[Inscribed and Circumscribed Circles]

One of the acute angles of a right triangle is $25^{\circ}$. Under what angle is each of its legs seen from the center of the circumscribed circle

#
The central angle is twice the corresponding inscribed angle.

## Solution

The center of the circle circumscribed around a right triangle coincides with the midpoint of the hypotenuse, and the inscribed angle is measured by half the angular measure of the corresponding arc of the circle.

## Answer

$50^{\circ}, 130^{\circ}$.

The radii of two concentric circles are in the ratio 7:4, and the width of the ring is 12. Find the radius of the smaller circle.

## Hint

The width of the ring between two concentric circles is equal to the difference of their radii.

## Solution

Let $R=7 x, r=4 x$ be the radii of the circles. Then

$$
R-r=3 x=12
$$

Therefore, $x=4$ and $r=4 x=16$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d7c3832c1c4b0d15493ag-31.jpg?height=306&width=323&top_left_y=659&top_left_x=868)

Answer

16.