Question

Prove Prove that perpendicular from vertex in an isosceles triangle bisects the opposite side

Answer 1

Here you go,

Consider PQR is an isosceles triangle such that PQ = PR and Pl is the bisector of ∠ P.

To prove : ∠PLQ = ∠PLR = 90°

and QL = LX

In ΔPLQ and ΔPLR

PQ = PR (given)

PL = PL (common)

∠QPL = ∠RPL ( PL is the bisector of ∠P)

ΔPLQ = ΔPLR ( SAS congruence criterion)

QL = LR (by cpct)

and ∠PLQ + ∠PLR = 180° ( linear pair)

2∠PLQ = 180°

∠PLQ = 180° / 2 = 90° ∴ ∠PLQ = ∠PLR = 90°

Thus, ∠PLQ = ∠PLR = 90° and QL = LR.

Hence, the bisector of the verticle angle an isosceles triangle bisects the base at right angle.

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