10.322. In a circle with center $O$, a chord $A B$ is drawn, intersecting the diameter at point $M$ and forming an angle of $60^{\circ}$ with the diameter. Find $O M$, if $A M=10 \, \text{cm}$, and $B M=4 \, \text{cm}$.
## Solution.

Draw $O P \perp A B$ (Fig. 10.111). Then $A P = B P = 7$ cm and $M P = 3$ cm. Since $\angle P M O = 60^{\circ}$, then $\angle M O P = 30^{\circ}$ and $O M = 2 M P = 6$ cm.

Answer: 6 cm.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0793.jpg?height=452&width=569&top_left_y=79&top_left_x=95)

Fig. 10.112

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0793.jpg?height=426&width=632&top_left_y=74&top_left_x=629)

Fig. 10.113