Question

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

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