Task 2. (10 points) A circle touches two adjacent sides $AB$ and $AD$ of square $ABCD$ and cuts off segments of length 4 cm from vertices $B$ and $D$ at the points of tangency. On the other two sides, the circle intersects and cuts off segments of 2 cm and 1 cm from the vertices, respectively. Find the radius of the circle.
# Solution.

![](https://cdn.mathpix.com/cropped/2024_05_06_b2236406d6db6e8875c1g-32.jpg?height=497&width=502&top_left_y=1462&top_left_x=777)

Fig. 1

Let $R$ be the radius of the circle, then $R+4$ is the side of the square. The segment $K L=4+R-2-1=R+1$. Consider the triangle $O N L$. By the Pythagorean theorem, the following equality holds:

$$
R^{2}=4^{2}+\left(\frac{1+R}{2}\right)^{2}
$$

This is a quadratic equation in terms of $R$, which has only one positive root 5.