03; P: †ev: 07; wm: †ev: 07, 05, 01; h: †ev: 11, 05, 02; e: †ev: 10, 06, 05; w`: †ev: 12]
DËi : mvaviY wbe©Pb : Ggb GKwU e„Ë AvuK‡Z n‡e hv GKwU
wbw`©ó e„ˇK ¯žk© K‡i Ges GKwU wbw`©ó mij‡iLv‡K GKwU
wbw`©ó we›`y‡Z ¯žk© K‡i|
E
F
O
Aby: 5. †Kv‡bv e„‡Ë Ggb GKwU ¯žk©K AvuK †hb Zv †Kv‡bv wbw`©ó
mij‡iLvi mgvšivj nq|
[Kz: †ev: 04, 06, iv: †ev: 00; Xv: †ev: 08, 01; w`: †ev: 11, 09; P: †ev: 02]
DËi : mvaviY wbe©Pb : †Kvb e„‡Ë Ggb GKwU ¯žk©K AvuK‡Z n‡e
†hb Zv †Kv‡bv wbw`©ó mij‡iLvi mgvšivj nq|
A
M
K
R
Q©
E
C
O
D
B
N
P
F
we‡kl wbe©Pb : g‡b Kwi, O †K›`ªwewkó MNP GKwU e„Ë Ges
AB GKwU wbw`©ó mij‡iLv| MNP e„‡Ë Ggb GKwU ¯žk©K
AvuK‡Z n‡e hv, AB mij‡iLvi mgvšivj n‡e|
A¼b : O we›`y †_‡K AB Gi Ici OR j¤^ AvuwK| OR, AB †iLv‡K
K we›`y‡Z Ges MNP e„ˇK M we›`y‡Z †Q` K‡i| RO †K ewa©Z
Ki‡j ZvI e„ËwUi P we›`y‡Z †Q` K‡i| MP †iLvi Ici M I P
we›`y‡Z h_vµ‡g CD I EF j¤^ Uvwb|
Zvn‡j, CD ev EF-B wb‡Y©q ¯žk©K n‡e|
cÖgvY : CD Ges EF h_vµ‡g OM Ges OP e¨vmv‡a©i Ici j¤^|
myZivs CD Ges EF DfqB h_vµ‡g M I P we›`y‡Z e„‡Ëi `yBwU ¯žk©K|
wKš‘ A¼bvbymv‡i, ∠OMD = ∠MKB = GK mg‡KvY|
wKšy Zviv Abyiƒc †KvY|
∴CD | | AB
Avevi, ∠OPF = ∠RKB = GK
C
B
we‡kl wbe©Pb : g‡b Kwi, O GKwU wbw`©ó e„‡Ëi †K›`ª Ges wbw`©ó
†iLv PQ Gi Dci¯’ A GKwU wbw`©ó we›`y| Ggb GKwU e„Ë
A¼b Ki‡Z n‡e hv A we›`y‡Z PQ †iLv‡K I cÖ`Ë e„ˇK †Kv‡bv
we›`y‡Z ¯žk© K‡i|
A¼b : O n‡Z PQ Gi Ici OD j¤^ A¼b Kwi| DO †K ewa©Z
Kivq Zv e…‡Ëi cwiwai mv‡_ E we›`y‡Z wgwjZ nq| E, A †hvM
Kwi| EA cÖ`Ë e„ˇK B we›`y‡Z †Q` K‡i| A we›`y‡Z AF ⊥ PQ
AvuwK| O, B †hvM K‡i ewa©Z Kivq Zv AF †K Q© we›`y‡Z †Q` K‡i|
Q-© B DwÏó e„‡Ëi †K›`ª| Q© †K †K›`ª K‡i QA
© ev QB
© Gi mgvb
e¨vmva© wb‡q ABC e„Ë A¼b Kwi| Zvn‡j ABC-B DwÏó e„Ë|
© P = 90°
cÖgvY : AF ⊥ PQ ∴ ∠QA
∴ e„ËwU A we›`y‡Z PQ †iLv‡K ¯žk© K‡i| B we›`ywU e„Ë؇qi mvaviY
we›`y Ges Zv‡`i †K›`ªØ‡qi ms‡hvRK mij‡iLvi Ici Aew¯’Z|
AZGe, e„Ë `yBwU B we›`y‡Z ci¯ži‡K ¯žk© K‡i‡Q|
∴ABC B DwÏó e„Ë| (cÖgvwYZ)
mg‡KvY|
wKšy Zviv GKvši †KvY|
P
D
A
Q
∴ EF | | AB
A_©vr, CD Ges EF ¯žk©KØq
AZGe
CD ev EF-B
DfqB AB †iLvi mgvšivj|
wb‡Y©q ¯žk©K| (cÖgvwYZ)
MwYZ (Avewk¨K) Q Extra- R¨vwgwZi mgvavb
512
Aby: 6. †Kv‡bv e„‡Ë Ggb GKwU ¯žk©K AvuK †hb Zv †Kv‡bv wbw`©ó
mij‡iLvi Ici j¤^ nq|[Xv: †ev: 11, P: †ev: 11, 09, 06, iv: †ev: 09, 07,
04; wm: †ev:, e: †ev: 07, 04; Kz: †ev: 11, 00; h: †ev: 11, 08, 03]
DËi : mvaviY wbe©Pb : †Kv‡bv e„‡Ë Ggb GKwU ¯žk©K AvuK‡Z
n‡e †hb Zv †Kv‡bv wbw`©ó mij‡iLvi Ici j¤^ nq|
K
M
C
A¼b : OA †h‡Kv‡bv e¨vmva© wbB Ges O we›`y‡Z ∠AOB = 120° AvuwK|
iwk¥ e„ËwU‡K B we›`y‡Z †Q` K‡i| OB †iLvi Ici B we›`y‡Z
Ges OA †iLvi Ici A we›`y‡Z `yBwU j¤^ Uvwb Ges g‡b Kwi, GB
j¤^Øq ci¯ži C we›`y‡Z wgwjZ nq|
Zvn‡j, AC I BC-B wb‡Y©q ¯žk©KØq, hv‡`i Ašf&©z³ ∠ACB = 60°|
cªgvY : PZzfR
©y OACB Gi, ∠AOB = 120° [ A¼b Abymv‡i ]
OB
∠OBC = 90° [Q OB ⊥ BC ]
A
E
B
O
L
Ges ∠OAC = 90° [Q OA ⊥ AC]
GLb, PZzfR
©y OACB-Gi,
DN
F
we‡kl wbe©Pb : g‡b Kwi, O †K›`ªwewkó ABC GKwU wbw`©ó e„Ë|
EF GKwU wbw`©ó mij‡iLv| ABC e„‡Ë Giƒc GKwU ¯žk©K
AvuK‡Z n‡e hv wbw`©ó mij‡iLv EF Gi Ici j¤^|
A¼b : EF †iLvi Ici †h‡Kv‡bv we›`y D wbB| O, D †hvM Kwi|
OD †iLvi O we›`y‡Z ∠DOB = ∠EDO AvuwK| OB iwk¥ e„ËwU‡K
B we›`y‡Z †Q` K‡i| BO †K ewa©Z Ki‡j Zv e„ËwU‡K A we›`y‡Z
†Q` K‡i| AB †iLvi A I B we›`y‡Z h_vµ‡g KL I MN `yÕwU j¤^
AvuwK| KL I MN j¤^Øq EF †K h_vµ‡g L I N we›`y‡Z †Q` K‡i|
Zvn‡j KL ev MN-B wb‡Y©q ¯žk©K|
cÖgvY : A¼bvbymv‡i, ∠EDO = ∠DOB.
wKšy Zviv GKvši †KvY|
∴ AB | | EF
Avevi, AB | | EF Ges MN †Q`K|
∴ ∠OBN = ∠BNF [GKvši †KvY e‡j ]
∴ ∠BNF = 90° [Q∠OBN = 90°]
A_©vr MN, EF Gi Ici j¤^|
Abyiƒcfv‡e cÖgvY Kiv hvq †h, KL, EF Gi Ici j¤^|
wKš‘ MN Ges KL h_vµ‡g e„‡Ëi OB Ges OA e¨vmv‡a©i Ici j¤^|
∴ MN Ges KL DfqB h_vµ‡g B Ges A we›`y‡Z e„‡Ëi `yBwU ¯žk©K|
AZGe, MN ev KL-B wb‡Y©q ¯žk©K| (cÖgvwYZ)
Aby: 7. †Kv‡bv e„‡Ë Ggb `yBwU ¯žk©K AvuK †hb Zv‡`i Ašfz³
©
†KvY 60° nq| [Kz: †ev: 12, 10, 08, 05; P: †ev:, e: †ev: 12, 09, 05; wm:
†ev: 12, 10, 08, 06, 03; h: †ev: 12, 09, 06, 04, 02; Xv: †ev: 09, 05;
iv: †ev: 12, 10, 08, 06, 03, 01; w`: †ev: 10]
DËi : mvaviY wbe©Pb : †Kv‡bv e„‡Ë Ggb `y&BwU ¯žk©K AvuK‡Z n‡e
†hb Zv‡`i Ašfz©³ †KvY 60° nq|
B
90°
D
O 120°
C
90°
A
we‡kl wbe©P© b : g‡b Kwi, O †K›`ªwewkó ABD GKwU e„Ë| ABD
e„‡Ë Giƒc `yBwU ¯žk©K AvuK‡Z n‡e hv‡`i Ašfz©³ †KvY 60° nq|
∠ACB + ∠AOB +∠OAC + ∠OBC = 360°
[Q PZzfz‡R©i Pvi †Kv‡Yi mgwó 360°]
ev, ∠ACB + 120° + 90° + 90° = 360°
ev, ∠ACB = 360° – 300°
∴ ∠ACB = 60°
Avevi, cÖ`Ë e„‡Ëi OB e¨vmva© Ges BC ⊥ OB. ∴ BC ¯žk©K|
Z`ª€c, cÖ`Ë e„‡Ëi OA e¨vmva© Ges AC ⊥ OA. ∴ AC ¯žk©K|
© ∠ACB = 60°
AZGe, AC B I BC-B wb‡Y©q m¤ž©KØq, hv‡`i Ašfz³
(cÖgvwYZ)
Aby: 10. GKwU e‡M©i Aše©„Ë I cwie„Ë AvuK|
A_ev, †Kv‡bv wbw`©ó PZzfz©‡Ri Aa©e„Ë A¼b Ki| [Xv: †ev: 12, 07;
Kz: †ev: 00; 02; e: †ev: 01; iv: †ev: 02; wm: †ev: 09; P: †ev: 10]
DËi : mvaviY wbe©Pb : GKwU e‡M©i Aše„Ë
© I cwie„Ë AvuK‡Z n‡e|
D
H
A
G
C
O
F
E
B
we‡kl wbe©Pb : g‡b Kwi, ABCD GKwU eM©| GB e‡M©i Aše„Ë
© I
cwie„Ë AvuK‡Z n‡e|
A¼b : A, C Ges B, D †hvM Kwi| AC I BD KY©Øq O we›`y‡Z
†Q` K‡i| O n‡Z AB Gi Ici OE j¤^ Uvwb| O-†K †K›`ª K‡i OE
Gi mgvb e¨vmva© wb‡q GKwU e„Ë AvuwK| e„ËwU AB, BC, CD I DA
evû‡K h_vµ‡g E, F, G I H we›`y‡Z ¯žk© K‡i| Zvn‡j, EFGH-B
wb‡Y©q Aše„Ë
© | Avevi, O-†K †K›`ª K‡i OA-Gi mgvb e¨vmva© wb‡q
GKwU e„Ë AvuwK| GB e„Ë ABCD e‡M©i wb‡Y©q cwie„Ë|
cÖgvY : †h‡nZz e‡M©i KY© Bnvi †KvY¸wj‡K mgwØLwÊZ K‡i Ges
O we›`y n‡Z AB, BC, CD, DA evûi `iZ¡ (j¤^`iZ¡) mgvb|
†m‡nZz, O †K †K›`ª K‡i OE e¨vmva© wb‡q e„Ë AvuK‡j e„ËwU AB,
BC, CD, DA evû‡K ¯žk© Ki‡e|
© |
AZGe, EFGH B wb‡Y©q Aše„Ë
Avevi, e‡M©i KY©Øq mgvb Ges Zviv ci¯ži‡K mgwØLwÊZ K‡i|
myZivs, OA = OB = OC = OD
myZivs, O †K †K›`ª K‡i OA e¨vmva© wb‡q Aw¼Z e„Ë A, B, C, D we›`y
w`‡q hv‡e|
AZGe, ABCD-B wb‡Y©q cwie„Ë| (cÖgvwYZ)
[we.`ª. ivRkvnx †evW©-2002 cÖkœc‡Î ïay Aše„Ë© AuvK‡Z †`qv n‡q‡Q;
Kzwgj−v †ev‡W© PZzf‡y© Ri ¯’‡j ABCD eM© †`Iqv n‡q‡Q| Zvi cwie„Ë
AuvK‡Z n‡e|]