10.067. Inside a circle with a radius of 15 cm, a point $M$ is taken at a distance of 13 cm from the center. Through point $M$, a chord of length 18 cm is drawn. Find the lengths of the segments into which point $M$ divides the chord.
Solution.

Draw $O C \perp A B$ (Fig. 10.66). Then $C B=\frac{1}{2} A B=9$ cm. From $\triangle O B C$

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-588.jpg?height=388&width=648&top_left_y=94&top_left_x=108)

Fig. 10.67

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-588.jpg?height=372&width=406&top_left_y=110&top_left_x=797)

Fig. 10.68

we find $O C=\sqrt{O B^{2}-B C^{2}}=12$ (cm), and from $\triangle O M C$ we get $M C=$ $=\sqrt{O M^{2}-O C^{2}}=5$ (cm). Therefore, $A M=9+5=14$ (cm), $M B=$ $=9-5=4$ (cm).

Answer: $\quad 14$ and 4 cm.