Question

Prove that the line segment joining the mid-point of the hypotenuse of a right triangle to
its opposite vertex is half of the hypotenuse

Answer 1

Answer:

solved

Step-by-step explanation:

let the hypotenuse length = z and the other sides lengths = x , y

and the line segment joining the mid-point of the hypotenuse = h

h² =  x² - z²/4

h² =   y²  - z²/4

adding

2h² =   x² + y² -  z²/2 = z²/2

h² = z²/4

h = z/2

Answer 2

Answer:

Step-by-step explanation:

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

∠PDA=∠PDB=90

∘

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)

∴PA=PC=PB