Area of two crescents

Let's call the sides of the triangle A, B and C (C is hypothenuse). Looking at the sketch we have:
area semi-circle diam. A $+$ area semi-circle diam. B $-$ (area large semi-circle diam. C $-$ area triangle) = blue shaded area

${pi \times A^2}/{4} + {pi \times B^2}/{4} - {pi \times C^2}/{4} + 100 = \text {blue shaded area}$
${pi}/{4}\times (A^2 +B^2 -C^2) + 100 = \text {blue shaded area}$ , triangle is square: $A^2 +B^2 = C^2$
$100 = \text {blue shaded area}$

The key here is that you're not given the proportions of the right triangle, therefore any right triangle will work.

Since you're given the area of the triangle, it would be very cumbersome to use the two legs (a and 200/a) in the calculations of area, plus very difficult to find the hypotenuse so we can use the area of the large semicircle. Not impossible, just difficult.

So one approach (being that any right triangle would work) would be to use a right triangle with very convenient numbers, such as 6-8-10 (so that we can easily find the radii). Once we get a good handle on the problem, we can then solve using the original given of 100 square units.

First, we'll analyze how to find the area of the crescents. One way to visualize it is to add the small semicircles to the triangle, then subtract the large semicircle.

Let's proceed with a stand-in triangle of 6-8-10, giving us radii of 3, 4, and 5.

Small semicircles will be 1/2 * pi * r^2, so with radii of 3 and 4, this gives us 25/2 pi. The triangle's area (using our stand-in 6-8-10 triangle) is 24, and lastly the large semicircle has a radius of 5, giving us an area of 25/2 pi.

Our calculation for the crescents is thus 25/2 pi + 24 - 25/2 pi!

This means that the area of the crescents is EQUAL to the area of the original triangle (100 in the original problem, not 24).

The answer is 100 square units.

Area of Similar shapes follows Pythagoras ka^2 + kb^2 = kc^2 (in this case k = 0.5pi)

Total area of 2 smaller semi-circles = area of larger semi circle

Blue Area = Triangle