### 10.386 The distances from the center of the circle inscribed in a right-angled triangle to the vertices of its acute angles are $\sqrt{5}$ and $\sqrt{10}$. Find the legs.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-236.jpg?height=437&width=992&top_left_y=1143&top_left_x=114)

$B M=\sqrt{O B^{2}-1}=3, C N=\sqrt{O C^{2}-1}=2 \Rightarrow A B=r+B M=4, A C=r+C N=3$ (AMON - square).

Answer: 3 and 4.