Inscribed Right Triangle

Inscribed Right Triangle

How to solve inscribed right triangle problems: property, proof, 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.

Proof

Inscribed Right Triangle: Proof

m[blue arc] = 180

The red angle is the inscribed angle
that intercepts the blue arc.
So m∠(red) = (1/2)⋅180 = 90.

So the given 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.
(blue point)

So the given triangle is a right triangle.

The measures of the triangle's interior angles are
30, x, and 90.

So it's a 30-60-90 triangle.

So x = 60.

Example 2

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

The side of the given triangle
passes through the center of the circle.
(blue point)

So the given triangle is a right triangle.

See the given right triangle.
The brown hypotenuse is, 13 + 13, 26.
And the leg is 10.

So this right triangle is similar to (5, 12, 13) right triangle.

Draw a (5, 12, 13) right triangle next to the given triangle.

The given triangle is × 2 bigger.

So x = 2⋅12.