[Inscribed, circumscribed, and exscribed circles; their radii] Pythagorean Theorem (direct and inverse).

Find the radius of the circumscribed circle of a right triangle if the radius of its inscribed circle is 3, and one of the legs is 10.
The hypotenuse is the diameter of the circumscribed circle.

## Solution

Let $a$ and $c$ be the other leg and the hypotenuse of the triangle. Then $c-a=10-2 \cdot 3=4, c^{2}-a^{2}=100$. Therefore, $c+a=25, c=1 / 2(25+5)=29 / 2$.

## Answer

 $29 / 4$.