iz'ukoyh ØeMveeJeueer 4.1 4-1 - WordPress.com · 2017-04-02 · . 2 7 = 1.8 5.4...

ef$eYegpe4

ØeMveeJeueer 4.1iz'ukoyh 4-1ØeMve 1. keâes<"keâeW ceW efoS MeyoeW ceW mes mener MeyoeW keâe ØeÙeesi e keâjles ngS, efjkeäle mLeeveeW keâes YeefjS~

(i) meYeer Je=òe ………… nesles nQ~ (meJeeËiemece, mece¤he)(ii) meYeer Jeie& ………… nesles nQ~ (mece¤he, meJeeËiemece)

(iii) meYeer ………… ef$eYegpe mece¤he nesles nQ~ (meceefÉyeeng, meceyeeng)(iv) YegpeeDeeW keâer meceeve mebKÙee Jeeues oes yengYegpe mece¤he nesles nQ, Ùeefo (a) Gvekesâ

mebiele keâesCe ………… neW leLee (b) Gvekeâer mebiele YegpeeSB ………… neW~(yejeyej, meceevegheeleer)

nue (i) meYeer�Je=òe mece¤he nesles�nQ�keäÙeeWefkeâ�Je=òeeW�kesâ�Deekeâej�meceeve�nesles�nQ�hejbleg�ceehe�ceW�veneR~

(ii) meYeer�Jeie& mece¤he nesles�nQ�keäÙeeWefkeâ�JeieeX�kesâ�Deekeâej�meceeve�nesles�nQ�hejbleg�ceehe�ceW�veneR~

(iii) meYeer meceyeeng ef$eYegpe mece¤he nesles nQ keäÙeeWefkeâ meceyeeng ef$eYegpeeW kesâ Deekeâej meceeve nesles nQhejbleg�ceehe�ceW�veneR~

(iv) YegpeeDeeW�keâer�meceeve�mebKÙee�Jeeues�oes�yengYegpe�mece¤he�nesles�nQ,�Ùeefo

(a) Gvekesâ�mebiele�keâesCe yejeyej neW~

(b) Gvekeâer�mebiele�YegpeeSB�meceevegheeeflekeâ�neW~

ØeMve 2. efvecveefueefKele ÙegiceeW kesâ oes efYebve-efYebve GoenjCe oerefpeS~(i) mece¤he Deeke=âefleÙeeB

(ii) Ssmeer Deeke=âefleÙeeB pees mece¤he veneR nQ~

nue (i) (a) meceyeeng�ef$eYegpeeW�keâe�Ùegice�mece¤he�Deeke=âefleÙeeB�nesleer�nQ~(b) JeieeX�kesâ�Ùegice�mece¤he�Deeke=âefleÙeeB�nesleer�nQ~

(ii) (a) Skeâ�ef$eYegpe�Deewj�Skeâ�ÛelegYeg&pe�Demeceeve�Deeke=âefleÙeeW�keâe�Ùegice�nw~(b) Skeâ�Jeie&�Deewj�Skeâ�meceuebye�Demeceeve�Deeke=âefleÙeeW�keâe�Ùegice�nw~

ØeMve 3. yeleeFS efkeâ efvecveefueefKele ÛelegYeg&pe mece¤he nQ Ùee veneR~

nue Deeke=âefle ceW efoS ieS oesveeW ÛelegYeg&pe meceeve veneR nQ, keäÙeeWefkeâ Gvekesâ mebiele keâesCe meceeve veneR nQ~Ùen�efÛe$eeW�mes�mhe<š�nw�efkeâ ∠ = °A 90 leLee ∠ ≠ °P 90

keâ#ee 10 ieefCele mebhetCe&�nue

1.5 mesceer

1.5 mesceer

S R

P Q

D C

A B

3 mesceer

3 mesceer

3mescee

r 3mesceer

1.5

mescee

r 1.5

mesceer

ef$eYegpe4

ØeMveeJeueer 4.2iz'ukoyh 4-2ØeMve 1. Deeke=âefle (i) Deewj (ii) ceW, DE BC|| nw~ (i) ceW EC Deewj (ii) ceW AD %eele keâerefpeS~

nue (i) eqÛe$e (i) ceW, DE BC|| (efoÙee nw)

⇒ AD

DB

AE

EC= (DeeOeejYetle meceevegheeeflekeâlee ØecesÙe mes)

⇒ 1.5

3

1=EC

(QAD = 1.5 mesceer, DB = 3 mesceer�Deewj AE = 1mesceer,�efoÙee�nw)

⇒ EC = =3

152

.mesceer

(ii) efÛe$e (ii) ceW, DE BC|| (efoÙee nw)

⇒ AD

DB

AE

EC= (DeeOeejYetle meceevegheeeflekeâlee ØecesÙe mes)

⇒ AD

72.= 1.8

5.4

(Q AE 1.8= mesceer, EC = 54. mesceer�Deewj BD = 72. mesceer,�efoÙee�nw)

⇒ AD = × =1.8 72

542 4

.

.. mesceer

ØeMve 2. efkeâmeer ∆PQR keâer YegpeeDeeW PQ Deewj PR hej ›eâceMe: efyebog E Deewj F efmLele nQ~efvecveefueefKele ceW mes ØelÙeskeâ efmLeefle kesâ efueS, yeleeFS efkeâ keäÙee EF QR|| nw

(i) PE = 39. mesceer, EQ = 3 mesceer, PF = 36. mesceer Deewj FR = 2 4. mesceer(ii) PE = 4 mesceer, QE = 45. mesceer, PF = 8 mesceer Deewj RF = 9 mesceer

(iii) PQ = 128. mesceer, PR = 2 56. mesceer, PE = 0 18. mesceer Deewj PF = 0 36. mesceer

nue (i) efÛe$e�ceW,PE

EQ= =39

3

.1.3,

eq$eYegpe

A

B C

1.5 mesceer 1 mesceer

3 mesceer

D E

A

B

C

7.2 mesceer

1.8 mesceerD

E

5.4 mesceer

(i) (ii)

ieefCele keâ#ee 10 eq$eYegpe

PF

FR= = =36

2 4

3

2

.

.1.5

⇒ PE

EQ

PF

FR≠

⇒ EF QR, kesâ meceeblej veneR nw keäÙeeWefkeâDeeOeejYetle meceevegheeeflekeâlee ØecesÙe mebleg<š venerRkeâjleer�nw~

(ii) efÛe$e�ceW, PE

EQ= 4

4.5= =40

45

8

9

Deewj PF

FR= 8

9

⇒ PE

EQ

PF

FR=

⇒ EF QR|| keäÙeeWefkeâ�DeeOeejYetle�meceevegheeeflekeâlee�ØecesÙe�mebleg<š�keâjleer�nw~

(iii) efÛe$e�ceW, PE

EQ

PE

PQ PE=

−=

−= =0.18

1.28 0.18

0.18

1.10

9

55

Deewj PF

FR

PF

PR PF=

−0.36

2.56 0.36

0.36

2.20

9

55−= =

⇒ PE

EQ

PF

FR=

⇒ EF QR|| keäÙeeWefkeâ�DeeOeejYetle�meceevegheeeflekeâlee�ØecesÙe�mebleg<š�keâjleer�nw~

ØeMve 3. oer ieF& Deeke=âefle ceW Ùeefo LM CB|| Deewj LN CD|| nes, lees efmeæ keâerefpeS efkeâAM

AB

AN

AD= nw~

nue ∆ ACB ceW, LM CB|| (efoÙee nw)


Q

P

R

E F

4 mesceer 8 mesceer

9 mesceer4.5 mesceer

Q

P

R

E F

0.18

mesceer 0.36

mesceer

1.28

mesceer 2.56

mesceer

Q

P

R

E F

3.9 mesceer 3.6 mesceer

2.4 mesceer3 mesceer

A

M

N

LC

B

D


⇒ AM

MB

AL

LC= …(i)

(DeeOeejYetle�meceevegheeeflekeâlee�ØecesÙe�mes)

∆ ACD ceW, LN CD|| (efoÙee nw)

⇒ AN

ND

AL

LC= …(ii)


meceer (i) Deewj (ii) mes,AM

MB

AN

ND= ⇒ MB

AM

ND

AN=

MB

AM

ND

AN+ = +1 1 (oesveeW he#eeW ceW 1 peesÌ[ves hej)

⇒ MB AM

AM

ND AN

AN

+ = +

⇒ AM

AM MB

AN

AN ND+=

+⇒ AM

AB

AN

AD=

Fefle efmeæced

ØeMve 4. oer ieF& Deeke=âefle ceW, DE AC|| Deewj DF AE|| nw~ efmeæ keâerefpeS efkeâ BF

FE

BE

EC= nw~

nue ∆ BAC ceW, DE AC|| (efoÙee nw)

⇒ BE

EC

BD

DA= …(i)


∆BAE ceW, DF AE|| (efoÙee nw)

⇒ BF

FE

BD

DA= …(ii)


meceer (i) Je (ii) mes,BF

FE

BE

EC= Fefle efmeæced

eq$eYegpe

B C

A

D

F E


ØeMve 5. Deeke=âefle ceW DE OQ|| Deewj DF OR|| nw~ oMee&FS efkeâ EF QR|| nw~

nue efÛe$e�ceW, DE OQ|| leLee DF OR|| (DeeOeejYetle�meceevegheeeflekeâlee�ØecesÙe�mes)

∆ PQO ceW, PE

EQ

PD

DO= …(i)

∆ POR ceW,PF

FR

PD

DO= …(ii)

meceer (i) Je (ii) mes, PE

EQ

PF

FR=

Deye ∆ PQR ceW,�(Thej�efmeæ�efkeâÙee�pee�Ûegkeâe�nw)PE

EQ

PF

FR= (Thej efmeæ efkeâÙee pee Ûegkeâe nw)

EF QR|| (DeeOeejYetle meceevegheeeflekeâlee ØecesÙe kesâ efJeueesce mes)

Fefle�efmeæced

ØeMve 6. Deeke=âefle ceW ›eâceMe: OP OQ, Deewj OR hej efmLele efyebog A B, Deewj C Fme Øekeâej nQ efkeâAB PQ|| Deewj AC PR|| nw~ oMee&FS efkeâ BC QR|| nw~

nue efÛe$e ceW, AB PQ|| (efoÙee nw)

⇒ OA

AP

OB

BQ= …(i)


efÛe$e ceW, AC PR|| (efoÙee nw)


P

Q R

E F

P

E F

D

O

Q R

Q R

P

A

B C

O


⇒ OA

AP

OC

CR= …(ii)


meceer (i) Je (ii) mes,OB

BQ

OC

CR=

⇒ BC QR|| (DeeOeejYetle meceevegheeeflekeâlee ØecesÙe kesâ efJeueesce mes)

ØeMve 7. ØecesÙe keâe ØeÙeesie keâjles ngS efmeæ keâerefpeS efkeâ Ske â ef$eYegpe keâer Skeâ Yegpee kesâ ceOÙe-efyebogmes neskeâj otmejer Yegpee kesâ meceeblej KeeRÛeer ieF& jsKee l eermejer Yegpee keâes meceefÉYeeefpele keâjleernw~ (Ùeeo keâerefpeS efkeâ Deehe Fmes keâ#ee IX ceW efmeæ keâj Ûegkesâ nQ~)

nue ∆ ABC ceW,

Q AB keâe�ceOÙe-efyebog D nw~

DeLee&led AD

DB= 1 …(i)

Q jsKee l BC||

jsKee l efyebog D mes nesleer ngF& KeeRÛeer pees AC kesâ efyebog E hejefceueleer�nw~�DeeOeejYetle�meceevegheeeflekeâlee�ØecesÙe�mes,

AD

DB

AE

EC=

∴ AE

EC= 1 [meceer (i) mes]

∴ AE EC=

⇒ AE

EC= 1

Dele: AC keâe ceOÙe-efyebog E nw~ Fefle efmeæced

ØeMve 8. ØecesÙe keâe ØeÙeesie keâjles ngS efmeæ keâerefpeS efkeâ Skeâ ef$eYegpe keâer efkeâvneR oes YegpeDeesW kesâceOÙe-efyebogDeeW keâes efceueeves Jeeueer jsKee leermeje r Yegpee kesâ meceeblej nesleer nw~ (Ùeeo keâerefpeSefkeâ Deehe keâ#ee IX ceW Ssmee keâj Ûegkesâ nQ)~

nue ∆ ABC ceW,�efyebog D leLee E ›eâceMe:�Yegpee AB Deewj AC kesâ�ceOÙe-eEyeog�nQ~

⇒ AD

DB= 1 leLee AE

EC= 1

eq$eYegpe

A

B C

D E l

B C

A

D E


⇒ AD

DB

AE

EC= ⇒ DE BC||

(DeeOeejYetle meceevegheeeflekeâlee kesâ efJeueesce ØecesÙe mes)

Fefle�efmeæced

ØeMve 9. ABCD Skeâ meceuebye nw efpemeceW AB DC|| nw leLee Fmekesâ efJekeâCe& hejmhej efyebog O hejØeefleÛÚso keâjles nQ~ oMee&FS efkeâ AO

BO

CO

DO= nw~

nue

nce�KeeRÛeles�nQ, EOF AB||

∆ ACD ceW, OE CD||

⇒ AE

ED

AO

OC= …(i)

∆ ABD ceW, OE BA|| (DeeOeejYetle meceevegheeeflekeâlee ØecesÙe mes)

⇒ DE

EA

DO

OB= (DeeOeejYetle meceevegheeeflekeâlee ØecesÙe mes)

⇒ AE

ED

OB

OD= …(ii)

meceer (i) Je (ii) mes,AO

OC

OB

OD=

DeLee&led AO

BO

CO

DO= Fefle efmeæced

ØeMve 10. Skeâ ÛelegYeg&pe ABCD kesâ efJekeâCe& hejmhej efyebog O hej Fme Øekeâej ØeefleÛÚso keâjles nQ efkeâAO

BO

CO

DO= nw~ oMee&FS efkeâ ABCD Skeâ meceuebye nw~

nue efÛe$e ceW, AO

BO

CO

DO= (efoÙee nw)

⇒ AO

OC

BO

OD= (efoÙee nw) …(i)

eEyeog O mes�nesles�ngS OE BA|| KeeRÛee~�jsKee OE AD, kesâ�eEyeog E hej�efceueleer�nw~

∆DAB ceW, EO AB||


D C

A B

FO

E


⇒ DE

EA

DO

OB=

⇒ AE

ED

BO

OD= …(ii)

meceer (i) leLee (ii) mes,AO

OC

AE

ED=

⇒ OE CD|| (DeeOeejYetle meceevegheeeflekeâlee ØecesÙe kesâ efJeueesce mes)

Deye, BA OE|| leLee OE CD||

⇒ AB CD||

Dele: ÛelegYeg&pe ABCD Skeâ meceuebye nw~ Fefle efmeæced

iz'ukoyh 4-3ØeMve 1. yeleeFS efkeâ Deeke=âefle ceW efoS ef$eYegpeeW kesâ ÙegiceeW ceW mes keâewve-keâewve mes Ùegice mece¤he nQ~ Gme

mece¤helee keâmeewšer keâes efueefKeS efpemekeâe ØeÙeesie De eheves Gòej osves ceW efkeâÙee nw leLee meeLener mece¤he ef$eYegpeeW keâes meebkesâeflekeâ ¤he ceW JÙekeäle keâerefpeS~

eq$eYegpe

D C

A B

EO

80° 40°

60°

80° 40°

60°

A

B C

P

Q R

(i)

A

B C

2 3

P

Q R4

6 5

(ii)

2.5

L

M P2

2.7 3

D

E F5

46

(iii)

ef$eYegpe4

ØeMveeJeueer 4.3

⇒ DE

EA

DO

OB=

⇒ AE

ED

BO

OD= …(ii)

meceer (i) leLee (ii) mes,AO

OC

AE

ED=

⇒ OE CD|| (DeeOeejYetle meceevegheeeflekeâlee ØecesÙe kesâ efJeueesce mes)

Deye, BA OE|| leLee OE CD||

⇒ AB CD||

Dele: ÛelegYeg&pe ABCD Skeâ meceuebye nw~ Fefle efmeæced

iz'ukoyh 4-3ØeMve 1. yeleeFS efkeâ Deeke=âefle ceW efoS ef$eYegpeeW kesâ ÙegiceeW ceW mes keâewve-keâewve mes Ùegice mece¤he nQ~ Gme

mece¤helee keâmeewšer keâes efueefKeS efpemekeâe ØeÙeesie De eheves Gòej osves ceW efkeâÙee nw leLee meeLener mece¤he ef$eYegpeeW keâes meebkesâeflekeâ ¤he ceW JÙekeäle keâerefpeS~

eq$eYegpe

D C

A B

EO

80° 40°

60°

80° 40°

60°

A

B C

P

Q R

(i)

A

B C

2 3

P

Q R4

6 5

(ii)

2.5

L

M P2

2.7 3

D

E F5

46

(iii)

nue (i) neB, ∆ ABC leLee ∆PQR ceW,

∠ = ∠ = °A P 60 , ∠ = ∠ = °B Q 80

leLee ∠ = ∠ = °C R 40

ÙeneB,�oesveeW�ef$eYegpe�kesâ�mebiele�keâesCe�yejeyej�nesles�nQ~∴ ∆ ∆ABC PQR~ (AAA mece¤helee mes)

(ii) neB, ∆ ABC leLee ∆ QRP ceW,AB

QR

BC

RP= = = =2

4

1

2

1

2,

2.5

5

leLee CA

PQ= =3

6

1

2

ÙeneB,�meYeer�mebiele�YegpeeSB�yejeyej�meceevegheele�ceW�nQ~∴ ∆ ∆ABC QRP~ (SSS mece¤helee mes)

(iii) veneR, ∆LMP leLee ∆FED ceW,MP

DE= =2

4

1

2,

LP

DF= =3

6

1

2


(iv)

Q R

P

70°

10

5

N L

M

70°

52.5

A

B C3

2.5

80°

D

F6

5

80°

E

(v)

D

E F

70°

80°

P

Q R

80° 30°

(vi)


nue (i) neB, ∆ ABC leLee ∆PQR ceW,

∠ = ∠ = °A P 60 , ∠ = ∠ = °B Q 80

leLee ∠ = ∠ = °C R 40

ÙeneB,�oesveeW�ef$eYegpe�kesâ�mebiele�keâesCe�yejeyej�nesles�nQ~∴ ∆ ∆ABC PQR~ (AAA mece¤helee mes)

(ii) neB, ∆ ABC leLee ∆ QRP ceW,AB

QR

BC

RP= = = =2

4

1

2

1

2,

2.5

5

leLee CA

PQ= =3

6

1

2

ÙeneB,�meYeer�mebiele�YegpeeSB�yejeyej�meceevegheele�ceW�nQ~∴ ∆ ∆ABC QRP~ (SSS mece¤helee mes)

(iii) veneR, ∆LMP leLee ∆FED ceW,MP

DE= =2

4

1

2,

LP

DF= =3

6

1

2


(iv)

Q R

P

70°

10

5

N L

M

70°

52.5

A

B C3

2.5

80°

D

F6

5

80°

E

(v)

D

E F

70°

80°

P

Q R

80° 30°

(vi)

leLee LM

EF= ≠27

5

1

2

.

DeLee&led MP

DE

LP

DF

LM

EF= ≠

ÙeneB,�meYeer�mebiele�YegpeeSB�meceevegheele�ceW�veneR�nQ~Fme�Øekeâej, ∆ LMP Deewj ∆ FED meceevegheele�ceW�veneR�nQ~

(iv) neB, ∆LMN leLee ∆RQP ceW,

∠ = ∠ = ° = = =M QMN

PQ70

5

25

50

1

2,

2.5

leLee ML

QR= =5

10

1

2

DeLee&led MN

PQ

ML

QR=

ÙeneB,�oes�meceerheJeleea�mebiele�YegpeeSB�meceevegheele�ceW�nQ�Deewj�Skeâ�keâesCe�meceeve�nw~

∴ ∆ ∆LMN RQP~ (SAS mece¤helee mes)

(v) veneR, ∆ ABC ceW, ∠A efoÙee�nw�hejbleg�meefcceefuele�Yegpee A veneR�oer�ngF&�nw~

(vi) neB, ∠ = ° ∠ = °D E70 80, leLee ∠ = °F 30

(Q ∆ DEF ceW, ∠ + ∠ + ∠ = °D E F 180 )

∠ = ° ∠ = °Q R80 30, , leye ∠ = °P 70

(Q∆QPR ceW, ∠ + ∠ + ∠ = °Q P R 180 )

ÙeneB, ∠ = ∠ ∠ = ∠ ∠ = ∠D P E Q F R, ,

∴ ∆ ∆DEF PQR~ (AAA mece¤helee mes)

ØeMve 2. Deeke=âefle ceW, ∆ ∆ ∠ = °ODC OBA BOC~ , 125 Deewj ∠ = °CDO 70 nw~∠ ∠DOC DCO, Deewj ∠OAB %eele keâerefpeS~

nue ∠ + ° = °DOC 125 180 (Q DOC Skeâ mejue jsKee nw)

⇒ ∠ = ° − ° = °DOC 180 125 55

∠ + ∠ + ∠ = °DCO CDO DOC 180

(Q∆ODC kesâ�leerveeW�keâesCeeW�keâe�Ùeesie = °180 )

eq$eYegpe

125°

70°

D

O

BA

C


leLee LM

EF= ≠27

5

1

2

.

DeLee&led MP

DE

LP

DF

LM

EF= ≠

ÙeneB,�meYeer�mebiele�YegpeeSB�meceevegheele�ceW�veneR�nQ~Fme�Øekeâej, ∆ LMP Deewj ∆ FED meceevegheele�ceW�veneR�nQ~

(iv) neB, ∆LMN leLee ∆RQP ceW,

∠ = ∠ = ° = = =M QMN

PQ70

5

25

50

1

2,

2.5

leLee ML

QR= =5

10

1

2

DeLee&led MN

PQ

ML

QR=

ÙeneB,�oes�meceerheJeleea�mebiele�YegpeeSB�meceevegheele�ceW�nQ�Deewj�Skeâ�keâesCe�meceeve�nw~

∴ ∆ ∆LMN RQP~ (SAS mece¤helee mes)

(v) veneR, ∆ ABC ceW, ∠A efoÙee�nw�hejbleg�meefcceefuele�Yegpee A veneR�oer�ngF&�nw~

(vi) neB, ∠ = ° ∠ = °D E70 80, leLee ∠ = °F 30

(Q ∆ DEF ceW, ∠ + ∠ + ∠ = °D E F 180 )

∠ = ° ∠ = °Q R80 30, , leye ∠ = °P 70

(Q∆QPR ceW, ∠ + ∠ + ∠ = °Q P R 180 )

ÙeneB, ∠ = ∠ ∠ = ∠ ∠ = ∠D P E Q F R, ,

∴ ∆ ∆DEF PQR~ (AAA mece¤helee mes)

ØeMve 2. Deeke=âefle ceW, ∆ ∆ ∠ = °ODC OBA BOC~ , 125 Deewj ∠ = °CDO 70 nw~∠ ∠DOC DCO, Deewj ∠OAB %eele keâerefpeS~

nue ∠ + ° = °DOC 125 180 (Q DOC Skeâ mejue jsKee nw)

⇒ ∠ = ° − ° = °DOC 180 125 55

∠ + ∠ + ∠ = °DCO CDO DOC 180

(Q∆ODC kesâ�leerveeW�keâesCeeW�keâe�Ùeesie = °180 )

eq$eYegpe

125°

70°

D

O

BA

C

⇒ ∠ + ° + ° = °DCO 70 55 180

⇒ ∠ + ° = °DCO 125 180

⇒ ∠ = ° − ° = °DCO 180 125 55

Deye, ∆ ∆ODC OBA~ . (efoÙee nw)

⇒ ∠ = ∠OCD OAB

⇒ ∠ = ∠OAB OCD = ∠ = °DCO 55

DeLee&led ∠ = °OAB 55

Dele: ∠ = ° ∠ = °DOC DCO55 55, leLee ∠ = °OAB 55

ØeMve 3. meceuebye ABCD, efpemeceW AB DC|| nw, kesâ efJekeâCe& AC Deewj BD hejmhej O hejØeefleÛÚso keâjles nQ~ oes ef$eYegpeeW keâer mece¤helee keâmeewšer keâe ØeÙeesie keâjles ngS oMee&FS efkeâOA

OC

OB

OD= nw~

nue m e c e u e b y e Û e l e g Y e g & p e ABCD k e s â e f J e k e â C e & AC D e e w j BD h e j m h e j e f y e b o g O h e j Ø e e f l e Û Ú s o k e â j l e s n Q ~

AB DC|| (efoÙee nw)

⇒ ∠ = ∠ ∠ = ∠1 3 2 4, (Skeâeblej keâesCe)

∠ = ∠DOC BOA (Meer<ee&efYecegKe keâesCe)

⇒ ∆ ∆OCD OAB~ (AAA mece¤helee mes)

⇒ OC

OA

OD

OB= (mece¤he ef$eYegpeeW keâer mebiele YegpeeDeeW kesâ Devegheele)

⇒ OA

OC

OB

OD= (JÙegl›eâce uesves hej)

ØeMve 4. Deeke=âefle ceW, QR

QS

QT

PR= leLee ∠ = ∠1 2 nw~ oMee&FS efkeâ ∆ ∆PQS TQR~ nw~


D C

A B

O

4 3

1 2

T

Q R

P

1

S

2


⇒ ∠ + ° + ° = °DCO 70 55 180

⇒ ∠ + ° = °DCO 125 180

⇒ ∠ = ° − ° = °DCO 180 125 55

Deye, ∆ ∆ODC OBA~ . (efoÙee nw)

⇒ ∠ = ∠OCD OAB

⇒ ∠ = ∠OAB OCD = ∠ = °DCO 55

DeLee&led ∠ = °OAB 55

Dele: ∠ = ° ∠ = °DOC DCO55 55, leLee ∠ = °OAB 55

ØeMve 3. meceuebye ABCD, efpemeceW AB DC|| nw, kesâ efJekeâCe& AC Deewj BD hejmhej O hejØeefleÛÚso keâjles nQ~ oes ef$eYegpeeW keâer mece¤helee keâmeewšer keâe ØeÙeesie keâjles ngS oMee&FS efkeâOA

OC

OB

OD= nw~

nue m e c e u e b y e Û e l e g Y e g & p e ABCD k e s â e f J e k e â C e & AC D e e w j BD h e j m h e j e f y e b o g O h e j Ø e e f l e Û Ú s o k e â j l e s n Q ~

AB DC|| (efoÙee nw)

⇒ ∠ = ∠ ∠ = ∠1 3 2 4, (Skeâeblej keâesCe)

∠ = ∠DOC BOA (Meer<ee&efYecegKe keâesCe)

⇒ ∆ ∆OCD OAB~ (AAA mece¤helee mes)

⇒ OC

OA

OD

OB= (mece¤he ef$eYegpeeW keâer mebiele YegpeeDeeW kesâ Devegheele)

⇒ OA

OC

OB

OD= (JÙegl›eâce uesves hej)

ØeMve 4. Deeke=âefle ceW, QR

QS

QT

PR= leLee ∠ = ∠1 2 nw~ oMee&FS efkeâ ∆ ∆PQS TQR~ nw~


D C

A B

O

4 3

1 2

T

Q R

P

1

S

2

nue efÛe$e ceW, ∠ = ∠1 2 (efoÙee nw)

⇒ PQ PR= (∆ PQR kesâ meceeve keâesCeeW keâer efJehejerle YegpeeSB)

efoÙee nw, QR

QS

QT

PR=

⇒ QR

QS

QT

PQ= (QPQ PR= Thej efmeæ efkeâÙee pee Ûegkeâe nw)

⇒ QS

QR

PQ

QT= (JÙegl›eâce uesves hej) …(i)

Deye, ∆ PQS Deewj ∆ TQR ceW, efoÙee nw

∠ = ∠PQS TQR (ØelÙeskeâ = ∠ 1)

leLee QS

QR

PQ

QT= [meceer (i) mes]

∴ ∆ ∆PQS TQR~ (SAS mece¤helee mes)

ØeMve 5. ∆PQR keâer YegpeeDeeW PR Deewj QR hej ›eâceMe: efyebog S Deewj T Fme Øekeâej efmLele nQ efkeâ∠ = ∠P RTS nw~ oMee&FS efkeâ ∆ ∆RPQ RTS~ nw~

nue ∆ PQR keâer YegpeeDeeW PR Deewj QR hej ›eâceMe: eEyeog S Deewj T Fme Øekeâej efmLele nQ efkeâ

∠ = ∠P RTS

DeLee&led ∠ = ∠RPQ RTS (efoÙee nw)

∠ = ∠PRQ SRT (ØelÙeskeâ = ∠ R)

∴ ∆ ∆RPQ RTS~ (AAA mece¤helee mes)

Fefle�efmeæced

veesš Ùeefo�ef$eYegpeeW�kesâ�oes�mebiele�keâesCe�meceeve�nQ,�lees�Fvekeâe�leermeje�mebiele�keâesCe�Yeer�meceeve�nesiee~

eq$eYegpe

R

P Q

S T


nue efÛe$e ceW, ∠ = ∠1 2 (efoÙee nw)

⇒ PQ PR= (∆ PQR kesâ meceeve keâesCeeW keâer efJehejerle YegpeeSB)

efoÙee nw, QR

QS

QT

PR=

⇒ QR

QS

QT

PQ= (QPQ PR= Thej efmeæ efkeâÙee pee Ûegkeâe nw)

⇒ QS

QR

PQ

QT= (JÙegl›eâce uesves hej) …(i)

Deye, ∆ PQS Deewj ∆ TQR ceW, efoÙee nw

∠ = ∠PQS TQR (ØelÙeskeâ = ∠ 1)

leLee QS

QR

PQ

QT= [meceer (i) mes]

∴ ∆ ∆PQS TQR~ (SAS mece¤helee mes)

ØeMve 5. ∆PQR keâer YegpeeDeeW PR Deewj QR hej ›eâceMe: efyebog S Deewj T Fme Øekeâej efmLele nQ efkeâ∠ = ∠P RTS nw~ oMee&FS efkeâ ∆ ∆RPQ RTS~ nw~

nue ∆ PQR keâer YegpeeDeeW PR Deewj QR hej ›eâceMe: eEyeog S Deewj T Fme Øekeâej efmLele nQ efkeâ

∠ = ∠P RTS

DeLee&led ∠ = ∠RPQ RTS (efoÙee nw)

∠ = ∠PRQ SRT (ØelÙeskeâ = ∠ R)

∴ ∆ ∆RPQ RTS~ (AAA mece¤helee mes)

Fefle�efmeæced

veesš Ùeefo�ef$eYegpeeW�kesâ�oes�mebiele�keâesCe�meceeve�nQ,�lees�Fvekeâe�leermeje�mebiele�keâesCe�Yeer�meceeve�nesiee~

eq$eYegpe

R

P Q

S T

ØeMve 6. Deeke=âefle ceW, Ùeefo ∆ ≅ ∆ABE ACD nw, lees oMee&FS efkeâ ∆ ∆ADE ABC~ nw~

nue efÛe$e ceW, ∆ ≅ ∆ABE ACD (efoÙee nw)

⇒ AB AC= leLee AE AD= (CPCT)

⇒ AB

AC= 1 leLee AD

AE= 1

⇒ AB

AC

AD

AE= (ØelÙeskeâ ∠1kesâ meceeve nw)

Deye ∆ ADE leLee ∆ ABC ceW,AD

AE

AB

AC= (efmeæ efkeâÙee pee Ûegkeâe nw)

DeLee&led AD

AB

AE

AC=

Deewj ∠ = ∠DAE BAC (ØelÙeskeâ ∠ A kesâ meceeve nw)

⇒ ∆ ∆ADE ABC~ (SAS mece¤helee mes)

Fefle�efmeæced

ØeMve 7. Deeke=âefle ceW, ∆ABC kesâ Meer<e&uebye AD Deewj CE hejmhej efyebog P hej ØeefleÛÚso keâjles nQ~oMee&FS efkeâ

(i) ∆ ∆AEP CDP~ (ii) ∆ ∆ABD CBE~

(iii) ∆ ∆AEP ADB~ (iv) ∆ ∆PDC BEC~

nue (i) efÛe$e ceW, ∠ = ∠AEP CDP (ØelÙeskeâ = °90 )


A

B C

D E

C

A

P

E B

D


ØeMve 6. Deeke=âefle ceW, Ùeefo ∆ ≅ ∆ABE ACD nw, lees oMee&FS efkeâ ∆ ∆ADE ABC~ nw~

nue efÛe$e ceW, ∆ ≅ ∆ABE ACD (efoÙee nw)

⇒ AB AC= leLee AE AD= (CPCT)

⇒ AB

AC= 1 leLee AD

AE= 1

⇒ AB

AC

AD

AE= (ØelÙeskeâ ∠1kesâ meceeve nw)

Deye ∆ ADE leLee ∆ ABC ceW,AD

AE

AB

AC= (efmeæ efkeâÙee pee Ûegkeâe nw)

DeLee&led AD

AB

AE

AC=

Deewj ∠ = ∠DAE BAC (ØelÙeskeâ ∠ A kesâ meceeve nw)

⇒ ∆ ∆ADE ABC~ (SAS mece¤helee mes)

Fefle�efmeæced

ØeMve 7. Deeke=âefle ceW, ∆ABC kesâ Meer<e&uebye AD Deewj CE hejmhej efyebog P hej ØeefleÛÚso keâjles nQ~oMee&FS efkeâ

(i) ∆ ∆AEP CDP~ (ii) ∆ ∆ABD CBE~

(iii) ∆ ∆AEP ADB~ (iv) ∆ ∆PDC BEC~

nue (i) efÛe$e ceW, ∠ = ∠AEP CDP (ØelÙeskeâ = °90 )


A

B C

D E

C

A

P

E B

D

Deewj ∠ = ∠APE CPD (Meer<ee&efYecegKe keâesCe)

⇒ ∠ ∆AEP CDP~ (AA mece¤helee mes)

(ii) efÛe$e ceW, ∠ = ∠ADB CEB (ØelÙeskeâ = °90 )

Deewj ∠ = ∠ABD CBE (ØelÙeskeâ = ∠ B)

⇒ ∆ ∆ABD CBE~ (AA mece¤helee mes)

(iii) efÛe$e ceW, ∠ = ∠AEP ADB (ØelÙeskeâ = °90 )

Deewj ∠ = ∠PAE DAB (GYeÙeefve<" keâesCe)

⇒ ∆ ∆AEP ADB~ (AA mece¤helee mes)

(iv) efÛe$e ceW, ∠ = ∠PDC BEC (ØelÙeskeâ = °90 )

Deewj ∠ = ∠PCD BCE (GYeÙeefve<" keâesCe)

⇒ ∆ ∆PDC BEC~ (AA mece¤helee mes)

ØeMve 8. meceeblej ÛelegYeg&pe ABCD keâer yeÌ{eF& ieF& Yegpee AD hej efmLele E Skeâ eEyeog nw leLee BE

Yegpee CD keâes F hej ØeefleÛÚso keâjleer nw~ oMee&FS efkeâ ∆ ∆ABE CFB~ nw~

nue meceeblej ÛelegYeg&pe ABCD keâer yeÌ{eF& ieF& Yegpee AD hej efmLele E Skeâ eEyeog nw leLee BE YegpeeCD keâes F hej�ØeefleÛÚso�keâjleer�nw~

meceeblej�ÛelegYeg&pe ABCD ceW,∠ = ∠A C (meceeblej ÛelegYeg&pe kesâ meccegKe keâesCe nQ) …(i)

Deye ∆ABE leLee ∆CFB ceW,

∠ = ∠EAB BCF [meceer (i) mes]

∠ = ∠ABE BFC (Skeâeblej keâesCe AB FC|| )

⇒ ∆ ∆ABE CFB~ (AAA mece¤helee mes)

ØeMve 9. oer ieF& Deeke=âefle ceW, ABC Deewj AMP oes mecekeâesCe ef$eYegpenQ, efpevekesâ keâesCe B Deewj M mecekeâesCe nQ~ efmeæ keâerefpeS efkeâ

(i) ∆ ∆ABC AMP~

(ii)CA

PA

BC

MP=

eq$eYegpe

E

A

D

B

F C

C

A PB

M


Deewj ∠ = ∠APE CPD (Meer<ee&efYecegKe keâesCe)

⇒ ∠ ∆AEP CDP~ (AA mece¤helee mes)

(ii) efÛe$e ceW, ∠ = ∠ADB CEB (ØelÙeskeâ = °90 )

Deewj ∠ = ∠ABD CBE (ØelÙeskeâ = ∠ B)

⇒ ∆ ∆ABD CBE~ (AA mece¤helee mes)

(iii) efÛe$e ceW, ∠ = ∠AEP ADB (ØelÙeskeâ = °90 )

Deewj ∠ = ∠PAE DAB (GYeÙeefve<" keâesCe)

⇒ ∆ ∆AEP ADB~ (AA mece¤helee mes)

(iv) efÛe$e ceW, ∠ = ∠PDC BEC (ØelÙeskeâ = °90 )

Deewj ∠ = ∠PCD BCE (GYeÙeefve<" keâesCe)

⇒ ∆ ∆PDC BEC~ (AA mece¤helee mes)

ØeMve 8. meceeblej ÛelegYeg&pe ABCD keâer yeÌ{eF& ieF& Yegpee AD hej efmLele E Skeâ eEyeog nw leLee BE

Yegpee CD keâes F hej ØeefleÛÚso keâjleer nw~ oMee&FS efkeâ ∆ ∆ABE CFB~ nw~

nue meceeblej ÛelegYeg&pe ABCD keâer yeÌ{eF& ieF& Yegpee AD hej efmLele E Skeâ eEyeog nw leLee BE YegpeeCD keâes F hej�ØeefleÛÚso�keâjleer�nw~

meceeblej�ÛelegYeg&pe ABCD ceW,∠ = ∠A C (meceeblej ÛelegYeg&pe kesâ meccegKe keâesCe nQ) …(i)

Deye ∆ABE leLee ∆CFB ceW,

∠ = ∠EAB BCF [meceer (i) mes]

∠ = ∠ABE BFC (Skeâeblej keâesCe AB FC|| )

⇒ ∆ ∆ABE CFB~ (AAA mece¤helee mes)

ØeMve 9. oer ieF& Deeke=âefle ceW, ABC Deewj AMP oes mecekeâesCe ef$eYegpenQ, efpevekesâ keâesCe B Deewj M mecekeâesCe nQ~ efmeæ keâerefpeS efkeâ

(i) ∆ ∆ABC AMP~

(ii)CA

PA

BC

MP=

eq$eYegpe

E

A

D

B

F C

C

A PB

M

nue (i) efÛe$e ceW, ∠ = ∠ABC AMP (ØelÙeskeâ = °90 , efoÙee nw)

Q ∠ = ∠BAC PAM (GYeÙeefve<" keâesCe ∠ A)

⇒ ∆ ∆ABC AMP~ (AA mece¤helee mes)

(ii) ∆ ∆ABC AMP~ ,AC

AP

BC

MP=

(mece¤he�ef$eYegpeeW�keâer�mebiele�YegpeeDeeW�kesâ�Devegheele�meceeve�nesles�nQ)

⇒ CA

PA

BC

MP= Fefle efmeæced

ØeMve 10. CD Deewj GH ›eâceMe: ∠ACB Deewj ∠EGF kesâ Ssmes meceefÉYeepekeâ nQ efkeâ eEyeog D

Deewj H ›eâceMe: ∆ABC Deewj ∆FEG keâer YegpeeDeeW AB Deewj FE hej efmLele nQ~ Ùeefo∆ ∆ABC FEG~ nw, lees oMee&FS efkeâ

(i)CD

GH

AC

FG= (ii) ∆ ∆DCB HGE~ (iii) ∆ ∆DCA HGF~

nue

Q ∆ ∆ABC FEG~ (efoÙee nw)

(i) ∆ACD leLee ∆FGH ceW,

∠ = ∠CAD GFH …(i)

Q ∆ ∆∴ ∠ = ∠⇒ ∠ = ∠

ABC FEG

CAB GFE

CAD GFH

~

∠ = ∠ACD FGH …(ii)

Q ∆ ∆∴ ∠ = ∠

⇒ ∠ = ∠

ABC FEG

ACB FGE

ACB FGE

~

1

2

1

2

⇒ ∠ = ∠ACD FGH

meceer (i) leLee (ii) mes,∆ ∆ACD FGH~ (QAAA mece¤helee mes)

∴ CD

GH

AC

FG=

(Q oes�mece¤he�ef$eYegpeeW�keâer�mebiele�YegpeeSB�meceevegheele�ceW�nesleer�nQ)


A

B C

D

E

F G

H


nue (i) efÛe$e ceW, ∠ = ∠ABC AMP (ØelÙeskeâ = °90 , efoÙee nw)

Q ∠ = ∠BAC PAM (GYeÙeefve<" keâesCe ∠ A)

⇒ ∆ ∆ABC AMP~ (AA mece¤helee mes)

(ii) ∆ ∆ABC AMP~ ,AC

AP

BC

MP=

(mece¤he�ef$eYegpeeW�keâer�mebiele�YegpeeDeeW�kesâ�Devegheele�meceeve�nesles�nQ)

⇒ CA

PA

BC

MP= Fefle efmeæced

ØeMve 10. CD Deewj GH ›eâceMe: ∠ACB Deewj ∠EGF kesâ Ssmes meceefÉYeepekeâ nQ efkeâ eEyeog D

Deewj H ›eâceMe: ∆ABC Deewj ∆FEG keâer YegpeeDeeW AB Deewj FE hej efmLele nQ~ Ùeefo∆ ∆ABC FEG~ nw, lees oMee&FS efkeâ

(i)CD

GH

AC

FG= (ii) ∆ ∆DCB HGE~ (iii) ∆ ∆DCA HGF~

nue

Q ∆ ∆ABC FEG~ (efoÙee nw)

(i) ∆ACD leLee ∆FGH ceW,

∠ = ∠CAD GFH …(i)

Q ∆ ∆∴ ∠ = ∠⇒ ∠ = ∠

ABC FEG

CAB GFE

CAD GFH

~

∠ = ∠ACD FGH …(ii)

Q ∆ ∆∴ ∠ = ∠

⇒ ∠ = ∠

ABC FEG

ACB FGE

ACB FGE

~

1

2

1

2

⇒ ∠ = ∠ACD FGH

meceer (i) leLee (ii) mes,∆ ∆ACD FGH~ (QAAA mece¤helee mes)

∴ CD

GH

AC

FG=

(Q oes�mece¤he�ef$eYegpeeW�keâer�mebiele�YegpeeSB�meceevegheele�ceW�nesleer�nQ)


A

B C

D

E

F G

H

(ii) ∆DCB leLee ∆HGE ceW,

∠ = ∠DBC HEG …(iii)

Q ∆ ∆∴ ∠ = ∠⇒ ∠ = ∠

ABC FEG

ABC FEG

DBC HEG

~

∠ = ∠DCB HGE …(iv)

Q ∆ ∆∴ ∠ = ∠

⇒ ∠ = ∠

ABC FEG

ACB FGE

ACB FGE

~

1

2

1

2

⇒ ∠ = ∠DCB HGE

meceer (iii) leLee (iv) mes,

∆ ∆DCB HGE~ (QAA mece¤helee mes)

(iii) ∆DCA leLee ∆HGF ceW,

∠ = ∠DAC HFG …(v)

Q ∆ ∆∴ ∠ = ∠⇒ ∠ = ∠⇒ ∠ = ∠

ABC FEG

CAB GFE

CAD GFH

DAC HFG

~

∠ = ∠DCA HGF …(vi)

Q ∆ ∆ ∴ ∠ = ∠

⇒ ∠ = ∠

⇒ ∠ = ∠

ABC FEG ACB FGE

ACB FGE

DCA HGF

~ ,

1

2

1

2

meceer (v) leLee (vi) mes,

∆ ∆DCA HGF~ (QAAA mece¤helee mes)

ØeMve 11. Deeke=âefle ceW, AB AC= Jeeues, Skeâ meceefÉyeeng ∆ABC keâer yeÌ{eF& ieF& Yegpee CB hejefmLele E Skeâ eEyeog nw~ Ùeefo AD BC⊥ Deewj EF AC⊥ nw, lees efmeæ keâerefpeS efkeâ∆ ∆ABD ECF~ nw~

eq$eYegpe

CE

B

F

A

D


(ii) ∆DCB leLee ∆HGE ceW,

∠ = ∠DBC HEG …(iii)

Q ∆ ∆∴ ∠ = ∠⇒ ∠ = ∠

ABC FEG

ABC FEG

DBC HEG

~

∠ = ∠DCB HGE …(iv)

Q ∆ ∆∴ ∠ = ∠

⇒ ∠ = ∠

ABC FEG

ACB FGE

ACB FGE

~

1

2

1

2

⇒ ∠ = ∠DCB HGE

meceer (iii) leLee (iv) mes,

∆ ∆DCB HGE~ (QAA mece¤helee mes)

(iii) ∆DCA leLee ∆HGF ceW,

∠ = ∠DAC HFG …(v)

Q ∆ ∆∴ ∠ = ∠⇒ ∠ = ∠⇒ ∠ = ∠

ABC FEG

CAB GFE

CAD GFH

DAC HFG

~

∠ = ∠DCA HGF …(vi)

Q ∆ ∆ ∴ ∠ = ∠

⇒ ∠ = ∠

⇒ ∠ = ∠

ABC FEG ACB FGE

ACB FGE

DCA HGF

~ ,

1

2

1

2

meceer (v) leLee (vi) mes,

∆ ∆DCA HGF~ (QAAA mece¤helee mes)

ØeMve 11. Deeke=âefle ceW, AB AC= Jeeues, Skeâ meceefÉyeeng ∆ABC keâer yeÌ{eF& ieF& Yegpee CB hejefmLele E Skeâ eEyeog nw~ Ùeefo AD BC⊥ Deewj EF AC⊥ nw, lees efmeæ keâerefpeS efkeâ∆ ∆ABD ECF~ nw~

eq$eYegpe

CE

B

F

A

D

nue efÛe$e ceW, ∆ ABC Skeâ meceefÉyeeng ef$eYegpe nw~ (efoÙee nw)

leLee AB AC= ⇒∠ = ∠B C …(i)

∆ ABD leLee ∆ ECF kesâ efueS

∠ = ∠ABD ECF [meceer (i) mes]

leLee ∠ = ∠ADB EFC [ØelÙeskeâ 90°]

⇒ ∆ ∆ABD ECF~ (AA mece¤helee mes)

ØeMve 12. Skeâ ∆ABC keâer YegpeeSB AB Deewj BC leLee ceeefOÙekeâe AD Skeâ DevÙe ∆PQR keâer›eâceMe: YegpeeDeeW PQ Deewj QR leLee ceeefOÙekeâe PM kesâ meceevegheeleer nQ (osefKeSDeeke=âefle)~ oMee&FS efkeâ ∆ ∆ABC PQR~ nw~

nue efoÙee nw ∆ ABC leLee ∆ PQR ceW,

AD leLee PM Gvekeâer�ceeefOÙekeâeSB�nQ~AB

PQ

BC

QR

AD

PM= = …(i)

efmeæ keâjvee nw ∆ ∆ABC PQR~

Gheheefòe efoÙee�nw,AB

PQ

BC

QR

AD

PM= = (efoÙee nw)

⇒ AB

PQ

BC

QR

AD

PM= =

1

21

2

∴ ∆ ∆ADB PMQ~ (SSS mece¤helee mes)

Deye, ∆ ABC leLee ∆ PQR ceW,AB

PQ

BC

QR= (efoÙee nw)

leLee ∠ = ∠B Q

∴ ∆ ∆ABC PQR~ (SAS mece¤helee mes)

ØeMve 13. Skeâ ∆ABC keâer Yegpee BC hej Skeâ eEyeog D Fme Øekeâej efmLele nw efkeâ∠ = ∠ADC BAC nw~ oMee&FS efkeâ CA CB CD2 = ⋅ nw~


CB

A

D RQ

P

M


nue efÛe$e ceW, ∆ ABC Skeâ meceefÉyeeng ef$eYegpe nw~ (efoÙee nw)

leLee AB AC= ⇒∠ = ∠B C …(i)

∆ ABD leLee ∆ ECF kesâ efueS

∠ = ∠ABD ECF [meceer (i) mes]

leLee ∠ = ∠ADB EFC [ØelÙeskeâ 90°]

⇒ ∆ ∆ABD ECF~ (AA mece¤helee mes)

ØeMve 12. Skeâ ∆ABC keâer YegpeeSB AB Deewj BC leLee ceeefOÙekeâe AD Skeâ DevÙe ∆PQR keâer›eâceMe: YegpeeDeeW PQ Deewj QR leLee ceeefOÙekeâe PM kesâ meceevegheeleer nQ (osefKeSDeeke=âefle)~ oMee&FS efkeâ ∆ ∆ABC PQR~ nw~


AD leLee PM Gvekeâer�ceeefOÙekeâeSB�nQ~AB

PQ

BC

QR

AD

PM= = …(i)

efmeæ keâjvee nw ∆ ∆ABC PQR~

Gheheefòe efoÙee�nw,AB

PQ

BC

QR

AD

PM= = (efoÙee nw)

⇒ AB

PQ

BC

QR

AD

PM= =

1

21

2

∴ ∆ ∆ADB PMQ~ (SSS mece¤helee mes)

Deye, ∆ ABC leLee ∆ PQR ceW,AB

PQ

BC

QR= (efoÙee nw)

leLee ∠ = ∠B Q

∴ ∆ ∆ABC PQR~ (SAS mece¤helee mes)

ØeMve 13. Skeâ ∆ABC keâer Yegpee BC hej Skeâ eEyeog D Fme Øekeâej efmLele nw efkeâ∠ = ∠ADC BAC nw~ oMee&FS efkeâ CA CB CD2 = ⋅ nw~


CB

A

D RQ

P

M

nue ∆ ABC leLee ∆DAC ceW,

∠ = ∠BAC ADC (efoÙee nw)

leLee ∠ = ∠ACB DCA (GYeÙeefve<" ∠C)

⇒ ∆ ∆ABC DAC~ (AAA mece¤helee mes)

⇒ AC

DC

BC

CA=

⇒ AC

CB

CD

CA=

⇒ CA

CD

CB

CA= .

⇒ CA CA CB CD× = ×⇒ CA CB CD

2 = × Fefle efmeæced

ØeMve 14. Skeâ ∆ABC keâer YegpeeSB AB Deewj AC leLee ceeefOÙekeâe AD Skeâ DevÙe ef$eYegpe keâerYegpeeDeeW PQ Deewj PR leLee ceeefOÙekeâe PM kesâ ›eâceMe: meceevegheeleer nQ~ oMee&FS efkeâ∆ ∆ABC PQR~ nw~


AD leLee PM Fvekeâer�ceeefOÙekeâeSB�nQ~AB

PQ

AC

PR

AD

PM= = …(i)

eqmeæ keâjvee nw ∆ ∆ABC PQR~

jÛevee AD keâes E lekeâ Fme Øekeâej yeÌ{eÙee efkeâ AD DE= leLee PM keâes N lekeâ Fme ØekeâejyeÌ{eÙee�efkeâ PM MN= ~ BE CE QN, , leLee RN keâes�efceueeÙee~

eq$eYegpe

B CD

A

1

2

D

E

A

B C

1 3

P

QR

N

M

24


nue ∆ ABC leLee ∆DAC ceW,

∠ = ∠BAC ADC (efoÙee nw)

leLee ∠ = ∠ACB DCA (GYeÙeefve<" ∠C)

⇒ ∆ ∆ABC DAC~ (AAA mece¤helee mes)

⇒ AC

DC

BC

CA=

⇒ AC

CB

CD

CA=

⇒ CA

CD

CB

CA= .

⇒ CA CA CB CD× = ×⇒ CA CB CD

2 = × Fefle efmeæced

ØeMve 14. Skeâ ∆ABC keâer YegpeeSB AB Deewj AC leLee ceeefOÙekeâe AD Skeâ DevÙe ef$eYegpe keâerYegpeeDeeW PQ Deewj PR leLee ceeefOÙekeâe PM kesâ ›eâceMe: meceevegheeleer nQ~ oMee&FS efkeâ∆ ∆ABC PQR~ nw~


AD leLee PM Fvekeâer�ceeefOÙekeâeSB�nQ~AB

PQ

AC

PR

AD

PM= = …(i)

eqmeæ keâjvee nw ∆ ∆ABC PQR~

jÛevee AD keâes E lekeâ Fme Øekeâej yeÌ{eÙee efkeâ AD DE= leLee PM keâes N lekeâ Fme ØekeâejyeÌ{eÙee�efkeâ PM MN= ~ BE CE QN, , leLee RN keâes�efceueeÙee~

eq$eYegpe

B CD

A

1

2

D

E

A

B C

1 3

P

QR

N

M

24

Gheheefòe ÛelegYeg&pe ABEC leLee PQNR meceeblej ÛelegYeg&pe nQ keäÙeeWefkeâ Fvekesâ efJekeâCe& Skeâ-otmejskeâes�›eâceMe:�eEyeog D leLee M hej�meceefÉYeeefpele�keâjles�nQ~

⇒ BE AC=leLee QN PR=

⇒ BE

QN

AC

PR=

⇒ BE

QN

AB

PQ= [meceer (i) mes]

DeLee&led AB

PQ

BE

QN= …(ii)

meceer (i) mes, AB

PQ

AD

PM

AD

PM

AE

PN= = =2

2

(Q efJekeâCe&�Skeâ-otmejs�keâes�ØeefleÛÚso�keâjles�nQ)

DeLee&led AB

PQ

AE

PN= …(iii)

meceer (ii) leLee (iii) mes,AB

PQ

BE

QN

AE

PN= =

⇒ ∆ ∆ABE PQN~

⇒ ∠ = ∠1 2 …(iv)

Fmeer�Øekeâej, ∆ ∆ACE PRN~

∠ = ∠3 4 …(v)

meceer (iv) leLee (v) keâes�peesÌ[ves�hej,

∠ + ∠ = ∠ + ∠1 3 2 4

⇒ ∠ = ∠A P

⇒ ∆ ∆ABC PQR~ (SAS mece¤helee mes)

Fefle efmeæced

ØeMve 15. uebyeeF& 6 ceer Jeeues Skeâ TOJee&Oej mlebYe keâer Yetefce hej ÚeÙee keâ er uebyeeF& 4 ceer nw, peyeefkeâGmeer meceÙe Skeâ ceerveej keâer ÚeÙee keâer uebyeeF& 28 ceer nw~ ceerveej keâer GBâÛeeF& %eele keâerefpeS~

nue efÛe$e (i) ceW, ceevee AB Skeâ TIJee&Oej mlebYe nw efpemekeâer GBâÛeeF& 6 ceer Deewj mlebYe keâer Yetefce hejÚeÙee�keâer�uebyeeF& 4 ceer�nw�Deewj�#eweflepe�mes θ keâesCe�yeveeleer�nw�DeLee&led BC = 4 ceer~

efÛe$e (ii) ceW, ceevee PM Skeâ ceerveej nw efpemekeâer GBâÛeeF& h ceer Deewj ceerveej keâer Yetefce hej ÚeÙeekeâer�uebyeeF& 28 ceer�nw~

DeLee&led NM = 28 ceer

∆ ACB leLee ∆ PNM ceW,

∠ = ∠ =C N θ



Gheheefòe ÛelegYeg&pe ABEC leLee PQNR meceeblej ÛelegYeg&pe nQ keäÙeeWefkeâ Fvekesâ efJekeâCe& Skeâ-otmejskeâes�›eâceMe:�eEyeog D leLee M hej�meceefÉYeeefpele�keâjles�nQ~

⇒ BE AC=leLee QN PR=

⇒ BE

QN

AC

PR=

⇒ BE

QN

AB

PQ= [meceer (i) mes]

DeLee&led AB

PQ

BE

QN= …(ii)

meceer (i) mes, AB

PQ

AD

PM

AD

PM

AE

PN= = =2

2

(Q efJekeâCe&�Skeâ-otmejs�keâes�ØeefleÛÚso�keâjles�nQ)

DeLee&led AB

PQ

AE

PN= …(iii)

meceer (ii) leLee (iii) mes,AB

PQ

BE

QN

AE

PN= =

⇒ ∆ ∆ABE PQN~

⇒ ∠ = ∠1 2 …(iv)

Fmeer�Øekeâej, ∆ ∆ACE PRN~

∠ = ∠3 4 …(v)

meceer (iv) leLee (v) keâes�peesÌ[ves�hej,

∠ + ∠ = ∠ + ∠1 3 2 4

⇒ ∠ = ∠A P

⇒ ∆ ∆ABC PQR~ (SAS mece¤helee mes)

Fefle efmeæced

ØeMve 15. uebyeeF& 6 ceer Jeeues Skeâ TOJee&Oej mlebYe keâer Yetefce hej ÚeÙee keâ er uebyeeF& 4 ceer nw, peyeefkeâGmeer meceÙe Skeâ ceerveej keâer ÚeÙee keâer uebyeeF& 28 ceer nw~ ceerveej keâer GBâÛeeF& %eele keâerefpeS~

nue efÛe$e (i) ceW, ceevee AB Skeâ TIJee&Oej mlebYe nw efpemekeâer GBâÛeeF& 6 ceer Deewj mlebYe keâer Yetefce hejÚeÙee�keâer�uebyeeF& 4 ceer�nw�Deewj�#eweflepe�mes θ keâesCe�yeveeleer�nw�DeLee&led BC = 4 ceer~

efÛe$e (ii) ceW, ceevee PM Skeâ ceerveej nw efpemekeâer GBâÛeeF& h ceer Deewj ceerveej keâer Yetefce hej ÚeÙeekeâer�uebyeeF& 28 ceer�nw~

DeLee&led NM = 28 ceer

∆ ACB leLee ∆ PNM ceW,

∠ = ∠ =C N θ


leLee ∠ = ∠ = °ABC PMN 90

∴ ∆ ∆ABC PMN~ (AAA mece¤helee mes)

⇒ AB

PM

BC

MN= ⇒ AB

BC

PM

MN=

⇒ 6

4 28= h ⇒ h = × =6 28

442 ceer

ØeMve 16. ∆AD Deewj PM, ∆ABC Deewj ∆PQR keâer ›eâceMe: ceeefOÙekeâeSB nQ, peyeefkeâ∆ ∆ABC PQR~ nw~ efmeæ keâerefpeS efkeâ AB

PQ

AD

PM= nw~

nue ∆ ABC leLee ∆ PQR keâer Yegpee BC leLee QR hej ›eâceMe: eEyeog D leLee M Fme Øekeâej nQ efkeâ

AD leLee PM, ∆ ABC leLee ∆ PQR keâer�ceeefOÙekeâeSB�nQ~

∆ ∆ABC PQR~ (efoÙee nw)

⇒ AB

PQ

BC

QR= = ∠ = ∠ ∠ = ∠ ∠ = ∠AC

PRA P B Q C R; , , …(i)

Deye BD CD BC= = 1

2

leLee QM RM QR= = 1

2…(ii)

( ,QD BC keâe�ceOÙe-eEyeog�nw�leLee M QR, ceOÙe-eEyeog�nw~)

meceer (i) mes, AB

PQ

BC

QR=

⇒ AB

PQ

BD

QM= 2

2[meceer (ii) mes]

⇒ AB

PQ

BD

QM=

Fme Øekeâej AB

PQ

BD

QM=

eq$eYegpe

C B

A

θ4 mesceer

6 mesceer

metŸe&

N M

P

θ28 mesceer

h mesceer

metŸe&

(i) (ii)


leLee ∠ = ∠ = °ABC PMN 90

∴ ∆ ∆ABC PMN~ (AAA mece¤helee mes)

⇒ AB

PM

BC

MN= ⇒ AB

BC

PM

MN=

⇒ 6

4 28= h ⇒ h = × =6 28

442 ceer

ØeMve 16. ∆AD Deewj PM, ∆ABC Deewj ∆PQR keâer ›eâceMe: ceeefOÙekeâeSB nQ, peyeefkeâ∆ ∆ABC PQR~ nw~ efmeæ keâerefpeS efkeâ AB

PQ

AD

PM= nw~

nue ∆ ABC leLee ∆ PQR keâer Yegpee BC leLee QR hej ›eâceMe: eEyeog D leLee M Fme Øekeâej nQ efkeâ

AD leLee PM, ∆ ABC leLee ∆ PQR keâer�ceeefOÙekeâeSB�nQ~

∆ ∆ABC PQR~ (efoÙee nw)

⇒ AB

PQ

BC

QR= = ∠ = ∠ ∠ = ∠ ∠ = ∠AC

PRA P B Q C R; , , …(i)

Deye BD CD BC= = 1

2

leLee QM RM QR= = 1

2…(ii)

( ,QD BC keâe�ceOÙe-eEyeog�nw�leLee M QR, ceOÙe-eEyeog�nw~)

meceer (i) mes, AB

PQ

BC

QR=

⇒ AB

PQ

BD

QM= 2

2[meceer (ii) mes]

⇒ AB

PQ

BD

QM=

Fme Øekeâej AB

PQ

BD

QM=

eq$eYegpe

C B

A

θ4 mesceer

6 mesceer

metŸe&

N M

P

θ28 mesceer

h mesceer

metŸe&

(i) (ii)

leLee ∠ = ∠ABD PQM (Q∠ = ∠B Q)

⇒ ∆ ∆ABD PQM~ (SAS mece¤helee mes)

⇒ AB

PQ

AD

PM= Fefle efmeæced

iz'ukoyh 4-4ØeMve 1. ceeve ueerefpeS ∆ ∆ABC DEF~ nw Deewj Fvekesâ #es$eheâue ›eâceMe: 64 mesceer 2 Deewj

121 mesceer 2 nQ~ Ùeefo EF = 15.4 mesceer 2 nes, lees BC %eele keâerefpeS~nue ∆ ∆ABC DEF~ (efoÙee�nw)

⇒ ar

ar

( )

( )

∆∆

=ABC

DEF

BC

EF

2

2

(mece¤he�ef$eYegpeeW�kesâ�#es$eheâueeW�kesâ�iegCe�mes)

⇒ 64

121

2

2= BC

EF

⇒ BC

EF

=

2 28

11⇒ BC

EF= 8

11

⇒ BC EF= ×8

11⇒ BC = ×8

11154. mesceer = 11.2 mesceer

ØeMve 2. Skeâ meceuebye ABCD efpemeceW AB DC|| nw, kesâ efJekeâCe& hejmhej eEyeog O hej ØeefleÛÚsokeâjles nQ~ Ùeefo AB CD= 2 nes, lees ∆AOB Deewj ∆COD kesâ #es$eheâueeW kesâ Devegheele%eele keâerefpeS~

nue ar

ar

( )

( )

∆∆

=AOB

COD

AB

CD

2

2

(mece¤he�ef$eYegpeeW�kesâ�#es$eheâue�kesâ�iegCe�mes)

= ( )2 2

2

CD

CD(QAB CD= 2 )

= × =4 4

1

2

2

CD

CD


B C

A

D Q R

P

M

D C

A B

O

ef$eYegpe4

ØeMveeJeueer 4.4

leLee ∠ = ∠ABD PQM (Q∠ = ∠B Q)


⇒ AB

PQ

AD

PM= Fefle efmeæced

iz'ukoyh 4-4ØeMve 1. ceeve ueerefpeS ∆ ∆ABC DEF~ nw Deewj Fvekesâ #es$eheâue ›eâceMe: 64 mesceer 2 Deewj

121 mesceer 2 nQ~ Ùeefo EF = 15.4 mesceer 2 nes, lees BC %eele keâerefpeS~nue ∆ ∆ABC DEF~ (efoÙee�nw)

⇒ ar

ar

( )

( )

∆∆

=ABC

DEF

BC

EF

2

2


⇒ 64

121

2

2= BC

EF

⇒ BC

EF

=

2 28

11⇒ BC

EF= 8

11

⇒ BC EF= ×8

11⇒ BC = ×8

11154. mesceer = 11.2 mesceer

ØeMve 2. Skeâ meceuebye ABCD efpemeceW AB DC|| nw, kesâ efJekeâCe& hejmhej eEyeog O hej ØeefleÛÚsokeâjles nQ~ Ùeefo AB CD= 2 nes, lees ∆AOB Deewj ∆COD kesâ #es$eheâueeW kesâ Devegheele%eele keâerefpeS~

nue ar

ar

( )

( )

∆∆

=AOB

COD

AB

CD

2

2


= ( )2 2

2

CD

CD(QAB CD= 2 )

= × =4 4

1

2

2

CD

CD


B C

A

D Q R

P

M

D C

A B

O

ØeMve 3. oer ieF& Deeke=âefle ceW, Skeâ ner DeeOeej BC hej oes ∆ABC Deewj ∆DBC yeves ngS nQ~ ÙeefoAD BC, keâes eEyeog O hej ØeefleÛÚso keâjs, lees oMee&FS efkeâ ar

ar

( )

( )

∆∆

=ABC

DBC

AO

DOnw~

nue

AL BC⊥ leLee DM BC⊥ KeeRÛee ∆ OLA leLee ∆ OMD ceW,

∠ = ∠ = °ALO DMO 90

leLee ∠ = ∠AOL DOM (Meer<ee&efYecegKe keâesCe)

∴ ∆ ∆OLA OMD~ (AA mece¤helee mes)

⇒ AL

DM

AO

DO= …(i)

Deye, ar

ar

( )

( )

( ) ( )

( ) ( )

∆∆

=× ×

× ×

ABC

DBC

BC AL

BC DM

1

21

2

= =AL

DM

AO

DO[meceer (i) mes]

∴ ar

ar

( )

( )

∆∆

=ABC

DBC

AO

DO

ØeMve 4. Ùeefo oes mece¤he ef$eYegpeeW kesâ #es$eheâue yejeyej neW, l ees efmeæ keâerefpeS efkeâ Jes ef$eYegpemeJeeËiemece nesles nQ~

nue ∆ ∆ABC PQR~ leLee ar ( )∆ =ABC ar(∆ PQR) (efoÙee nw)

eq$eYegpe

A C

B D

OL

M

A C

B D

O

B C

A

Q R

P


ØeMve 3. oer ieF& Deeke=âefle ceW, Skeâ ner DeeOeej BC hej oes ∆ABC Deewj ∆DBC yeves ngS nQ~ ÙeefoAD BC, keâes eEyeog O hej ØeefleÛÚso keâjs, lees oMee&FS efkeâ ar

ar

( )

( )

∆∆

=ABC

DBC

AO

DOnw~

nue

AL BC⊥ leLee DM BC⊥ KeeRÛee ∆ OLA leLee ∆ OMD ceW,

∠ = ∠ = °ALO DMO 90

leLee ∠ = ∠AOL DOM (Meer<ee&efYecegKe keâesCe)

∴ ∆ ∆OLA OMD~ (AA mece¤helee mes)

⇒ AL

DM

AO

DO= …(i)

Deye, ar

ar

( )

( )

( ) ( )

( ) ( )

∆∆

=× ×

× ×

ABC

DBC

BC AL

BC DM

1

21

2

= =AL

DM

AO

DO[meceer (i) mes]

∴ ar

ar

( )

( )

∆∆

=ABC

DBC

AO

DO

ØeMve 4. Ùeefo oes mece¤he ef$eYegpeeW kesâ #es$eheâue yejeyej neW, l ees efmeæ keâerefpeS efkeâ Jes ef$eYegpemeJeeËiemece nesles nQ~

nue ∆ ∆ABC PQR~ leLee ar ( )∆ =ABC ar(∆ PQR) (efoÙee nw)

eq$eYegpe

A C

B D

OL

M

A C

B D

O

B C

A

Q R

P

DeLee&led ar

ar

( )

( )

∆∆

=ABC

PQR1

⇒ AB

PQ

BC

QR

CA

PR

2

2

2

2

2

21= = =


⇒ AB PQ BC QR= =, leLee CA PR= (SSS Devegheeeflekeâ ceeheob[ mes)

⇒ ∆ ≅ ∆ABC PQR . Fefle efmeæced

ØeMve 5. Skeâ ∆ABC keâer YegpeeDeeW AB BC, Deewj CA kesâ ceOÙe-eEyeog ›eâceMe: D E, Deewj F nQ~∆DEF Deewj ∆ABC kesâ #es$eheâueeW keâe Devegheele %eele keâerefpeS~

nue ∆ ABC keâer YegpeeDeeW AB BC, leLee CA kesâ ceOÙe-eEyeog ›eâceMe: D E, leLee F keâes uesles ngS Skeâ

∆DEF yeveÙee~

ÙeneB, DF BC DE CA= =1

2

1

2,

leLee EF AB= 1

2…(i)

(QD E, leLee F ›eâceMe:�YegpeeDeeW AB BC, , leLee CA kesâ�ceOÙe-eEyeog�nQ~)

⇒ DF

BC

DE

CA

EF

AB= = = 1

2(SSS Devegheeeflekeâ ceeheob[ mes)

⇒ ∆ ∆DEF CAB~

⇒ ar

ar

( )

( )

∆∆

=DEF

CAB

DE

CA

2

2

=

=

1

2 1

4

2

2

CA

CA[meceer (i) mes]


⇒ ar

ar

( )

( )

∆∆

=DEF

ABC

1

4[ ( ) ( )]Qar ar∆ = ∆CAB ABC

Dele:�DeYeer<š�Devegheele 1 : 4 nw~


B C

A

E

D F


DeLee&led ar

ar

( )

( )

∆∆

=ABC

PQR1

⇒ AB

PQ

BC

QR

CA

PR

2

2

2

2

2

21= = =


⇒ AB PQ BC QR= =, leLee CA PR= (SSS Devegheeeflekeâ ceeheob[ mes)

⇒ ∆ ≅ ∆ABC PQR . Fefle efmeæced

ØeMve 5. Skeâ ∆ABC keâer YegpeeDeeW AB BC, Deewj CA kesâ ceOÙe-eEyeog ›eâceMe: D E, Deewj F nQ~∆DEF Deewj ∆ABC kesâ #es$eheâueeW keâe Devegheele %eele keâerefpeS~

nue ∆ ABC keâer YegpeeDeeW AB BC, leLee CA kesâ ceOÙe-eEyeog ›eâceMe: D E, leLee F keâes uesles ngS Skeâ

∆DEF yeveÙee~

ÙeneB, DF BC DE CA= =1

2

1

2,

leLee EF AB= 1

2…(i)

(QD E, leLee F ›eâceMe:�YegpeeDeeW AB BC, , leLee CA kesâ�ceOÙe-eEyeog�nQ~)

⇒ DF

BC

DE

CA

EF

AB= = = 1

2(SSS Devegheeeflekeâ ceeheob[ mes)

⇒ ∆ ∆DEF CAB~

⇒ ar

ar

( )

( )

∆∆

=DEF

CAB

DE

CA

2

2

=

=

1

2 1

4

2

2

CA

CA[meceer (i) mes]


⇒ ar

ar

( )

( )

∆∆

=DEF

ABC

1

4[ ( ) ( )]Qar ar∆ = ∆CAB ABC

Dele:�DeYeer<š�Devegheele 1 : 4 nw~


B C

A

E

D F

ØeMve 6. efmeæ keâerefpeS efkeâ oes mece¤he ef$eYegpeeW kesâ #es$eheâ ueeW keâe Devegheele Fvekeâer mebieleceeefOÙekeâeDeeW kesâ Devegheele keâe Jeie& neslee nw~

nue

efÛe$e�ceW, AD ABC, ∆ keâer�ceeefOÙekeâe�Deewj PM PQR, ∆ keâer�ceeefOÙekeâe�nw~

ÙeneB, D leLee M ›eâceMe: BC Deewj QR keâe�ceOÙe-eEyeog�nw~

Deye, ∆ ∆ABC PQR~

⇒ ∠ = ∠B Q (mebiele keâesCe) …(i)

AB

PQ

BC

QR=

⇒ AB

PQ

BD

QM= 2

2

(QD BC, keâe�ceOÙe-eEyeog�leLee M QR, keâe�ceOÙe-eEyeog�nw~)

⇒ AB

PQ

BD

QM= …(ii)

∆ABD leLee ∆PQM ceW,

∠ = ∠ABD PQM [meceer (i) mes]

leLee AB

PQ

BD

QM= [meceer (ii) mes]


⇒ AB

PQ

AD

PM= …(iii)

Deye, ar

ar

( )

( )

∆∆

=ABC

PQR

AB

PQ

2

2


⇒ ar

ar

( )

( )

∆∆

=ABC

PQR

AD

PM

2

2[meceer (iii) mes]

ØeMve 7. eqmeæ keâerefpeS efkeâ Skeâ Jeie& keâer efkeâmeer Yegpee hej y eveeS ieS meceyeeng ef$eYegpe keâe #es$eheâueGmeer Jeie& kesâ Skeâ efJekeâCe& hej yeveeS ieS meceyeeng ef$e Yegpe kesâ #es$eheâue keâe DeeOee neslee nw~

eq$eYegpe

B C

A

DQ R

P

M


ØeMve 6. efmeæ keâerefpeS efkeâ oes mece¤he ef$eYegpeeW kesâ #es$eheâ ueeW keâe Devegheele Fvekeâer mebieleceeefOÙekeâeDeeW kesâ Devegheele keâe Jeie& neslee nw~

nue

efÛe$e�ceW, AD ABC, ∆ keâer�ceeefOÙekeâe�Deewj PM PQR, ∆ keâer�ceeefOÙekeâe�nw~

ÙeneB, D leLee M ›eâceMe: BC Deewj QR keâe�ceOÙe-eEyeog�nw~

Deye, ∆ ∆ABC PQR~

⇒ ∠ = ∠B Q (mebiele keâesCe) …(i)

AB

PQ

BC

QR=

⇒ AB

PQ

BD

QM= 2

2

(QD BC, keâe�ceOÙe-eEyeog�leLee M QR, keâe�ceOÙe-eEyeog�nw~)

⇒ AB

PQ

BD

QM= …(ii)

∆ABD leLee ∆PQM ceW,

∠ = ∠ABD PQM [meceer (i) mes]

leLee AB

PQ

BD

QM= [meceer (ii) mes]


⇒ AB

PQ

AD

PM= …(iii)

Deye, ar

ar

( )

( )

∆∆

=ABC

PQR

AB

PQ

2

2


⇒ ar

ar

( )

( )

∆∆

=ABC

PQR

AD

PM

2

2[meceer (iii) mes]

ØeMve 7. eqmeæ keâerefpeS efkeâ Skeâ Jeie& keâer efkeâmeer Yegpee hej y eveeS ieS meceyeeng ef$eYegpe keâe #es$eheâueGmeer Jeie& kesâ Skeâ efJekeâCe& hej yeveeS ieS meceyeeng ef$e Yegpe kesâ #es$eheâue keâe DeeOee neslee nw~

eq$eYegpe

B C

A

DQ R

P

M

nue

ABCD Skeâ�Jeie&�nw�efpemekeâer�Yegpee�keâer�uebyeeF& = a

leye,�efJekeâCe& BD a= 2

∆ PAB leLee ∆QBD meceyeeng�ef$eYegpe�nw~.

⇒ ∆ ∆PAB QBD~ (meceyeeng ef$eYegpe mece¤he nesles nQ)

⇒ ar

ar

( )

( )

∆∆

=PAB

QBD

AB

BD

2

2(mece¤he ef$eYegpeeW kesâ #es$eheâue kesâ iegCe mes)

= =a

a

2

22

1

2( )

⇒ ar ar( ) ( )∆ = ∆PAB QBD1

2Fefle efmeæced

mener�Gòej�ÛegefveS�Deewj�Deheves�Gòej�keâe�DeewefÛelÙe�oerefpeS~

ØeMve 8. ABC Deewj BDE oes meceyeeng ef$eYegpe Fme Øekeâej nQ efkeâ D Yegpee BC keâe ceOÙe-eEyeog nw~∆ABC Deewj ∆BDE kesâ #es$eheâueeW keâe Devegheele nw(a) 2 : 1 (b) 1 : 2 (c) 4 : 1 (d) 1 : 4

nue (c) ÙeneB, AB BC CA a= = = (ceevee)

Q∆ ABC Skeâ�meceyeeng�ef$eYegpe�nw~

BD a= 1

2(Q BC keâe ceOÙe-eEyeog D nw)

Deye, ∆ ∆ABC BDE~ (Q oesveeW meceyeeng ef$eYegpe nQ)

⇒ ar

ar

( )

( )

∆∆

=ABC

BDE

AB

BD

2

2(mece¤he ef$eYegpeeW kesâ #es$eheâueeW kesâ iegCe mes)

=

=a

a

2

1

2

4

1

Dele: ∆ ABC Deewj ∆BDE kesâ #es$eheâueeW�keâe�Devegheele = 4 1:


D C

A B

Q

a 2

a 2

2

P

a a

a


nue

ABCD Skeâ�Jeie&�nw�efpemekeâer�Yegpee�keâer�uebyeeF& = a

leye,�efJekeâCe& BD a= 2

∆ PAB leLee ∆QBD meceyeeng�ef$eYegpe�nw~.

⇒ ∆ ∆PAB QBD~ (meceyeeng ef$eYegpe mece¤he nesles nQ)

⇒ ar

ar

( )

( )

∆∆

=PAB

QBD

AB

BD

2


= =a

a

2

22

1

2( )

⇒ ar ar( ) ( )∆ = ∆PAB QBD1

2Fefle efmeæced

mener�Gòej�ÛegefveS�Deewj�Deheves�Gòej�keâe�DeewefÛelÙe�oerefpeS~

ØeMve 8. ABC Deewj BDE oes meceyeeng ef$eYegpe Fme Øekeâej nQ efkeâ D Yegpee BC keâe ceOÙe-eEyeog nw~∆ABC Deewj ∆BDE kesâ #es$eheâueeW keâe Devegheele nw(a) 2 : 1 (b) 1 : 2 (c) 4 : 1 (d) 1 : 4

nue (c) ÙeneB, AB BC CA a= = = (ceevee)

Q∆ ABC Skeâ�meceyeeng�ef$eYegpe�nw~

BD a= 1

2(Q BC keâe ceOÙe-eEyeog D nw)

Deye, ∆ ∆ABC BDE~ (Q oesveeW meceyeeng ef$eYegpe nQ)

⇒ ar

ar

( )

( )

∆∆

=ABC

BDE

AB

BD

2

2(mece¤he ef$eYegpeeW kesâ #es$eheâueeW kesâ iegCe mes)

=

=a

a

2

1

2

4

1

Dele: ∆ ABC Deewj ∆BDE kesâ #es$eheâueeW�keâe�Devegheele = 4 1:


D C

A B

Q

a 2

a 2

2

P

a a

a

ØeMve 9. oes mece¤he ef$eYegpeeW keâer YegpeeSB 4 9: kesâ Devegheele ceW nQ~ Fve ef$eYegpeeW kesâ #es$eheâueeW k eâeDevegheele nw(a) 2 : 3 (b) 4 : 9 (c) 81 : 16 (d) 16 : 81

nue (d) oes mece¤he ef$eYegpeeW kesâ #es$eheâueeW keâe Devegheele Gvekeâer mebiele YegpeeDeeW kesâ JeieeX kesâ Devegheelekesâ�yejeyej�neslee�nw~

∴ ef$eYegpeeW�kesâ�#es$eheâueeW�keâe�Devegheele =

=4

916 81

2

:

iz'ukoyh 4-5ØeMve 1. kegâÚ ef$eYegpeesW keâer YegpeeSB veerÛes oer ieF& nQ~ efve Oee&efjle keâerefpeS efkeâ FveceW mes keâewve-keâewve mes

ef$eYegpe mecekeâesCe ef$eYegpe nQ? Fme efmLeefle ceW keâCe& k eâer uebyeeF& Yeer efueefKeS~(i) 7 mesceer, 24 mesceer , 25 mesceer

(ii) 3 mesceer, 8 mesceer, 6 mesceer(iii) 50 mesceer, 80 mesceer, 100 mesceer(iv) 13 mesceer, 2 mesceer , 5 mesceer

nue nce peeveles nQ efkeâ mecekeâesCe ef$eYegpe ceW, oes Úesšer YegpeeDeeW kesâ JeieeX keâe Ùeesie, leermejer Yegpee kesâJeie& kesâ yejeyej neslee nw~

(i) ÙeneB, ( ) ( )7 24 49 5762 2+ = += =625 25 2( )

Dele: oer ieF& YegpeeSB 7 mesceer, 24 mesceer Deewj 25 mesceer mecekeâesCe ef$eYegpe yeveeleer nQ DeewjkeâCe&�keâer�uebyeeF& 25 mesceer�nw~

(ii) ÙeneB, ( ) ( )3 6 9 36 452 2+ = + =

leLee ( )8 642 =

45 64≠Q oesveeW�ceeve�yejeyej�veneR�nQ~

Dele:�oer�ieF&�YegpeeSB 3 mesceer, 8 mesceer�Deewj 6 mesceer�mecekeâesCe�ef$eYegpe�veneR�yeveeleer�nQ~

(iii) ÙeneB, ( ) ( )50 80 2500 6400 89002 2+ = + =

leLee ( )100 100002 =

Q oesveeW�ceeve�yejeyej�veneR�nQ~

∴ oer�ieF&�YegpeeSB 50 mesceer, 80 mesceer�leLee 100 mesceer�mecekeâesCe�ef$eYegpe�veneR�yeveeleer�nQ~

(iv) ÙeneB, ( ) ( )12 5 144 252 2+ = += =169 13 2( )

Dele:�oer�ieF&�YegpeeSB 13 mesceer, 12 mesceer�leLee�5�mesceer�mecekeâesCe�ef$eYegpe�yeveeleer�nQ~

eq$eYegpe

ef$eYegpe4

ØeMveeJeueer 4.5

ØeMve 9. oes mece¤he ef$eYegpeeW keâer YegpeeSB 4 9: kesâ Devegheele ceW nQ~ Fve ef$eYegpeeW kesâ #es$eheâueeW k eâeDevegheele nw(a) 2 : 3 (b) 4 : 9 (c) 81 : 16 (d) 16 : 81

nue (d) oes mece¤he ef$eYegpeeW kesâ #es$eheâueeW keâe Devegheele Gvekeâer mebiele YegpeeDeeW kesâ JeieeX kesâ Devegheelekesâ�yejeyej�neslee�nw~

∴ ef$eYegpeeW�kesâ�#es$eheâueeW�keâe�Devegheele =

=4

916 81

2

:

iz'ukoyh 4-5ØeMve 1. kegâÚ ef$eYegpeesW keâer YegpeeSB veerÛes oer ieF& nQ~ efve Oee&efjle keâerefpeS efkeâ FveceW mes keâewve-keâewve mes

ef$eYegpe mecekeâesCe ef$eYegpe nQ? Fme efmLeefle ceW keâCe& k eâer uebyeeF& Yeer efueefKeS~(i) 7 mesceer, 24 mesceer , 25 mesceer

(ii) 3 mesceer, 8 mesceer, 6 mesceer(iii) 50 mesceer, 80 mesceer, 100 mesceer(iv) 13 mesceer, 2 mesceer , 5 mesceer

nue nce peeveles nQ efkeâ mecekeâesCe ef$eYegpe ceW, oes Úesšer YegpeeDeeW kesâ JeieeX keâe Ùeesie, leermejer Yegpee kesâJeie& kesâ yejeyej neslee nw~

(i) ÙeneB, ( ) ( )7 24 49 5762 2+ = += =625 25 2( )

Dele: oer ieF& YegpeeSB 7 mesceer, 24 mesceer Deewj 25 mesceer mecekeâesCe ef$eYegpe yeveeleer nQ DeewjkeâCe&�keâer�uebyeeF& 25 mesceer�nw~

(ii) ÙeneB, ( ) ( )3 6 9 36 452 2+ = + =

leLee ( )8 642 =

45 64≠Q oesveeW�ceeve�yejeyej�veneR�nQ~

Dele:�oer�ieF&�YegpeeSB 3 mesceer, 8 mesceer�Deewj 6 mesceer�mecekeâesCe�ef$eYegpe�veneR�yeveeleer�nQ~

(iii) ÙeneB, ( ) ( )50 80 2500 6400 89002 2+ = + =

leLee ( )100 100002 =

Q oesveeW�ceeve�yejeyej�veneR�nQ~

∴ oer�ieF&�YegpeeSB 50 mesceer, 80 mesceer�leLee 100 mesceer�mecekeâesCe�ef$eYegpe�veneR�yeveeleer�nQ~

(iv) ÙeneB, ( ) ( )12 5 144 252 2+ = += =169 13 2( )

Dele:�oer�ieF&�YegpeeSB 13 mesceer, 12 mesceer�leLee�5�mesceer�mecekeâesCe�ef$eYegpe�yeveeleer�nQ~

eq$eYegpe


ØeMve 2. PQR Skeâ mecekeâesCe ef$eYegpe nw efpemekeâe ∠P mecekeâesCe nw leLee QR hej eEyeog M FmeØekeâej efmLele nw efkeâ PM QR⊥ nw~ oMee&FS efkeâ PM QM MR2 = ⋅ nw~

nue ∆PQR leLee ∆MPQ ceW,

∠ + ∠ = ∠ + ∠1 2 2 4 (ØelÙeskeâ = °90 )

⇒ ∠ = ∠1 4

Fmeer Øekeâej, ∠ = ∠2 3

leLee ∠ = ∠PMR PMQ (ØelÙeskeâ 90°)

∆ ∆QPM PRM~ (AAA mece¤helee)

⇒ ar

ar

( )

( )

∆∆

=QPM

PRM

PM

RM

2


⇒

1

21

2

2

2

( ) ( )

( ) ( )

QM PM

RM PM

PM

RM

×

×= (ef$eYegpe keâe #es$eheâue = ×1

2DeeOeej × GBâÛeeF&)

⇒ QM

RM

PM

RM=

2

2⇒ PM QM RM

2 = ×

Ùee PM QM MR2 = ×

ØeMve 3. Deeke=âefle ceW, ABD Skeâ mecekeâesCe ef$eYegpe nw efpemekeâe ∠A mecekeâesCe nw leLee AC BD⊥nw~ oMee&FS efkeâ

(i) AB BC BD2 = ⋅ (ii) AC BC DC2 = ⋅ (iii) AD BD CD2 = ⋅

nue pewmee efkeâ Ghejeskeäle ØeMve ceW efmeæ efkeâÙee pee Ûegkeâe nw~∆ ∆ ∆ABC DAC DBA~ ~


P Q

R

1

34 2

M

AB

D

C


(i) ∆ ∆ABC DBA~

leye, ar

ar

( )

(

∆∆

=ABC

DBA

AB

DB

2


⇒

1

21

2

( ) ( )

( ) ( )

BC AC

BD AC

×

×= AB

DB

2

2

(eq$eYegpe�keâe�#es$eheâue = ×1


⇒ AB BC BD2 = ⋅

(ii) ∆ ∆ABC DAC~

⇒ ar

ar

( )

( )

∆∆

=ABC

DAC

AC

DC

2


⇒

1

21

2

( ) ( )

( ) ( )

BC AC

DC AC

×

×= AC

DC

2

2



⇒ AC BC DC2 = ⋅

(iii) ∆ ∆DAC DBA~

⇒ ar

ar

( )

( )

∆∆

=DAC

DBA

DA

DB

2

2

⇒

1

21

2

( ) ( )

( ) ( )

CD AC

BD AC

×

×= AD

BD

2

2



⇒ AD BD CD2 = ⋅ Fefle efmeæced

ØeMve 4. ABC Skeâ meceefÉyeeng ef$eYegpe nw efpemekeâe ∠C mecekeâesCe nw~ efmeæ keâerefpeS efkeâAB AC2 22= nw~

nue Q ∆ ABC Skeâ�meceefÉyeeng�ef$eYegpe�nw�efpemekeâe ∠C mecekeâesCe�nw~

leLee AC BC= …(i)

heeFLeeieesjme�ØecesÙe�mes,

AB AC BC2 2 2= + = +AC AC

2 2 = 2 2AC

[QBC AC= meceer (i) mes]

eq$eYegpe

C B

A


ØeMve 5. ABC Skeâ meceefÉyeeng ef$eYegpe nw efpemeceW AC BC= nw~ Ùeefo AB AC2 22= nw, leesefmeæ keâerefpeS efkeâ ABC Skeâ mecekeâesCe ef$eYegpe nw~

nue QABC Skeâ�meceefÉyeeng�ef$eYegpe�nw�efpemeceW AC BC=

∆ABC ceW,AC BC= …(i)

leLee AB AC2 22= …(ii)

Deye, AC BC AC AC2 2 2 2+ = + [meceer (i) mes]

= =2 2 2AC AB [meceer (ii) mes]

DeLee&led AC BC AB2 2 2+ =

ØeMve 6. Skeâ meceyeeng ∆ABC keâer Yegpee 2a nw~ Gmekesâ ØelÙeskeâ Meer<e&uebye keâer uebyeeF& %eele keâerefpeS~

nue ∆ABC Skeâ�meceyeeng�ef$eYegpe�nw�efpemekeâer�Yegpee 2 a nw~

AD BC⊥ KeeRÛee (jÛevee mes)

peneB, AD Meer<e&uebye�nw~

∆ADB leLee ∆ADC ceW,

AD AD= (GYeÙeefve<")

leLee ∠ = ∠ = °ADB ADC 90

∆ ≅ ∆ADB ADC (RHS meJeeËiemecelee)

⇒ BD CD BC a= = =1

2

Deye, ∆ ABD ceW�heeFLeeieesjme�ØecesÙe�mes,

AB AD BD2 2 2= +

⇒ ( )2 2 2 2a AD a= +

⇒ AD a2 23=

⇒ AD a= 3

Dele:�ØelÙeskeâ�Meer<e&�uebye�keâer�uebyeeF& 3 a nw~

ØeMve 7. efmeæ keâerefpeS efkeâ Skeâ meceÛelegYeg&pe keâer YegpeeDee W kesâ JeieeX keâe Ùeesie Gmekesâ efJekeâCeeX kesâJeieeX kesâ Ùeesie kesâ yejeyej neslee nw~

nue ABCD Skeâ�meceÛelegYeg&pe�nw�efpemeceW AB BC CD DA a= = = = (ceevee)

Deewj�efJekeâCe& AC leLee BD Skeâ-otmejs�keâes�eEyeog O hej�mecekeâesCe�hej�meceefÉYeeefpele�keâjles�nQ~


C

A B

B C

A

D


∆OAB ceW, ∠ = °AOB 90

OA AC= 1

2leLee OB BD= 1

2

∆AOB ceW�heeFLeeieesjme�ØecesÙe�mes,

OA OB AB2 2 2+ =

⇒ 1

2

1

2

2 22

AC BD AB

+

=

⇒ AC BD AB2 2 24+ =

leLee 4 2 2 2AB AC BD= +

⇒ AB BC CD DA AC BD2 2 2 2 2 2+ + + = + ( )Q AB BC CD DA= = =

Fefle�efmeæced

ØeMve 8. oer ieF& Deeke=âefle ceW ∆ABC kesâ DeYÙeblej ceW efmLele keâesF& eEyeog O nw leLeeOD BC OE AC⊥ ⊥, Deewj OF AB⊥ nw oMee&FS efkeâ

(i) OA OB OC OD OE OF AF BD CE2 2 2 2 2 2 2 2 2+ + − − − = + +

(ii) AF BD CE AE CD BF2 2 2 2 2 2+ + = + +

nue ∆ABC ceW,�eEyeog O mes�jsKee OB OC, leLee OA keâes�efceueeÙee~

(i) mecekeâesCe ∆ OFA ceW,

eq$eYegpe

D C

A B

O

B C

A

D

O

E

F

B C

A

D

O

E

F


OA OF AF2 2 2= + (heeFLeeieesjme ØecesÙe mes)

⇒ OA OF AF2 2 2− = …(i)

Fmeer Øekeâej ∆OBD ceW, OB OD BD2 2 2− = …(ii)

leLee ∆OCE ceW, OC OE CE2 2 2− = …(iii)

meceer (i), (ii) leLee (iii) keâes�peesÌ[ves�hej,

OA OB OC OD OE OF2 2 2 2 2 2+ + − − − = + +AF BD CE

2 2 2

(ii) meceer (i) mes,

OA OB OC OD OE OF2 2 2 2 2 2+ + − − − = + +AF BD CE

2 2 2 …(iv)

Fmeer�Øekeâej,

OA OB OC OD OE OF2 2 2 2 2 2+ + − − − = + +BF CD AE

2 2 2 …(v)

meceer (iv) Je (v) mes,

AF BD CE AE CD BF2 2 2 2 2 2+ + = + +

ØeMve 9. 10 ceer uebyeer Skeâ meerÌ{er Skeâ oerJeej hej efškeâeves hej Ye tefce mes 8 ceer keâer GBâÛeeF& hej efmLeleSkeâ efKeÌ[keâer lekeâ hengBÛeleer nw~ oerJeej kesâ DeeOee j mes meerÌ{er kesâ efveÛeues efmejs keâer otjer %eelekeâerefpeS~

nue ceevee B efKeÌ[keâer keâer efmLeefle nw Deewj CB meerÌ{er keâer uebyeeF& nw~

leye AB = 8 ceer (efKeÌ[keâer keâer GBâÛeeF&)

BC = 10 ceer (meerÌ{er keâer uebyeeF&)

ceevee�oerJeej�kesâ�DeeOeej�mes�meerÌ{er�kesâ�efveÛeues�efmejs�keâer�otjer x ceer�nw~

∆ ABC ceW,�heeFLeeieesjme�ØecesÙe�mes,

BC AB CA2 2 2= +

⇒ ( ) ( )10 82 2 2= + x

⇒ 100 64 2= + x

⇒ x2 100 64= −

⇒ x2 36=

⇒ x = 36

⇒ x = 6

Dele:�oerJeej�kesâ�DeeOeej�mes�meerÌ{er�kesâ�efveÛeues�efmejs�keâer�otjer = 6 ceer


AC

B

10 ceer 8 ceer

x ceer


ØeMve 10. 18 ceer GBâÛes Skeâ TOJee&Oej KebYes kesâ Thejer efmejs mes Skeâ leej keâe Skeâ efmeje pegÌ[e ngDee nwleLee leej keâe otmeje efmeje Skeâ KetBšs mes pegÌ[e ngDee nw~ KebYes kesâ DeeOeej mes KetBšs keâesefkeâleveer otjer hej ieeÌ[e peeS efkeâ leej levee jns peyeefkeâ leej keâer uebyeeF& 24 ceer nw?

nue ceevee AB Skeâ TOJee&Oej KebYee nw efpemekeâer GBâÛeeF& 18 ceer Deewj leej keâer uebyeeF& BC = 24 ceer nw~

ceevee�KebYes�kesâ�DeeOeej�mes�KetBšs�keâer�otjer = x ceer

∆ ABC ceW,�heeFLeeieesjme�ØecesÙe�mes,

DeLee&led AC AB BC2 2 2+ =

∴ x2 2 218 24+ =( ) ( )

⇒ x2 2 224 18= −( ) ( )

= −576 324

= 252

⇒ x = 252 ceer

⇒ x = 6 7 ceer

Dele:�KebYes�kesâ�DeeOeej�mes�KetBšs�keâer�otjer = 6 7 ceer�nw~

ØeMve 11. Skeâ nJeeF& penepe Skeâ nJeeF& De[d[s mes Gòej keâer Deesj 1000 efkeâceer/Iebše keâer Ûeeue mesGÌ[lee nw~ Fmeer meceÙe Skeâ DevÙe nJeeF& penepe Gmeer nJeeF& De[d[s mes heefMÛece keâer Deesj1200 efkeâceer/Iebše keâer Ûeeue mes GÌ[lee nw~ 1

1

2Iebšs kesâ yeeo oesveeW nJeeF& penepeeW kesâ yeerÛe

keâer otjer efkeâleveer nesieer?

nue ceevee oesveeW nJeeF& penepeeW kesâ yeerÛe keâer otjer x efkeâceer nw~ Skeâ nJeeF& penepe BC otjer Gòej

efoMee�ceW 112

Iebšs�ceW 1000 efkeâceer/Iebše�keâer�Ûeeue�mes�GÌ[lee�nw~

∴ BC = ×10003

2efkeâceer (Q otjer = Ûeeue × meceÙe)

= 1500 efkeâceer

otmeje nJeeF& penepe BA otjer heefMÛece efoMee ceW 112

Iebšs ceW 1200 efkeâceer/Iebše keâer Ûeeue mes

GÌ[lee�nw~

eq$eYegpe

AC

B

24 c eer18 c eer

K ebY ee

x c eerK e@Btö e

l eej


∴ BA = ×12003

2= 1800 eqkeâceer (Q otjer = Ûeeue × meceÙe )

mecekeâesCe ∆ ABC ceW,�heeFLeeieesjme�ØecesÙe�mes,

AC AB BC2 2 2= + = +( ) ( )1800 15002 2

= +3240000 2250000 = 5490000

⇒ AC = 5490000 ceer ⇒ AC = 300 61 ceer

Dele:�oesveeW�nJeeF&�penepeeW�kesâ�yeerÛe�keâer�otjer 300 61 ceer�nw~

ØeMve 12. oes KebYes efpevekeâer GBâÛeeFÙeeB 6 ceer Deewj 11 ceer nQ leLee Ùes meceleue Yetefce hej KeÌ[s nQ~ ÙeefoFvekesâ DeeOeejeW kesâ yeerÛe keâer otjer 12 ceer nw, lees Fvekesâ Thejer efmejeW kesâ yeerÛe keâer otjer %eelekeâerefpeS~

nue ceevee BC Deewj AD oes�KebYes�nQ�efpevekeâer�GBâÛeeFÙeeB�›eâceMe: 6 ceer�Deewj 11 ceer�nQ~�leye,

CE BC BE BC AD= − = −= −11 6 = 5 mesceer

ceevee�KebYeeW�kesâ�Thejer�efmejeW�kesâ�yeerÛe�keâer�otjer, DC x= ceer

∆DEC ceW,�heeFLeeieesjme�ØecesÙe�mes

DC DE CE2 2 2= +

⇒ x2 2 212 5 169= + =( ) ( )

⇒ x = 13Dele:�KebYeeW�kesâ�efmejeW�kesâ�yeerÛe�keâer�otjer = 13 ceer


12 c eer

6 c eer

11 c eer

5 c eerx c eer

D E

A B

12 c eer

C

BA

C

hen u ee n J eeF& p en ep e

GÚ ej

otm eje n J eeF& p en ep e heefM¤ec e

GÚ ej

heefM¤ec e hetjy e

oef#eC e


ØeMve 13. Skeâ ∆ABC efpemekeâe ∠C mecekeâesCe nw, keâer YegpeeDeeW CA Deewj CB hej ›eâceMe: eEyeog DDeewj E efmLele nQ~ efmeæ keâerefpeS efkeâ AE BD AB DE2 2 2 2+ = + nw~

nue Skeâ ∆ ABC efpemekeâe ∠C mecekeâesCe nw, KeeRÛee~ YegpeeDeeW CA Deewj CB hej ›eâceMe: eEyeog D

Deewj E efmLele nQ~ ED BD, leLee EA keâes efceueeÙee~

mecekeâesCe ∆ ACE ceW,

AE CA CE2 2 2= + (heeFLeeieesjme ØecesÙe mes) …(i)

leLee�mecekeâesCe ∆BCD ceW,

BD BC CD2 2 2= + …(ii)

meceer (i) Je (ii) keâes�peesÌ[ves�hej,

AE BD CA CE BC CD2 2 2 2 2 2+ = + + +( ) ( )= + + +( ) ( )BC CA CD CE

2 2 2 2

(Q∆ ABC ceW, BA BC CA2 2 2= + leLee ∆ECD ceW, DE CD CE

2 2 2= + )

= +BA DE2 2 (heeFLeeieesjme ØecesÙe mes)

∴ AE BD AB DE2 2 2 2+ = + Fefle efmeæced

ØeMve 14. efoS ieS efÛe$e ceW, ∆ABC kesâ Meer<e& A mes BC hej [eueeieÙee uebye BC keâes eEyeog D hej Fme Øekeâej ØeefleÛÚsokeâjlee nw efkeâ DB CD= 3 nw~ efmeæ keâerefpeS efkeâ2 22 2 2AB AC BC= + nw~

nue

efoÙee nw, DB CD= 3

⇒ CD BC= 1

4…(i)

eq$eYegpe

AC

B

D

E

A

C DB

A

CD

B


leLee DB BC= 3

4

∆ABD ceW, AB DB AD2 2 2= + …(ii)

∆ACD ceW, AC CD AD2 2 2= + (heeFLeeieesjme ØecesÙe mes) …(iii)

meceer (iii) keâes�meceer (ii) ceW�mes�Ieševes�hej,

AB AC DB CD2 2 2 2− = −

=

−

3

4

1

4

2 2

BC BC

= −9

16

1

16

2 2BC BC = 1

2

2BC

⇒ 2 22 2 2AB AC BC− = ⇒ 2 22 2 2

AB AC BC= + Fefle efmeæced

ØeMve 15. efkeâmeer meceyeeng ∆ABC keâer Yegpee BC hej Skeâ eEyeog D Fme Øekeâej efmLele nw efkeâBD BC= 1

3nw~ efmeæ keâerefpeS efkeâ 9 72 2AD AB= nw~

nue Skeâ meceyeeng ∆ ABC KeeRÛee efpemekeâer Yegpee BC hej Skeâ eEyeog D Fme Øekeâej efmLele nw efkeâ

BD BC= 1

3Deewj AE BC⊥ KeeRÛeer~

AB BC CA a= = = (ceevee)

(meceyeeng ef$eYegpe�kesâ�iegCe�mes)

BD BC a= =1

3

1

3

⇒ CD BC a= =2

3

2

3

Q AE BC⊥

⇒ BE EC a= = 1

2

(∴meceyeeng�ef$eYegpe�ceW,�GBâÛeeF& AE,�Yegpee BC keâer�uebye�meceefÉYeepekeâ�nw)


B C

A

D E


DE BE BD a a a= − = − =1

2

1

3

1

6

∆ ADE ceW,�heeFLeeieesjme�ØecesÙe�mes

AD AE DE2 2 2= +

= − +AB BE DE2 2 2

(QmecekeâesCe ∆ABE ceW, AE AB BE2 2 2= − )

= −

+

a a a2

2 21

2

1

6

= − +a a a2 2 21

4

1

36

= − +( )36 9 1

36

2a

= =28

36

7

9

2 2a AB

⇒ 9 72 2AD AB= Fefle efmeæced

ØeMve 16. efkeâmeer meceyeeng ef$eYegpe ceW, efmeæ keâerefpeS efkeâ Gmekeâer Skeâ Yegpee kesâ Jeie& keâe leerve iegveeGmekesâ Skeâ Meer<e&uebye kesâ Jeie& kesâ Ûeej iegves kesâ yejeyej neslee nw~

nue Skeâ�meceyeeng ∆ ABC KeeRÛee�efpemekeâer�Yegpee a nw~

leLee AD BC⊥

ceevee AD x=

Deye, BD CD BC a= = =1

2

1

2

(Skeâ�meceyeeng�ef$eYegpe ceW,�Meer<e&uebye AD,�Yegpee BC keâer�meceefÉYeepekeâ�nw)mecekeâesCe ∆ ABD ceW,

AB AD BD2 2 2= +

⇒ a x a2 2

21

2= +

⇒ a x a

2 2 21

4= +

⇒ 4 42 2 2a x a= + ⇒ 3 42 2

a x= Fefle efmeæced

eq$eYegpe

B C

A

D

x

a


ØeMve 17. mener Gòej Ûegvekeâj Gmekeâe DeewefÛelÙe oerefpeS~∆ABC ceW, AB = 6 3 mesceer, AC = 12 mesceer Deewj BC = 6 mesceer nw~ ∠B nw(a) 120° (b) 60° (c) 90° (d) 45°

nue (b) efoÙee�nw, BC = 6 mesceer�leLee AB = 6 3 mesceer�leLee AC = 12 mesceer

Deye, AB BC2 2 2 26 3 6+ = +( ) ( )

= +108 36 = 144 = =( ) ( )12 2 2AC

⇒∆ABC ceW�efpemekeâe ∠B mecekeâesCe�nw~

⇒ ∠ = °B 90

BC AB<⇒ ∠A< ∠C

⇒∠A keâe ceeve 45° mes DeefOekeâ veneR nes mekeâlee nw⇒ ∠ = °A 30 ⇒ ∠ = ° − ° = °B 90 30 60

iz'ukoyh 4-6 (,sfPNd)*

ØeMve 1. Deeke=âefle ceW PS ∠QPR keâe meceefÉYeepekeâ nw~ efmeæ keâerefpeS efkeâ QS

SR

PQ

PR= nw~

nue efoÙee nw ∆ PQR ceW, PS,∠QPR keâe�meceefÉYeepekeâ�nw~

efmeæ keâjvee nw QS

SR

PQ

PR=

jÛevee RT SP|| KeeRÛee�Deewj QP keâes T lekeâ�Deeies�yeÌ{eÙee~

Gheheefòe QRT SP|| Deewj�FvnW�Skeâ�efleÙe&keâ�jsKee PR ØeefleÛÚsefole�keâjleer�nw~

∴ ∠ = ∠1 2 (Deble: Skeâeblej keâesCe) …(i)

QRT SP|| Deewj�efleÙe&keâ�jsKee QT FvnW�ØeefleÛÚsefole�keâjleer�nw~

∴ ∠ = ∠3 4 (mebiele keâesCe) …(ii)

hejbleg ∠ = ∠1 3 (efoÙee nw)

∴ ∠ = ∠2 4 [meceer (i) Je (ii) mes]


Q SR

P

ef$eYegpe4

ØeMveeJeueer 4.6 (SsefÛÚkeâ)

ØeMve 17. mener Gòej Ûegvekeâj Gmekeâe DeewefÛelÙe oerefpeS~∆ABC ceW, AB = 6 3 mesceer, AC = 12 mesceer Deewj BC = 6 mesceer nw~ ∠B nw(a) 120° (b) 60° (c) 90° (d) 45°

nue (b) efoÙee�nw, BC = 6 mesceer�leLee AB = 6 3 mesceer�leLee AC = 12 mesceer

Deye, AB BC2 2 2 26 3 6+ = +( ) ( )

= +108 36 = 144 = =( ) ( )12 2 2AC

⇒∆ABC ceW�efpemekeâe ∠B mecekeâesCe�nw~

⇒ ∠ = °B 90

BC AB<⇒ ∠A< ∠C

⇒∠A keâe ceeve 45° mes DeefOekeâ veneR nes mekeâlee nw⇒ ∠ = °A 30 ⇒ ∠ = ° − ° = °B 90 30 60

iz'ukoyh 4-6 (,sfPNd)*

ØeMve 1. Deeke=âefle ceW PS ∠QPR keâe meceefÉYeepekeâ nw~ efmeæ keâerefpeS efkeâ QS

SR

PQ

PR= nw~

nue efoÙee nw ∆ PQR ceW, PS,∠QPR keâe�meceefÉYeepekeâ�nw~

efmeæ keâjvee nw QS

SR

PQ

PR=

jÛevee RT SP|| KeeRÛee�Deewj QP keâes T lekeâ�Deeies�yeÌ{eÙee~

Gheheefòe QRT SP|| Deewj�FvnW�Skeâ�efleÙe&keâ�jsKee PR ØeefleÛÚsefole�keâjleer�nw~

∴ ∠ = ∠1 2 (Deble: Skeâeblej keâesCe) …(i)

QRT SP|| Deewj�efleÙe&keâ�jsKee QT FvnW�ØeefleÛÚsefole�keâjleer�nw~

∴ ∠ = ∠3 4 (mebiele keâesCe) …(ii)

hejbleg ∠ = ∠1 3 (efoÙee nw)

∴ ∠ = ∠2 4 [meceer (i) Je (ii) mes]


Q SR

P

∴ PT PR= …(iii)

(Q meceeve�keâesCeeW�keâer�efJehejerle�YegpeeSB�meceeve�nesleer�nQ~)Deye, ∆QRT ceW,

PS RT|| (jÛevee mes)

∴ QS

SR

PQ

PT= (DeeOeejYetle meceevegheeeflekeâlee ØecesÙe mes)

⇒ QS

SR

PQ

PR= [meceer (iii) mes]

Fefle efmeæced

ØeMve 2. oer ieF& Deeke=âefle ceW D, ∆ABC kesâ keâCe& AC hejefmLele Skeâ eEyeog nw peyeefkeâ BD AC⊥ leLeeDM BC⊥ Deewj DN AB⊥ nw~ efmeæ keâerefpeS efkeâ(i) DM DN MC2 = ⋅

(ii) DN DM AN2 = ⋅

nue efoÙee nw ∆ ABC kesâ keâCe& CA hej Skeâ eEyeog D Fme Øekeâej

nw�efkeâ DM BC⊥ Deewj DN AB⊥ nw~

Deye, NM keâes efceueeÙee~ ceevee BD Deewj MN, eEyeog O hejØeefleÛÚso�keâjleer�nQ~

eq$eYegpe

Q S R

P

3 1

2

4

T

A

N

BMC

D

2

21

1

2

O

A

N

BMC

D


∴ PT PR= …(iii)

(Q meceeve�keâesCeeW�keâer�efJehejerle�YegpeeSB�meceeve�nesleer�nQ~)Deye, ∆QRT ceW,

PS RT|| (jÛevee mes)

∴ QS

SR

PQ

PT= (DeeOeejYetle meceevegheeeflekeâlee ØecesÙe mes)

⇒ QS

SR

PQ

PR= [meceer (iii) mes]

Fefle efmeæced

ØeMve 2. oer ieF& Deeke=âefle ceW D, ∆ABC kesâ keâCe& AC hejefmLele Skeâ eEyeog nw peyeefkeâ BD AC⊥ leLeeDM BC⊥ Deewj DN AB⊥ nw~ efmeæ keâerefpeS efkeâ(i) DM DN MC2 = ⋅

(ii) DN DM AN2 = ⋅

nue efoÙee nw ∆ ABC kesâ keâCe& CA hej Skeâ eEyeog D Fme Øekeâej

nw�efkeâ DM BC⊥ Deewj DN AB⊥ nw~

Deye, NM keâes efceueeÙee~ ceevee BD Deewj MN, eEyeog O hejØeefleÛÚso�keâjleer�nQ~

eq$eYegpe

Q S R

P

3 1

2

4

T

A

N

BMC

D

2

21

1

2

O

A

N

BMC

D

Gheheefòe

(i) ∆DMC leLee ∆NDM ceW,

∠ = ∠DMC NDM (ØelÙeskeâ 90°)

∠ = ∠MCD DMN (ceevee)

ceevee MCD = ∠1

leye, ∠ = ° − ∠MDC 90 1

( )Q∠ + ∠ + ∠ = °MCD MDC DMC 180

∴ ∠ = ° − ° − ∠ODM 90 90 1( )= ∠1⇒ ∠ = ∠DMN 1

∴ ∆ ∆DMC NDM~ (AA mece¤helee mes)

∴ DM

ND

MC

DM=

(mece¤he�ef$eYegpeeW�keâer�mebiele�Yegpee�meceevegheeleer�nesleer�nw)

⇒ DM MC ND2 = ⋅

(ii) ∆DNM leLee ∆NAD ceW,

∠ = ∠NDM AND (ØelÙeskeâ 90°)

∠ = ∠DNM NAD (ceevee)

ceevee ∠ = ∠NAD 2

leye, ∠ = ° − ∠NDA 90 2

( )Q∠ + ∠ + ∠ = °NDA DAN DNA 180

∴ ∠ = ° − ° − ∠ = ∠ODN 90 90 2 2( )

∴ ∠ = ∠DNO 2

∴ ∆DNM NAD~∆ (AA mece¤helee mes)

∴ DN

NA

DM

ND=

⇒ DN

AN

DM

DN= ⇒ DN DM AN2 = ×

ØeMve 3. oer ieF& Deeke=âefle ceW, ABC Skeâ ef$eYegpe nw efpemeceW∠ > °ABC 90 nw leLee AD CB⊥ nw~ efmeækeâerefpeS efkeâAC AB BC BC BD2 2 2 2= + + ⋅ nw~

nue efÛe$e ceW, efoÙee nw efkeâ ABC Skeâ ef$eYegpe nw efpemeceW∠ > °ABC 90 nw�leLee AD BC⊥ yeÌ{eF&�peeleer�nw~


CBD

A


Gheheefòe

(i) ∆DMC leLee ∆NDM ceW,

∠ = ∠DMC NDM (ØelÙeskeâ 90°)

∠ = ∠MCD DMN (ceevee)

ceevee MCD = ∠1

leye, ∠ = ° − ∠MDC 90 1

( )Q∠ + ∠ + ∠ = °MCD MDC DMC 180

∴ ∠ = ° − ° − ∠ODM 90 90 1( )= ∠1⇒ ∠ = ∠DMN 1

∴ ∆ ∆DMC NDM~ (AA mece¤helee mes)

∴ DM

ND

MC

DM=

(mece¤he�ef$eYegpeeW�keâer�mebiele�Yegpee�meceevegheeleer�nesleer�nw)

⇒ DM MC ND2 = ⋅

(ii) ∆DNM leLee ∆NAD ceW,

∠ = ∠NDM AND (ØelÙeskeâ 90°)

∠ = ∠DNM NAD (ceevee)

ceevee ∠ = ∠NAD 2

leye, ∠ = ° − ∠NDA 90 2

( )Q∠ + ∠ + ∠ = °NDA DAN DNA 180

∴ ∠ = ° − ° − ∠ = ∠ODN 90 90 2 2( )

∴ ∠ = ∠DNO 2

∴ ∆DNM NAD~∆ (AA mece¤helee mes)

∴ DN

NA

DM

ND=

⇒ DN

AN

DM

DN= ⇒ DN DM AN2 = ×

ØeMve 3. oer ieF& Deeke=âefle ceW, ABC Skeâ ef$eYegpe nw efpemeceW∠ > °ABC 90 nw leLee AD CB⊥ nw~ efmeækeâerefpeS efkeâAC AB BC BC BD2 2 2 2= + + ⋅ nw~

nue efÛe$e ceW, efoÙee nw efkeâ ABC Skeâ ef$eYegpe nw efpemeceW∠ > °ABC 90 nw�leLee AD BC⊥ yeÌ{eF&�peeleer�nw~


CBD

A

Gheheefòe mecekeâesCe ∆ ABC ceW,

Q ∠ = °D 90

∴ AC AD DC2 2 2= + (heeFLeeieesjme ØecesÙe mes)

= + +AD BD BC2 2( ) ( )QDC DB BC= +

= + + + ⋅( )AD DB BC BD BC2 2 2 2

[ ( ) ]Q a b a b ab+ = + +2 2 2 2

= + + ⋅AB BC BC BD2 2 2

[QmecekeâesCe ∆ADB ceW, ∠ = ° = +D AB AD DB90 2 2 2, (heeFLeeieesjme�ØecesÙe�mes)]

Fefle efmeæced

ØeMve 4. oer ieF& Deeke=âefle ceW, ABC Skeâ ef$eYegpe nw efpemeceW ∠ < °ABC 90 nw leLee AD BC⊥nw~ efmeæ keâerefpeS efkeâ AC AB BC BC BD2 2 2 2= + − ⋅ nw~

nue efÛe$e�ceW,�efoÙee�nw ABC Skeâ�ef$eYegpe�nw�efpemeceW ∠ < °ABC 90 Deewj AD BC⊥ nw~

Gheheefòe mecekeâesCe ∆ ADC ceW,

∴ ∠ = °D 90


= + −AD BC BD2 2( ) (QBC BD DC= + )

= + + − ⋅AD BC BD BC BD2 2 2 2 [ ( ) ]Q a b a b ab− = + −2 2 2 2

= + + − ⋅( )AD BD BC BC BD2 2 2 2

= + − ⋅AB BC BC BD2 2 2

(∴ mecekeâesCe ∆ ADB ceW, ∠ = °D 90 , AB AD BD2 2 2= + ) (heeFLeeieesjme�ØecesÙe�mes)

ØeMve 5. oer ieF& Deeke=âefle ceW, AD ∆ABC keâer SkeâceeefOÙekeâe nw leLee AM BC⊥ nw~ efmeæ keâerefpeSefkeâ

(i) AC AD BC DMBC2 2

2

2= + ⋅ +

eq$eYegpe

B C

A

D

B C

A

M D


Gheheefòe mecekeâesCe ∆ ABC ceW,

Q ∠ = °D 90


= + +AD BD BC2 2( ) ( )QDC DB BC= +

= + + + ⋅( )AD DB BC BD BC2 2 2 2

[ ( ) ]Q a b a b ab+ = + +2 2 2 2

= + + ⋅AB BC BC BD2 2 2

[QmecekeâesCe ∆ADB ceW, ∠ = ° = +D AB AD DB90 2 2 2, (heeFLeeieesjme�ØecesÙe�mes)]

Fefle efmeæced

ØeMve 4. oer ieF& Deeke=âefle ceW, ABC Skeâ ef$eYegpe nw efpemeceW ∠ < °ABC 90 nw leLee AD BC⊥nw~ efmeæ keâerefpeS efkeâ AC AB BC BC BD2 2 2 2= + − ⋅ nw~

nue efÛe$e�ceW,�efoÙee�nw ABC Skeâ�ef$eYegpe�nw�efpemeceW ∠ < °ABC 90 Deewj AD BC⊥ nw~

Gheheefòe mecekeâesCe ∆ ADC ceW,

∴ ∠ = °D 90


= + −AD BC BD2 2( ) (QBC BD DC= + )

= + + − ⋅AD BC BD BC BD2 2 2 2 [ ( ) ]Q a b a b ab− = + −2 2 2 2

= + + − ⋅( )AD BD BC BC BD2 2 2 2

= + − ⋅AB BC BC BD2 2 2

(∴ mecekeâesCe ∆ ADB ceW, ∠ = °D 90 , AB AD BD2 2 2= + ) (heeFLeeieesjme�ØecesÙe�mes)

ØeMve 5. oer ieF& Deeke=âefle ceW, AD ∆ABC keâer SkeâceeefOÙekeâe nw leLee AM BC⊥ nw~ efmeæ keâerefpeSefkeâ

(i) AC AD BC DMBC2 2

2

2= + ⋅ +

eq$eYegpe

B C

A

D

B C

A

M D

(ii) AB AD BC DMBC2 2

2

2= − ⋅ +

(iii) AC AB AD BC2 2 2 221

2+ = +

nue efoÙee nw AD ABC, ∆ keâer�ceeefOÙekeâe�nw�Deewj AM BC⊥

Gheheefòe

(i) mecekeâesCe ∆ AMC ceW,

Q ∠ = °M 90

∴ AC AM MC2 2 2= + (heeFLeeieesjme ØecesÙe mes)

= + +AM MD DC2 2( ) ( )QMC MD DC= +

= + + + ⋅( )AM MD DC MD DC2 2 2 2

[ ( ) ]Q a b a b ab+ = + +2 2 2 2

= + + ⋅AD DC DC MD2 2 2

[QmecekeâesCe ∆ AMD ceW ∠ = °M 90 , AM MD AD2 2 2+ = (heeFLeeieesjme�ØecesÙe�mes)]

= +

+

⋅AD

BC BCDM2

2

22

2

[Q2DC BC= (AD, ∆ABC keâer�ceeefOÙekeâe�nw)]

∴ AC ADBC

BC DM2 22

2= +

+ ⋅ …(i)

(ii) mecekeâesCe ∆ AMB ceW,

Q ∠ = °M 90

∴ AB AM MB2 2 2= + (heeFLeeieesjme ØecesÙe mes)

= + −AM BD MD2 2( ) ( )QBD BM MD= +

= + + − ⋅AM BD MD BD MD2 2 2 2

[ ( ) ]Q a b a b ab− = + −2 2 2 2

= + + − ⋅( )AM MD BD BD MD2 2 2 2

= + − ⋅AD BD BD MD2 2 2

[Q mecekeâesCe ∆ AMD ceW ∠ = °M 90 , AM MD AD2 2 2+ = (heeFLeeieesjme�ØecesÙe�mes)]

= −

⋅ +

ADBC

DMBC2

2

22 2

(Q 2BD BC= , AD, ∆ABC keâer�ceeefOÙekeâe�nw)

∴ AB AD BC DMBC2 2

2

2= − ⋅ +

…(ii)



(ii) AB AD BC DMBC2 2

2

2= − ⋅ +

(iii) AC AB AD BC2 2 2 221

2+ = +

nue efoÙee nw AD ABC, ∆ keâer�ceeefOÙekeâe�nw�Deewj AM BC⊥

Gheheefòe

(i) mecekeâesCe ∆ AMC ceW,

Q ∠ = °M 90

∴ AC AM MC2 2 2= + (heeFLeeieesjme ØecesÙe mes)

= + +AM MD DC2 2( ) ( )QMC MD DC= +

= + + + ⋅( )AM MD DC MD DC2 2 2 2

[ ( ) ]Q a b a b ab+ = + +2 2 2 2

= + + ⋅AD DC DC MD2 2 2

[QmecekeâesCe ∆ AMD ceW ∠ = °M 90 , AM MD AD2 2 2+ = (heeFLeeieesjme�ØecesÙe�mes)]

= +

+

⋅AD

BC BCDM2

2

22

2

[Q2DC BC= (AD, ∆ABC keâer�ceeefOÙekeâe�nw)]

∴ AC ADBC

BC DM2 22

2= +

+ ⋅ …(i)

(ii) mecekeâesCe ∆ AMB ceW,

Q ∠ = °M 90

∴ AB AM MB2 2 2= + (heeFLeeieesjme ØecesÙe mes)

= + −AM BD MD2 2( ) ( )QBD BM MD= +

= + + − ⋅AM BD MD BD MD2 2 2 2

[ ( ) ]Q a b a b ab− = + −2 2 2 2

= + + − ⋅( )AM MD BD BD MD2 2 2 2

= + − ⋅AD BD BD MD2 2 2

[Q mecekeâesCe ∆ AMD ceW ∠ = °M 90 , AM MD AD2 2 2+ = (heeFLeeieesjme�ØecesÙe�mes)]

= −

⋅ +

ADBC

DMBC2

2

22 2

(Q 2BD BC= , AD, ∆ABC keâer�ceeefOÙekeâe�nw)

∴ AB AD BC DMBC2 2

2

2= − ⋅ +

…(ii)


(iii) meceer (i) leLee (ii) keâes�peeÌ[ves�hej,

AC AB AD BC2 2 2 221

2+ = + ( ) Fefle efmeæced

ØeMve 6. efmeæ keâerefpeS efkeâ Skeâ meceeblej ÛelegYeg&pe kesâ efJekeâCeeX kesâ JeieeX keâe Ùeesie Gmekeâer YegpeeDeeW kesâJeieeX kesâ Ùeesie kesâ yejeyej neslee nw~

nue efoÙee nw ABCD Skeâ�meceeblej�ÛelegYeg&pe�nw�efpemekesâ�efJekeâCe& AC Deewj BD nQ~

Deye, AM DC⊥ Deewj BN DC⊥ leLee DC keâes�eEyeog N lekeâ�yeÌ{eÙee~

Gheheefòe mecekeâesCe ∆ AMD leLee ∆BNC ceW,

AD BC= (meceeblej ÛelegYeg&pe keâer efJehejerle YegpeeSB meceeve nesleer nw)

AM BN=

(Ùes�Skeâ�ner�meceeblej�ÛelegYeg&pe�kesâ�Skeâ�ner�DeeOeej�hej�uebyeeW�keâer�uebyeeFÙeeB�nQ~)

∴ ∆ ≅ ∆AMD BNC (RHS meJeeËiemecelee mes)

∴ MD NC= (CPCT mes)…(i)

mecekeâesCe ∆BND ceW,

Q ∠ = °N 90

∴ BD BN DN2 2 2= + (heeFLeeieesjme ØecesÙe mes)

= + +BN DC CN2 2( ) ( )QDN DC CN= +

= + + + ⋅BN DC CN DC CN2 2 2 2

[ ( ) ]Q a b a b ab+ = + +2 2 2 2

= + + + ⋅( )BN CN DC DC CN2 2 2 2

= + + ⋅BC DC DC CN2 2 2 …(ii)

(QmecekeâesCe ∆BNC ceW ∠ = °N 90 )

BN CN BC2 2 2+ = (heeFLeeieesjme ØecesÙe mes)

mecekeâesCe ∆ AMC ceW, ∠ = °M 90

∴ AC AM MC2 2 2= + ( )QMC DC DM= −

= + −AM DC DM2 2( ) [ ( ) ]Q a b a b ab− = + −2 2 2 2

eq$eYegpe

D M N

BA


(iii) meceer (i) leLee (ii) keâes�peeÌ[ves�hej,

AC AB AD BC2 2 2 221

2+ = + ( ) Fefle efmeæced

ØeMve 6. efmeæ keâerefpeS efkeâ Skeâ meceeblej ÛelegYeg&pe kesâ efJekeâCeeX kesâ JeieeX keâe Ùeesie Gmekeâer YegpeeDeeW kesâJeieeX kesâ Ùeesie kesâ yejeyej neslee nw~

nue efoÙee nw ABCD Skeâ�meceeblej�ÛelegYeg&pe�nw�efpemekesâ�efJekeâCe& AC Deewj BD nQ~

Deye, AM DC⊥ Deewj BN DC⊥ leLee DC keâes�eEyeog N lekeâ�yeÌ{eÙee~

Gheheefòe mecekeâesCe ∆ AMD leLee ∆BNC ceW,

AD BC= (meceeblej ÛelegYeg&pe keâer efJehejerle YegpeeSB meceeve nesleer nw)

AM BN=

(Ùes�Skeâ�ner�meceeblej�ÛelegYeg&pe�kesâ�Skeâ�ner�DeeOeej�hej�uebyeeW�keâer�uebyeeFÙeeB�nQ~)

∴ ∆ ≅ ∆AMD BNC (RHS meJeeËiemecelee mes)

∴ MD NC= (CPCT mes)…(i)

mecekeâesCe ∆BND ceW,

Q ∠ = °N 90

∴ BD BN DN2 2 2= + (heeFLeeieesjme ØecesÙe mes)

= + +BN DC CN2 2( ) ( )QDN DC CN= +

= + + + ⋅BN DC CN DC CN2 2 2 2

[ ( ) ]Q a b a b ab+ = + +2 2 2 2

= + + + ⋅( )BN CN DC DC CN2 2 2 2

= + + ⋅BC DC DC CN2 2 2 …(ii)

(QmecekeâesCe ∆BNC ceW ∠ = °N 90 )

BN CN BC2 2 2+ = (heeFLeeieesjme ØecesÙe mes)

mecekeâesCe ∆ AMC ceW, ∠ = °M 90

∴ AC AM MC2 2 2= + ( )QMC DC DM= −

= + −AM DC DM2 2( ) [ ( ) ]Q a b a b ab− = + −2 2 2 2

eq$eYegpe

D M N

BA

= + + − ⋅AM DC DM DC DM2 2 2 2

= + + − ⋅( )AM DM DC DC DM2 2 2 2 = + − ⋅AD DC DC DM2 2 2

[QmecekeâesCe ∆ AMD ceW ∠ = °M 90 , AD AM DM2 2 2= + (heeFLeeieesjme�ØecesÙe�mes)]

= + − ⋅AD AB DC CN2 2 2 …(iii)

[QDC AB= , meceeblej�ÛelegYegg&pe�keâer�efJehejerle�YegpeeSB�leLee BM CN= meceer (i) mes]

meceer (ii) Je (iii) keâes�peesÌ[ves�hej,

AC BD AD AB BC DC2 2 2 2 2 2+ = + + +( ) ( )

= + + +AB BC CD DA2 2 2 2 Fefle efmeæced

ØeMve 7. oer ieF& Deeke=âefle ceW, Skeâ Je=òe keâer oes peerJeeSB AB DeewjCD hejmhej eEyeog P hej ØeefleÛÚso keâjleer nQ~ efmeækeâerefpeS efkeâ(i) ∆ ∆APC DPB~

(ii) AP PB CP DP⋅ = ⋅

nue efÛe$e ceW, efoÙee nw efkeâ oes peerJeeSB AB Deewj CD Skeâ-otmejs keâes eEyeog P hej ØeefleÛÚso keâjleer nQ~

Gheheefòe

(i) ∆ APC leLee ∆DPB ceW,

∠ = ∠APC DPB (Meer<ee&efYecegKe keâesCe)

∠ = ∠CAP BDP (Skeâ ner Je=òeKeb[ kesâ keâesCe)

∴ ∆ ∠APC DPB~ (AA mece¤helee mes)…(i)

(ii) ∆ ∆APC DPB~ [Yeeie (i) ceW efmeæ efkeâÙee pee Ûegkeâe nw]

∴ AP

DP

CP

BP=

(Qmece¤he�ef$eYegpeeW�keâer�mebiele�YegpeeSB�meceevegheeeflekeâ�nesleer�nQ~)

⇒ AP BP CP DP⋅ = ⋅⇒ AP PB CP DP⋅ = ⋅ Fefle efmeæced

ØeMve 8. oer ieF& Deeke=âefle ceW, Skeâ Je=òe keâer oes peerJeeSB AB Deewj CD yeÌ{eves hej hejmhej eEyeog P hejØeefleÛÚso keâjleer nQ~ efmeæ keâerefpeS efkeâ(i) ∆ ∆PAC PDB~ (ii) PA PB PC PD⋅ = ⋅


B

D

C

P

A

B

DC

A

P


= + + − ⋅AM DC DM DC DM2 2 2 2

= + + − ⋅( )AM DM DC DC DM2 2 2 2 = + − ⋅AD DC DC DM2 2 2

[QmecekeâesCe ∆ AMD ceW ∠ = °M 90 , AD AM DM2 2 2= + (heeFLeeieesjme�ØecesÙe�mes)]

= + − ⋅AD AB DC CN2 2 2 …(iii)

[QDC AB= , meceeblej�ÛelegYegg&pe�keâer�efJehejerle�YegpeeSB�leLee BM CN= meceer (i) mes]

meceer (ii) Je (iii) keâes�peesÌ[ves�hej,

AC BD AD AB BC DC2 2 2 2 2 2+ = + + +( ) ( )

= + + +AB BC CD DA2 2 2 2 Fefle efmeæced

ØeMve 7. oer ieF& Deeke=âefle ceW, Skeâ Je=òe keâer oes peerJeeSB AB DeewjCD hejmhej eEyeog P hej ØeefleÛÚso keâjleer nQ~ efmeækeâerefpeS efkeâ(i) ∆ ∆APC DPB~

(ii) AP PB CP DP⋅ = ⋅

nue efÛe$e ceW, efoÙee nw efkeâ oes peerJeeSB AB Deewj CD Skeâ-otmejs keâes eEyeog P hej ØeefleÛÚso keâjleer nQ~

Gheheefòe

(i) ∆ APC leLee ∆DPB ceW,

∠ = ∠APC DPB (Meer<ee&efYecegKe keâesCe)

∠ = ∠CAP BDP (Skeâ ner Je=òeKeb[ kesâ keâesCe)

∴ ∆ ∠APC DPB~ (AA mece¤helee mes)…(i)

(ii) ∆ ∆APC DPB~ [Yeeie (i) ceW efmeæ efkeâÙee pee Ûegkeâe nw]

∴ AP

DP

CP

BP=

(Qmece¤he�ef$eYegpeeW�keâer�mebiele�YegpeeSB�meceevegheeeflekeâ�nesleer�nQ~)

⇒ AP BP CP DP⋅ = ⋅⇒ AP PB CP DP⋅ = ⋅ Fefle efmeæced

ØeMve 8. oer ieF& Deeke=âefle ceW, Skeâ Je=òe keâer oes peerJeeSB AB Deewj CD yeÌ{eves hej hejmhej eEyeog P hejØeefleÛÚso keâjleer nQ~ efmeæ keâerefpeS efkeâ(i) ∆ ∆PAC PDB~ (ii) PA PB PC PD⋅ = ⋅


B

D

C

P

A

B

DC

A

P

nue efÛe$e ceW, efoÙee nw efkeâ Skeâ Je=òe keâer oes peerJeeSB AB Deewj CD yeÌ{eves hej hejmhej eEyeog P hejØeefleÛÚso�keâjleer�nQ~

Gheheefòe

(i) nce peeveles nQ efkeâ Ûe›eâerÙe ÛelegYeg&pe ceW, yee¢e keâesCe, Deble: meccegKe keâesCeeW kesâ yejeyej neslee nw~

∴ ∠ = ∠PAC PDB …(i)

leLee ∠ = ∠PCA PBD …(ii)

meceer (i) leLee (ii) mes,

∆ ∆PAC PDB~ (Q AA mece¤helee mes)

(ii) ∆ ∆PAC PDB~ [Yeeie (i) ceW�efmeæ�efkeâÙee�pee�Ûegkeâe�nw]

∴ PA

PD

PC

PB=

(Q mece¤he�ef$eYegpeeW�keâer�mebiele�YegpeeSB�meceevegheeeflekeâ�nesleer�nQ~)

⇒ PA PB PC PD⋅ = ⋅

ØeMve 9. oer ieF& Deeke=âefle ceW, ∆ABC keâer Yegpee BC hej Skeâ eEyeog DFme Øekeâej efmLele nw efkeâ BD

CD

AB

AC= nw~ efmeæ keâerefpeS efkeâ

∠AD, ∠BAC keâe meceefÉYeepekeâ nw~

nue efoÙee nw ∆ ABC keâer�Yegpee BC hej�eEyeog D Fme�Øekeâej�nw�efkeâ

BD

CD

AB

AC=

Deye, BA keâes Deeies E lekeâ yeÌ{eÙee Deewj AE AC= keâešer~ CE

keâes�efceueeÙee~

Gheheefòe BD

CD

AB

AC= (efoÙee nw)

⇒ BD

CD

AB

AE=

[Q AC AE= (jÛevee mes)]

∴ ∆BCE ceW,AD CE||

(DeeOeejYetle�meceevegheeeflekeâlee�kesâ�efJeueesce�ØecesÙe�mes)

∴ ∠ = ∠BAD AEC (mebiele keâesCe)…(i)

leLee ∠ = ∠CAD ACE (Deble: Skeâeblej keâesCe)…(ii)

Q AC AE= (jÛevee mes)

∴ ∠ = ∠AEC ACE …(iii)

eq$eYegpe

B D C

A

B D C

A

E


nue efÛe$e ceW, efoÙee nw efkeâ Skeâ Je=òe keâer oes peerJeeSB AB Deewj CD yeÌ{eves hej hejmhej eEyeog P hejØeefleÛÚso�keâjleer�nQ~

Gheheefòe

(i) nce peeveles nQ efkeâ Ûe›eâerÙe ÛelegYeg&pe ceW, yee¢e keâesCe, Deble: meccegKe keâesCeeW kesâ yejeyej neslee nw~

∴ ∠ = ∠PAC PDB …(i)

leLee ∠ = ∠PCA PBD …(ii)

meceer (i) leLee (ii) mes,

∆ ∆PAC PDB~ (Q AA mece¤helee mes)

(ii) ∆ ∆PAC PDB~ [Yeeie (i) ceW�efmeæ�efkeâÙee�pee�Ûegkeâe�nw]

∴ PA

PD

PC

PB=

(Q mece¤he�ef$eYegpeeW�keâer�mebiele�YegpeeSB�meceevegheeeflekeâ�nesleer�nQ~)

⇒ PA PB PC PD⋅ = ⋅

ØeMve 9. oer ieF& Deeke=âefle ceW, ∆ABC keâer Yegpee BC hej Skeâ eEyeog DFme Øekeâej efmLele nw efkeâ BD

CD

AB

AC= nw~ efmeæ keâerefpeS efkeâ

∠AD, ∠BAC keâe meceefÉYeepekeâ nw~

nue efoÙee nw ∆ ABC keâer�Yegpee BC hej�eEyeog D Fme�Øekeâej�nw�efkeâ

BD

CD

AB

AC=

Deye, BA keâes Deeies E lekeâ yeÌ{eÙee Deewj AE AC= keâešer~ CE

keâes�efceueeÙee~

Gheheefòe BD

CD

AB

AC= (efoÙee nw)

⇒ BD

CD

AB

AE=

[Q AC AE= (jÛevee mes)]

∴ ∆BCE ceW,AD CE||

(DeeOeejYetle�meceevegheeeflekeâlee�kesâ�efJeueesce�ØecesÙe�mes)

∴ ∠ = ∠BAD AEC (mebiele keâesCe)…(i)

leLee ∠ = ∠CAD ACE (Deble: Skeâeblej keâesCe)…(ii)

Q AC AE= (jÛevee mes)

∴ ∠ = ∠AEC ACE …(iii)

eq$eYegpe

B D C

A

B D C

A

E

(ef$eYegpe�keâer�meceeve�YegpeeDeeW�kesâ�efJehejerle�keâesCe�meceeve�nesles�nQ~)

meceer (i), (ii) leLee (iii) mes,

∠ = ∠BAD CAD

DeLee&led AD, ∠BAC keâe meceefÉYeepekeâ nw~

ØeMve 10. efoS ieS efÛe$e ceW, veeefpecee Skeâ veoer keâer Oeeje ceW ceÚefueÙeeB hekeâÌ[ jner nw~ Gmekeâer ceÚueerhekeâÌ[ves Jeeueer ÚÌ[ keâe efmeje heeveer keâer melen mes 1.8 ceer Thej nw leLee [esjer kesâ efveÛeuesefmejs mes ueiee keâeBše heeveer kesâ melen hej Fme Øekeâej efmLele nw efkeâ Gmekeâer veeefpecee mes otjer3.6 ceer nw Deewj ÚÌ[ kesâ efmejs kesâ "erkeâ veerÛes heeveer kesâ melen hej efmLele eEyeog mes Gmekeâer otjer2.4 ceer nw~ Ùen ceeveles ngS efkeâ Gmekeâer [esjer (Gmekeâer ÚÌ[ kesâ efmejs mes keâeBšs lekeâ) leveerngF& nw, Gmeves efkeâleveer [esjer yeenj efvekeâeueer ngF& nw~ Ùeefo Jen [esjer keâes 5 mesceer/mes keâer ojmes Deboj KeeRÛes, lees 12 meskebâ[ kesâ yeeo veeefpecee keâer keâeBšs mes #eweflepe otjer efkeâleveer nesieer?

nue yeenj�efvekeâueer�[esjer�keâer�uebyeeF&

= +( ) ( )1.8 2.42 2

(heeFLeeieesjme�ØecesÙe�mes)

= +3.24 5.76 = 3 ceer

Dele:�veeefpecee�ves 3 ceer�[esjer�yeenj�efvekeâeueer�ngF&�nw~

12 meskebâ[�ceW�KeeRÛeer�ieF&�[esjer�keâer�uebyeeF&

= × =5 12 60 mesceer = 0.6 ceer

∴Mes<e�yeenj�yeÛeer�[esjer�keâer�uebyeeF& = −3.0 0.6=2.4 ceer

BD AD AB2 2 2= −

(heeFLeeieesjme�ØecesÙe�mes)= −( ) ( )2.4 1.82 2 = − =5.76 3.24 2.52

⇒ BD = 2.52 = 1.59 ceer (ueieYeie)

Dele: 12 meskebâ[�yeeo�veeefpecee�keâer�keâeBšs�keâer�#eweflepe�otjer�mes

= + =1.2 1.59 2.79 ceer (ueieYeie)


2.4 c eer 1.2 c eer

1.8 c eer

1.8 c eer

BC

A

2.4 c eer

1.8 c eer2.4 c eer

BC

A

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