# Inscribed Right Triangle

How to solve inscribed right triangle problems: property, examples, and their solutions.

## Property

If the side of an inscribed triangle

passes through the center of the circle,

then that triangle is a right triangle.

## Example 1

The side of the given triangle

passes through the center of the circle.

So the given triangle is a right triangle.

So the brown angle is a right angle.

The interior angles are 30º, *x*º, and 90º.

So this is a 30-60-90 triangle.

30-60-90 triangle

So *x* = 60.

## Example 2

The side of the given triangle

passes through the center of the circle.

So the given triangle is a right triangle.

So the brown angle is a right angle.

The hypotenuse of the right triangle is,

13 + 13, 26.

Starting from the shortest side,

the sides of the given right triangle are

(10, *x*, 26).

Then the related Pythagorean triple is

(5, 12, 13).

So draw a (5, 12, 13) right triangle

next to the given triangle.

These two triangles are similar.

Pythagorean triples

These two triangles are similar,

their sides are proportional.

So *x*/12 = 10/5.

Similar triangles

10/5 = 2

So *x*/12 = 2.

Multiply 12 on both sides.

Then *x* = 24.