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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.

You can use the free Mathway calculator and problem solver 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.

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.

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 calculator and problem solver 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.

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