10.234. A circle with a radius of 13 cm touches two adjacent sides of a square with a side length of $18 \mathrm{~cm}$. Into which two segments does the circle divide each of the other two sides of the square?
## Solution.

Draw the radius $O K$ (Fig. 10.42); then $K L=\sqrt{K O^{2}-L O^{2}}$. $L O=B E=18-13=5(\mathrm{~cm}) \Rightarrow K L=\sqrt{13^{2}-5^{2}}=12$ (cm), $B K=13-12=1$ (cm). We have found that the side of the square is divided into segments of 1 and 17 cm.

Answer: 1 and 17 cm.