Question

In a right angled triangle, prove that the line segment joining the mid-point of the hypotenuse to the opposite vertex is half the hypotenuse.

Answer 1

Let $P$ be the mid point of the hypotenuse of the right $△ABC$ right angled at $B$

Draw a line parallel to $BC$ from $P$ meeting $B$ at $O$

Join $PB$

In $△PAD$ and $△PBD$

$\angle PDA=\angle PDB{90}^{\circ }$ each due to conv of mid point theorem

$PD=PD$ ....... (common)

$AD=DB$ ...... (As $D$ is mid point of $AB$)

So $△$ $PAD$ and $PBD$ are congruent by SAS rule

$PA=PB$ ...... (C.P.C.T)

As $PA=PC$ .... (Given as $P$ is mid-point)

$\therefore PA=PC=PB$