1.46. 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.
1.46. Let $r$ be the desired radius. In $\triangle A O K$ (Fig. 1.37) we have $O K=\sqrt{r^{2}-9}$, and in $\triangle O B N$ we have

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-434.jpg?height=463&width=465&top_left_y=1397&top_left_x=675)

Fig. 1.37
$O N^{2}+B N^{2}=O B^{2}$, 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., $r=\sqrt{10} \text{ cm}$.

Answer: $\sqrt{10}$ cm.