10.169. A rectangle is inscribed in a circle of radius $R$, the area of which is half the area of the circle. Determine the sides of the rectangle.
Solution.

Let the sides of the rectangle be $a$ and $b$. Then the diagonal of the rectangle, being the diameter of the circle, is $\sqrt{a^{2}+b^{2}}$, i.e., $a^{2}+b^{2}=4 R^{2}$. The area of the circle $\pi R^{2}=2 a b$. We obtain the system: $\left\{\begin{array}{l}a^{2}+b^{2}=4 R^{2}, \\ \pi R^{2}=2 a b .\end{array}\right.$ From which $b=\frac{R \sqrt{\pi+4} \pm R \sqrt{4-\pi}}{2}, a=\frac{R \sqrt{\pi+4} \mp R \sqrt{4-\pi}}{2}$.

Answer: $\frac{R \sqrt{\pi+4} \pm R \sqrt{4-\pi}}{2}$.