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More Lessons for Geometry

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In this lesson, we will learn

### Thales' theorem

### Using the Theorem

**Inscribed Right Triangles (Right Triangles Inside of Circles)**

Thales' Theorem: 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.

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.

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**˚**.

*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˚

Thales' Theorem: 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.

This proof uses the bow theorem.

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