Inscribed Right Triangle

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.

If the side of an inscribed triangle
passes through the center of the circle,

then that triangle is a right triangle.

Example 1

Find the value of x. The measures of the interior angles: 30, x.

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, xº, and 90º.

So this is a 30-60-90 triangle.

30-60-90 triangle

So x = 60.

Example 2

Find the value of x. Radius: 13, Sides: 10, x.

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.