## Pythagorean Theorem (direct and inverse). Inscribed and circumscribed circles

Area of a triangle (through height and base)
[ Area of a triangle (through semiperimeter and radius of the inscribed or exscribed circle).

In a right triangle, the point of tangency of the inscribed circle divides the hypotenuse into segments of 5 and

12. Find the legs of the triangle.
Express the legs in terms of the radius of the inscribed circle and use the Pythagorean theorem.

## Solution

Let the radius of the inscribed circle be $r$. Then the legs of the triangle are $5+r$ and $12+r$. By the Pythagorean theorem, $(5+r)^{2}+(12+r)^{2}=17^{2}$, from which $r=3$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-27.jpg?height=257&width=477&top_left_y=1535&top_left_x=802)

## Answer

8 and 15.