Question

use analytical geometry to prove that the mid point of the hypotenuse of a right angled triangle is equidistant from its vertices

Answer 1

Let P be the mid point of the hypo. of the right triangle ABC, right angled at B.

Draw a  line parallel to BC   from P meeting AB at D.

Join PB.

 in triangles,PAD and PBD,

angle PDA= angle PDB (90 each due to conv of  mid point theorem)

PD=PD(common)

AD=DB( as D is mid point of AB)

so triangles PAD  and PBD are congruent by SAS rule.

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

but

 PA=PC(given as P is mid point )

So,

 PA=PC=PB