Question

In the figure, semicircles are constructed on the hypotenuse and legs of a right angle triangle. If S1, S2 are the area of shaded regions and S is the area of right angled triangle, which of the following is true and why ?

1. S1 + S2 = S
2. S1 + S2 = 2S
3. S1 + S2 = 3S

Answer 1

Semi-Circle Lunes
Shaded portions are called lunes

Area of two lunes=Area of triangle
S1+S2=S

So Ans is 1
ie S1+S2=S

Answer 2

Here 1/2 pi (b/2)^2 = 1/2 pi(a/2)^2 + 1/2 pi(c/2)^2 . So the area of larger semicircle is the sum of other two semicircles.That is
S + A = S1+S2 + A where A is the area of common portions.
S1+S2 = S