[Central Angle. Arc Length and Circumference]

Two concentric circles are given. A tangent to the smaller circle divides the arc length of the larger circle in the ratio $1: 5$. Find the ratio of the areas of the circles bounded by these circles.

#
The length of an arc of a circle is proportional to the magnitude of the corresponding central angle.

## Solution

Let $A$ and $B$ be the points of intersection of the given tangent with the larger circle, $C$ be the point of tangency with the smaller circle, $O$ be the center of the circles, and $r$ and $R$ be their radii $(r<R)$.

Since the length of the arc is proportional to the corresponding central angle, then

$$
\angle A O B=\frac{1}{6} \cdot 2 \pi=\frac{\pi}{3}
$$

From the right triangle $O C B$, we find that

$$
\frac{r}{R}=\frac{C O}{O B}=\cos \angle C O B=\cos \frac{1}{2} \angle A O B=\cos 30^{\circ}=\frac{\sqrt{3}}{2}
$$

Therefore, the ratio of the areas of the given circles is

$$
\frac{\pi r^{2}}{\pi R^{2}}=\left(\frac{r}{R}\right)^{2}=\frac{3}{4}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-39.jpg?height=406&width=440&top_left_y=1984&top_left_x=818)

## Answer