10.044. In a circle, two chords $A B=a$ and $A C=b$ are drawn. The length of arc $A C$ is twice the length of arc $A B$. Find the radius of the circle.
Solution.

Let $\angle B O A=\alpha$, then $\angle A C B=\frac{\alpha}{2}, a^{2}=2 R^{2}-2 R^{2} \cdot \cos \alpha$ ( $R$-the radius to be found) (Fig. 10.44). By the Law of Sines $\frac{a}{\sin \frac{\alpha}{2}}=\frac{b}{\sin \alpha}=2 R$, $\sin \alpha=\frac{b}{2 R}, a^{2}=2 R^{2}-2 R^{2} \cdot \frac{1}{2 R} \sqrt{4 R^{2}-b^{2}}, \frac{a^{2}}{2 R^{2}}=1-\sqrt{1-\frac{b^{2}}{4 R^{2}}} ;$ $R^{2}=\frac{a^{4}}{4 a^{2}-b^{2}} ; R=\frac{a^{2}}{\sqrt{4 a^{2}-b^{2}}}$.

Answer: $\frac{a^{2}}{\sqrt{4 a^{2}-b^{2}}}$.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-577.jpg?height=334&width=484&top_left_y=93&top_left_x=94)

Fig. 10.45

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-577.jpg?height=336&width=442&top_left_y=94&top_left_x=809)

Fig. 10.46