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CBSE Class 9 Mathematics Question 9 of 21

Triangles — Question 1

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Question 1

In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that :

(i) OB = OC

(ii) AO bisects ∠A

Answer

Given :

AB = AC

In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that : NCERT Class 9 Mathematics CBSE Solutions.

OB is the bisectors of ∠B

⇒ ∠ABO = ∠OBC = 12∠B\dfrac{1}{2}∠B.

OC is the bisectors of ∠C

⇒ ∠ACO = ∠OCB = 12∠C\dfrac{1}{2}∠C.

(i) It is given that in triangle ABC, AB = AC

⇒ ∠ACB = ∠ABC

Dividing both sides of equation by 2, we get :

⇒∠ACB2=∠ABC2\Rightarrow \dfrac{∠ACB}{2} = \dfrac{∠ABC}{2}

⇒ ∠OCB = ∠OBC

We know that,

Sides opposite to equal angles of a triangle are also equal.

⇒ OB = OC.

Hence, proved that OB = OC.

(ii) In Δ OAB and Δ OAC,

⇒ AO = AO (Common)

⇒ OB = OC (Proved above)

⇒ AB = AC (Proved above)

∴ Δ OAB ≅ Δ OAC (By S.S.S. congruence rule)

We know that,

Corresponding parts of congruent triangles are equal.

⇒ ∠BAO = ∠CAO (By C.P.C.T.)

∴ AO bisects ∠A

Hence, proved that AO bisects ∠A.