10.031. A circle with a radius of 2 is inscribed in a rhombus, which is divided by its diagonal into two equilateral triangles. Find the side of the rhombus.
## Solution.

Let $a$ be the side of the rhombus, and $d_{1}$ and $d_{2}$ be its diagonals (Fig. 10.31). Since the height of the rhombus is equal to the diameter of the circle, its area is $a$. On the other hand, $S=\frac{1}{2} d_{1} d_{2}$, from which, considering that $d_{1}=a$, we find $d_{2}=8$. But $a^{2}=\left(\frac{d_{1}}{2}\right)^{2}+\left(\frac{d_{2}}{2}\right)^{2}$ or $a^{2}=\frac{a^{2}}{4}+16$, i.e., $a=\frac{8 \sqrt{3}}{3}$.

Answer: $8 \sqrt{3} / 3$.