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 A_{1}, A_{2} and A_{3}.

To prove A_{3} = A_{1} + A_{2}

In ∆RST,

by Pythagoras theorem,

RT^{2} = RS^{2} + ST^{2}

= RT^{2} = y^{2} + x^{2}

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

⇒ A_{1} + A_{2} = A_{3}

Hence proved.

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