Thales’ theorem tells us that a triangle inscribed in a circle, where the hypotenuse corresponds to the diameter of the circle, is always a right triangle. This theorem can be proved using the sum of interior angles.

Here, we will look at a summary of Thales’ theorem. Then, we will learn to apply this theorem to solve some practice problems.

## Summary of Thales’ theorem

Thales’ theorem is considered as a special case of the inscribed angles theorem. This theorem tells us that, if we have a triangle inscribed in a circle as shown in the following diagram, the angle formed at vertex B is always a right angle.

Therefore, if three points A, B, and C are located on the circumference of a circle, where line AC is the diameter of the circle, then the angle ∠ABC is a right angle (90°).

This theorem can be proved by drawing a bisector from the center to vertex B and then using the angles of the isosceles triangles formed and the fact that the sum of interior angles of any triangle is equal to 180°. You can look at the full proof in this article.

## Thales’ theorem – Examples with answers

The following examples show the application of Thales’ theorem. Each example has its respective solution, but it is recommended that you try to solve the examples yourself before looking at the solution.

**EXAMPLE 1**

Segment AC is the diameter of the circle. What is the measure of angle Z?

##### Solution

The segment AC is the diameter of the circle, so we can use Thales’ theorem. Therefore, we know that the angle at vertex B is a 90° angle. Since the sum of interior angles in any triangle is equal to 180°, we have:

90°+50°+Z=180°

140°+Z=180°

Z=180°-140°

Z=40°

The measure of angle Z is 40°.

**EXAMPLE 2**

If M is the center of the circle, determine the measure of angle Z.

##### Solution

Since point M is the center of the circle, we know that segment XZ is the diameter of the circle. Therefore, we can use Thales’ theorem.

By the theorem, we know that the angle at vertex Y is a right angle, that is, a 90° angle. We can use the sum of interior angles of a triangle to get the measure of Z:

90°+50°+Z=180°

140°+Z=180°

Z=180°-140°

Z=40°

The measure of angle Z is 40°.

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**EXAMPLE 3**

Segment AC is the diameter of the circle. What is the length of the diameter?

##### Solution

Using Thales’ theorem, we can determine that triangle ABC is a right triangle. This means that angle B is a right angle. Therefore, we can use the Pythagorean theorem to find the length of the diameter:

$latex {{c}^2}={{a}^2}+{{b}^2}$

$latex {{c}^2}={{12}^2}+{{13}^2}$

$latex {{c}^2}=144+169$

$latex {{c}^2}=313$

$latex c=17.7$

The length of segment AC is 17.7 units.

**EXAMPLE 4**

Determine the length of segment AB in the diagram below.

##### Solution

Segment AC is the diameter of the circle. Then, applying Thales’ theorem, we can determine that triangle ABC is a right triangle. Therefore, we use the Pythagorean theorem to find the length of segment AB:

$latex {{a}^2}+{{b}^2}={{c}^2}$

$latex {{a}^2}+{{8}^2}={{10}^2}$

$latex {{a}^2}+64=100$

$latex {{a}^2}=100-64$

$latex {{a}^2}=36$

$latex a=6$

The length of segment AB is 6 units.

**EXAMPLE 5**

If C is the center of the circle, what is the measure of angle ∠ABC?

##### Solution

Since point C is the center of the circle, we know that segment AD is the diameter. Therefore, we can apply Thales’ theorem.

Two of the sides of the triangles ABC and BCD are equal since they correspond to the radii of the circle. Therefore, two angles are also equal and the triangles must be isosceles triangles. So, we have:

∠CBD = ∠CDB = 60°

Applying Thales’ theorem, we have:

∠ABD = 90°

So, we have:

∠ABC = 90°-60° = 30°

Therefore, the measure of angle ∠ABC is 30°.

## Thales’ theorem – Practice problems

Apply what you have learned about Thales’ theorem and solve the following problems. If you need help with this, you can look at the solved examples above or the summary of Thales’ theorem.

## See also

Interested in learning more about Thale’s theorem? Take a look at these pages:

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