# Circle Theorem Examples

Related Topics:

More Lessons for Geometry
Math Worksheets
In this lesson, we will learn

- a Circle Theorem called
the Thales' Theorem or Triangle inscribed in semicircle or Angle inscribed in semicircle or “90 degrees in Semicircle” Theorem.
- how to use the Thales' Theorem to find missing angles.
- how to prove the Thales' Theorem.

## Thales' theorem

The Thales' theorem states

**Every angle subtended at the circumference by the diameter of a circle is a right angle (90˚). **

or

The diameter of a circle always subtends a right angle to any point on the circle.

or

The angle inscribed in a semicircle is 90**˚**.

*POQ* is the diameter. ∠*PAQ = ∠PBQ = ∠PCQ *= 90˚.

## Using the Theorem

*Example: *

*O* is the centre of the circle. Find the value of *x*

* Solution: *

∠*ABC* = 90˚ ( angle in a semicircle = 90˚)

63˚ + 90˚ + *x* = 180˚ ( sum of angles in a triangle )

*x* = 27˚

Inscribed Right Triangles (Right Triangles Inside of Circles)

This video introduces you to Thales' Theorem that if the longest side of a triangle inscribed within a circle is the same length as the diameter of a circle, then that triangle is a right triangle, as well as the converse: if a right triangle is inscribed within a circle, the length of its hypotenuse is the diameter of the circle.

We can use the Thales' Theorem to find missing angles.

## Proving the Theorem

Proof of the Thales' Theorem

Proof showing that a triangle inscribed in a circle having a diameter as one side is a right triangle. This proof uses the

bow theorem.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.