Question

Prove that the median from the vertex of an isosceles triangle is the bisector of vertical angle.

Answer 1

Consider $PQR$ is an isosceles triangle such that

$PQ=PR$ and $PL$ is the bisector of $\angle P$.

To prove: $\angle PLQ=\angle PLR={90}^o$

and $QL=LR$

In $△PLQ$ and $△PLR$

$PQ=PR$ ..... (given)

$PL=PL$ ..... (common)

$\angle QPL=\angle RPL$ ..... $(PL$ is the bisector of $\angle P)$

$△PLQ\cong △PLR$ ...... (SAS congruence criterion)

$QL=LR$ ........ (By cpct)

and $\angle PLQ+\angle PLR={180}^o$ ..... (Linear pair)

$2\angle PLQ={180}^o$

$\angle PLQ=\frac{{180}^o}{2}={90}^o$

Therefore, $\angle PLQ=\angle PLR={90}^o$

Thus,$\angle PLQ=\angle PLR={90}^o$ and $QL=LR$