10.203. From a point on the circumference, two chords of lengths 9 and $17 \mathrm{~cm}$ are drawn. Find the radius of the circle if the distance between the midpoints of these chords is 5 cm.
Solution.

From point $B$ of the circle (Fig. 10.13), chords $A B=9$ cm and $B K=17$ cm, $M N=5$ cm are drawn. Then $M N$ is the midline of $\triangle A B K$, $A K=2 M N=10$ cm. The desired radius $R$ will be found as the radius of the circumcircle of $\triangle A B K: R=\frac{A B \cdot A K \cdot B K}{4 S_{\triangle A B K}}$. The semiperimeter of the triangle $p=\frac{A B+A K+B K}{2}=18$ cm. $S_{\triangle A B K}=\sqrt{(18-9)(18-10)(18-7) \cdot 18}=36$ cm$^2$. $R=10 \frac{5}{8}$ cm.

Answer: $10 \frac{5}{8}$ cm.