Find the radius

The line segment AB is tangent to the circle at point A.
The line segment OA is the radius of the circle to the point of tangency.
A fundamental property of circles is that the radius drawn to the point of tangency is perpendicular to the tangent line.
Therefore, $\angle OAB = 90^\circ$.
This means that $\triangle OAB$ is a right-angled triangle with the right angle at A.

In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In $\triangle OAB$:

• Hypotenuse = OB = 13 cm
• One leg = AB = 12 cm
• Other leg = OA = r (the radius)

According to the Pythagorean theorem:

Subtract $144\, \text{cm}^2$ from both sides:

Take the square root of both sides:

The radius is $5\, \text{cm}$.