5. The radii of the circles that form the circular ring shown in the diagram on the right are $r$ and $2r$

Determine the ratio of the areas of the shaded and unshaded parts of the circular ring.
Solution. Let $A, B, C$ be the common points of

![](https://cdn.mathpix.com/cropped/2024_06_05_a6c386d4b461b4a2c2ceg-4.jpg?height=282&width=276&top_left_y=777&top_left_x=1150)
the larger circle and the tangents $t^{\prime}$ and $t^{\prime \prime}$ (diagram on the right) and let $T$ be the point where $t$ touches the smaller circle. Since $O B=2 O T$, we conclude that $\Varangle O B T=30^{\circ}$, and from this it follows that $\measuredangle A B C=60^{\circ}$. Similarly, it is proven that

![](https://cdn.mathpix.com/cropped/2024_06_05_a6c386d4b461b4a2c2ceg-4.jpg?height=305&width=329&top_left_y=1088&top_left_x=1150)
$\measuredangle B A C=\measuredangle B C A=60^{\circ}$. Therefore, triangle $A B C$ is equilateral, and thus the arcs in the larger circle are equal. Clearly, the shaded part is equal to one-third of the entire ring, so the areas of the shaded and unshaded parts are in the ratio $1: 2$.