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

The radii of the inscribed and circumscribed circles of a right triangle are 2 and 5, respectively.

Find the legs of the triangle.
The hypotenuse is the diameter of the circumscribed circle.

## Solution

Let $a$ and $b$ be the lengths of the legs. Since the hypotenuse is 10, then $a+b=10+2 \cdot 2=14, a^{2}+b^{2}=100$, from which $\{a$, $b\}=\{6,8\}$

![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-11.jpg?height=366&width=784&top_left_y=1809&top_left_x=637)

## Answer

6 and 8.

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