Question

prove that the area of a square described on hypotenuse of a right angle triangle is equal to sum of the area of square described on the other two sides

Answer 1

A right angled ∆ABC, right angled at B



To Prove- AC²=AB²+BC²



Construction: draw perpendicular BD onto the side AC .


Proof:

We know that if a perpendicular is drawn from the vertex of a right angle of a right angled triangle to the hypotenuse, than triangles on both sides of the perpendicular are similar to the whole triangle and to each other.


We have

△ADB∼△ABC. (by AA similarity)


Therefore, AD/ AB=AB/AC


(In similar Triangles corresponding sides are proportional)



AB²=AD×AC……..(1)


Also, △BDC∼△ABC


Therefore, CD/BC=BC/AC


(in similar Triangles corresponding sides are proportional)


Or, BC²=CD×AC……..(2)



Adding the equations (1) and (2) we get,



AB²+BC²=AD×AC+CD×AC


AB²+BC²=AC(AD+CD)


( From the figure AD + CD = AC)


AB²+BC²=AC . AC



Therefore, AC²=AB²+BC²

This theroem is known as Pythagoras theroem...