Question

prove that in an isosceles triangle the vertex,the median,the bisector are coincident...............

Answer 1

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.