Question

Prove that the area of the semi-circle drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the semi-circles drawn on the other two sides of the triangle.

Answer 1

Let RST be a right triangle at S and RS = y, ST = x.

Three semi-circles are draw on the sides RS, ST and RT, respectively A₁, A₂ and A₃.

To prove A₃ = A₁ + A₂

In ∆RST,

by Pythagoras theorem,

RT² = RS² + ST²

= RT² = y² + x²

∴ Area of semi-circle drawn on RT,

⇒ A₁ + A₂ = A₃

Hence proved.