4. On each side of a right-angled triangle, a semicircle is drawn with that side as a diameter. The areas of the three semicircles are $x^{2}, 3 x$ and 180 where $x^{2}$ and $3 x$ are both less than 180 . What is the area of the smallest semicircle?
SolUTION
036

The areas of a square of side $2 r$ and its inscribed circle are in the ratio $4 r^{2}: \pi r^{2}=4: \pi$. Therefore, the area of the square on each side of the triangle and the area of the corresponding semicircle are in the ratio $4: \frac{\pi}{2}=8: \pi$. This is independent of side length and we therefore may use the Pythagoras' equation on the areas of the semicircles. This gives $x^{2}+3 x=180$. The solutions to this equation are $x=-15$ (which is impossible since this would give an area of $3 \times-15=-45$ ) and $x=12$, meaning the required area is 36 .