4. The radii of the concentric circles that form the circular ring shown in the diagram on the right have lengths $r$ and $2r$. Determine the ratio of the areas of the shaded and unshaded parts of the circular ring.
Solution. Let $A, B$ and $C$ be the common points of the larger circle and the tangents $t_{1}$ and $t_{2}$ to the smaller circle, and let $T$ be the point where $t_{1}$ touches the smaller circle. Since $\overline{O B}=2 \overline{O T}$, it follows that $\measuredangle O B T=30^{\circ}$, and therefore $\measuredangle B A C=60^{\circ}$. Specifically,

$$
\measuredangle B A C=\measuredangle O A B+\measuredangle O A C=2 \measuredangle O A B=2 \measuredangle O A T=2 \measuredangle O B T
$$

Note that in the first equality, we used the fact that the line OA is the axis of symmetry of the entire geometric configuration. From this symmetry, we have that $\overline{A B}=\overline{A C}$.

Thus, triangle $A B C$ is equilateral, and the segments of the larger circle are equal. Given that $T$ is the midpoint of segment $A B$, we conclude that the smaller circle is inscribed in triangle $A B C$. Consequently, the shaded part of the circular ring is one-third of the entire ring, and the areas of the shaded and unshaded parts of

![](https://cdn.mathpix.com/cropped/2024_06_05_aa5de80eeceb04c02556g-9.jpg?height=380&width=380&top_left_y=423&top_left_x=1069)
this ring are in the ratio $1: 2$.