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

![](https://cdn.mathpix.com/cropped/2024_05_06_b2236406d6db6e8875c1g-27.jpg?height=514&width=508&top_left_y=1491&top_left_x=845)

Fig. 1

Let $X$ be the desired segment, then $X+10$ is the side of the square. The segment $K L=10+X-4-2=X+4$. Consider $\triangle O N L$. By the Pythagorean theorem, the following equality holds:

$$
10^{2}=X^{2}+\left(\frac{X+4}{2}\right)^{2}
$$

This is a quadratic equation in terms of $X$, which has only one positive root 8.

Answer. 8.