10.239. Find the radius of the circle, in the segment of which, corresponding to a chord of length $6 \mathrm{~cm}$, a square with a side of $2 \mathrm{~cm}$ is inscribed.
Solution.

Let $r$ be the desired radius. In $\triangle A O K$ (Fig. 10.46), we have $O K=\sqrt{r^{2}-9}$, and in $\triangle O B N$, we have $O N^{2}+B N^{2}=O B^{2}$ or $(O K+2)^{2}+1=r^{2}$. Therefore, $r^{2}-9+4 \sqrt{r^{2}-9}+4+1=r^{2}$, from which $r^{2}-9=1$, i.e., $r=\sqrt{10}$ cm.

Answer: $\sqrt{10}$ cm.