Question

Prove that the medians bisecting the equal sides of an isosceles triangle are equal.

Answer 1

Given: In $\Delta ABC$, $D$ and $E$ are mid-points of $AB$ and $AC$ respectively.
To prove: $BE=CD$
Proof: Since triangle $ABC$ is an isosceles triangle, then
$AB=AC$ …(i)
and $\angle ABC=\angle ACB$ …(ii)
$D$ and $E$ are mid-points of $AB$ and $AC$ respectively,
then $DB=DA$ and $EC=AE$ …(iii)
Now, in $\Delta BCD$ and $\Delta BCE$
$BC=BC$ (common)
$\angle DBC=\angle ECB$ by (ii)
$BD=CE$ by (iii)
$\Delta BCD=\Delta BCE$ (by SAS congruency rule)
$BE=CD$
Hence proved.