Question

In the figure below, PQ = PR. Prove that the point P is one the bisector of ∠ABC.

Answer 1

Given, PQ = PR and ∠BRP =∠BQP = 90°

Construction: Join BP

In ΔBPQ and ΔBPR:

∠PQB = ∠PRB = 90° (Given)

PB = PB (Common side)

PQ = PR (Given)

As the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and one other side of another right-angled triangle, ΔBPQ ≅ ΔBPR.

Congruent parts of congruent triangles are congruent.

∴ ∠PBQ = ∠PBR

Thus, BP is the bisector of ∠QBR.

Hence, P lies on the bisector of ∠ABC.