Question

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

Hope This Helps :)

Answer 2

Step-by-step explanation:

Let ΔABC be a right angle triangle at angle B. Let P be the midpoint of hypotenuse AC. Draw a circle with center P and AC as a diameter Since, ∠ABC = 90o , therefore the circle passes through B. Read more on Sarthaks.com - https://www.sarthaks.com/114587/prove-that-segment-joining-point-hypotenuse-right-triangle-opposite-vertex-hypotenuse