2. The radius of the circle inscribed in the triangle is 2. One of the sides of the triangle is divided by the point of tangency into segments equal to 1 and 8. Find the other sides of the triangle.

Answer. 10 and 17.
Let K, L, M be the points of tangency of the inscribed circle with the sides AB, BC, AC of triangle $ABC$ respectively; thus, $AK=8, BK=1$. Denote $CL=x$. By the equality of the segments of tangents drawn from a point to a circle, we get $\mathrm{BL}=1, \mathrm{AM}=8$, $\mathrm{CM}=\mathrm{x}$. We express the area of the triangle in two ways (using Heron's formula and as the semiperimeter multiplied by the radius of the inscribed circle) and obtain the equation:

$$
2(x+9)=\sqrt{(x+9) \cdot x \cdot 8 \cdot 1} \Leftrightarrow \sqrt{x+9}=\sqrt{2 x} \Leftrightarrow x=9
$$

Thus, the other two sides of the triangle are 10 and 17.