Inscribed Right Triangle

How to use the property of the inscribed right triangle: property, 2 examples, and their solutions.

Property

Property

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

then the triangle is a right triangle.

Example 1

Example

Solution

The given triangle is an inscribed triangle.

The side of the given triangle
passes through the center of the circle.

Then the given triangle is a right triangle.

So the brown angle is a right angle.

The interior angles of the triangle are
30º, xº and 90º.

So this triangle is a 30-60-90 triangle.

So x = 60.

So 60 is the answer.

Example 2

Example

Solution

The given triangle is an inscribed triangle.

The side of the given triangle
passes through the center of the circle.

Then the given triangle is a right triangle.

So the brown angle is a right angle.

This hypotenuse is the diameter.
The left radius is 13.

So the right radius is also 13.

See this right triangle.

The sides are (10, x, 13 + 13) = (10, x, 26).

(10, x, 26) looks like the multiple of (5, 12, 13).

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

Pythagorean Triple

These two triangles are similar.

Then their sides are proportional.

So x/12 = 10/5.

Similar Triangles

10/5 = 2

x/12 = 2

Multiply 12 to both sides.

Then x = 24.

So x = 24.