Task 2. (10 points) Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$, and the points of tangency cut off segments of length 4 cm from vertex $A$. Two tangents are drawn from point $C$ to this circle. Find the side of the square if the angle between the tangents is $60^{\circ}$.
# Solution.

![](https://cdn.mathpix.com/cropped/2024_05_06_b2236406d6db6e8875c1g-02.jpg?height=439&width=559&top_left_y=1294&top_left_x=820)

Fig. 1

The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. If radii of the circle are drawn to the points of tangency, a square with a side of 4 cm is formed. Then $O A=4 \sqrt{2}$, $O C=\frac{O K}{\sin 30^{\circ}}=8$, so the diagonal of the square $A B C D A C=8-4 \sqrt{2}$, and the side of the square $A B=\frac{A C}{\sqrt{2}}=4(\sqrt{2}-1)$.

Answer. $4(\sqrt{2}-1)$.