$\left.\begin{array}{c}\text { Diameter, chords, and secants } \\ {[\text { Pythagorean Theorem (direct and inverse). }}\end{array}\right]$

The length of side $B C$ of triangle $A B C$ is 12 cm. A circle is circumscribed around the triangle with a radius of 10 cm. Find the lengths of sides $A B$ and $A C$ of the triangle, given that the radius $O A$ of the circle bisects side $B C$ into two equal segments.
The radius of a circle passing through the midpoint of a chord, which is not a diameter, is perpendicular to this chord.

## Solution

Let $M$ be the midpoint of the chord $B C$. Since $B C=12>10=2 O C$, $B C$ is not the diameter of the circle. It is known that the radius of a circle passing through the midpoint of a chord, which is not a diameter, is perpendicular to this chord, so $OM$ is the height of the isosceles triangle $VOC$.

The height $A M$ of triangle $A B C$ is also its median, so triangle $A B C$ is isosceles.

From the right triangle $OMV$, we find that

$$
O M=\sqrt{O B^{2}-B M^{2}}=\sqrt{100-36}=\sqrt{64}=8
$$

Therefore, $A M=O A-O M=10-8=2$. Hence,

$$
A B=B C=\sqrt{B M^{2}+A M^{2}}=\sqrt{36+4}=\sqrt{40}=2 \sqrt{10}
$$

## Answer

$2 \sqrt{10}$

The length of side $A B$ of triangle $A B C$ is 24 cm. A circle with a radius of 13 cm is circumscribed around the triangle. Find the lengths of sides $A C$ and $B C$ of the triangle, given that the radius $O C$ of the circle bisects side $A B$ into two equal segments.

## Hint

The radius of a circle passing through the midpoint of a chord, which is not a diameter, is perpendicular to this chord.

## Answer

$4 \sqrt{13}$.