In the larger of two concentric circles, a chord equal to 32 is drawn and touches the smaller circle. Find the radius of each circle if the width of the resulting ring is 8.

#
Apply the theorem of the products of segments of intersecting chords.

## Solution

Let $O$ be the center of the circles, $AB$ be the given chord of the larger circle, $M$ be the point of tangency with the smaller circle, and $r$ be the radius of the smaller circle. Then $M$ is the midpoint of $AB$ (since $OM \perp AB$).

First method.

By the Pythagorean theorem in the right triangle AMO:

$$
OA^2 = OM^2 + MA^2 \text{, or } (r+8)^2 = r^2 + 16^2 \text{.}
$$

From this equation, we find that $r=12$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d1936174aeda27dcc10dg-10.jpg?height=412&width=400&top_left_y=114&top_left_x=835)

Answer

12 and 20.