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.

Question

Philoid · Accepted Answer

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

We know that,

Area of a semi-circle with radius,

∴ Area of semi-circle drawn on RT,

Now, area of semi-circle drawn on RS,

And area of semi-circle drawn on ST,

On adding Equation (ii) and (iii), we get

⇒ A₁ + A₂ = A₃

Hence proved.