The radii of two circles are 2 and 4. Their common internal tangents are perpendicular to each other. Find the length of each of them.
For each of the given circles, the quadrilateral formed by the indicated tangents and radii drawn to the points of tangency is a square.

## Solution

For each of the given circles, the quadrilateral formed by the tangents and radii drawn to the points of tangency is a square. Therefore, the length of the desired tangent is equal to the sum of the radii, i.e., the sides of the squares.

![](https://cdn.mathpix.com/cropped/2024_05_06_9e7ee8b57660b9721937g-01.jpg?height=423&width=748&top_left_y=938&top_left_x=655)

## Answer

6. 

3 Problem $\underline{52637}$ Topics: [ $\frac{\text { Inscribed quadrilaterals (other) }}{\text { Diameter, basic properties }}$ [Inscribed quadrilateral with perpendicular diagonals]

In the inscribed quadrilateral $A B C D$, diagonal $A C$ is perpendicular to diagonal $B D$ and bisects it. Find the angles of the quadrilateral if $\angle B A D=\alpha$.

## Hint

The perpendicular bisector of a chord passes through the center of the circle.

## Solution

Since diagonal $A C$ is a diameter, then

$$
\angle A B C=\angle A D C=90^{\circ}
$$

Since $\angle B A D$ and $\angle B C D$ are opposite angles of the inscribed quadrilateral, then

$$
\angle B C D=90^{\circ}-\angle B A D=90^{\circ}-\alpha
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_9e7ee8b57660b9721937g-01.jpg?height=501&width=458&top_left_y=2402&top_left_x=800)
[Inscribed angle subtending a diameter]