NCERT Solutions For Class 9 Math Chapter – 7 Exercise – 7.3

NCERT Solutions For Class 9 Math Chapter – 7 Exercise  7.3

 

Q1. ABC and DBC are two isosceles triangles on the same base  BC and vertices A and D are on the same side of BC If AD is extended to intersect BC at P, Show that

  • ABD ∆ACD
  • ABP ACP
  • AP bisects ∟A as well as ∟D.
  • AP is the perpendicular bisector of BC.

Solution:-

Ncert solutions class 9 chapter 7-17

  • In triangle ABD and ACD.

AB = AC

BD = CD

AD = DA

∆ABD  ∆ACD [SSS Rule]

∟BAD = ∟CAD and ∟ABD = ∟ACD  [ CPCT]

 

  • In triangle ABP and ACP

AB = AC

AP = PA

∟BAP

= ∟CAP

∆ABP  ∆ACP

BP = PC ∟BPA = ∟CPA

 

  • ∟BAP = ∟CAP

AP bisects ∟A.

∟BAD + ∟ABD = ∟CAD = ∟ACD

∟BDP = ∟CDP

So , DP bisects ∟D.

AP bisects ∟A as well as ∟D.

 

  • ∟BPA + ∟CPA = 1800

2∟BPA = 1800

∟BPA = 900

AP bisects BC.

 

Q2. AD is an altitude of an isosceles  triangle ABC  in which AB = AC Show that

  • AD bisects BC.
  • AD bisects

Solution:-

Ncert solutions class 9 chapter 7-18

∆ABD  ∆ACD [RHS Rule]

  • BD = CD

AD bisects BC.

 

  • ∟BAD = ∟CAD

AD bisects ∟BAC      ∟A.

 

Q3. Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ∆PQR Show that:

  • ABM PQN
  • ABC

Solution:-

 

 

Ncert solutions class 9 chapter 7-19

 

 

 

 

  • M and N are mid – points of BC and QR respectively, As AM and PN are medians.

BM = 1/2BC and QN = 1/2QR

Also , BC = QR

1/2BC = 1/2QR = BM = QN

Consider triangles ABM and PQN,

AB = PQ

AM = PN

BM = QN

∆ABM  ∆PQN [SSS Rule]

 

  • From Result (i)

∟ABM = ∟PQN

∟ABC = ∟PQR  (iii)

AB = PQ

BC = QR

∟ABC = ∟PQR

∆ABC  ∆PQR. [SAS rule]

 

Q4. BE and CF are two equals altitude of a triangle ABC Using RHS Congruence rule , Prove that  the triangle ABC is isosceles.

Solution:-

Ncert solutions class 9 chapter 7-20

∆BFC  ∆CEB [RHS Rule]

∟FBC = ∟ECB

∟ABC = ∟ACB

AC  = AB

[ Sides opposite to equal angels of a triangle are equal.]

∆ABC is an isosceles triangle.

 

Q5. ABC is an isosceles triangle with AB = AC Draw AP perpendicular BC to show that ∟B = ∟C.

Solution:-

Ncert solutions class 9 chapter 7-21

AB = BC

AP = PA

∟APB =  ∟APC

∆APB  ∆ACP

∟B = ∟C.