A circle with a radius of 25 cm is circumscribed about an isosceles triangle.

A circle with a radius of 25 cm is circumscribed about an isosceles triangle. The distance from the center of the circle to the base is 7 cm. Find the area of the triangle.

It is known from the condition that a circle with a radius of 25 cm is described around an isosceles triangle. It is also known that the distance from the center of the circle to the base is 7 cm. Find the area of ​​the triangle.

We will start with what we denote by O – the cent of the circumscribed circle around ABC, that is, we can write that OB = 25 cm.

The height of the triangle can be written as h = 25 + 7 = 32 cm.

Let L denote the point of intersection of the height with the base of the triangle. Let’s apply the Pythagorean theorem to the resulting right-angled triangle LOC:

LC ^ 2 = OC ^ 2 – LO ^ 2 = 25 ^ 2 – 7 ^ 2 = 625 – 49 = 576;

LC = 24 cm.

Using the condition of the problem – triangle ABC is isosceles, we obtain the length of the base:

AC = 2 * LC = 48 cm.

S = 1/2 * AC * h = 1/2 * 48 * 32 = 768 cm ^ 2.