Question

In the right triangle , , , and angle is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?

Answer 1

We can reflect $ABC$ over line $AC$. Thus forms the triangle ${AB}^{1}C$ and a circle and out of semicircle. Let us call the center of the circle $O$.

We can see that circle $O$ is the circle of ${AB}^{1}C$. We can see the formula for finding the radius of the circle to solve the problem. Area of a triangle $=$ Semi $-$ perimeter $\times$ radius

Area of ${AB}^{1}C$ $=12\times 5=60$ semi perimeter $=5+13=18$

Simplifying $\frac{60}{18}=\frac{10}{3}$