Thales’ theorem is a special case of the inscribed angles theorem. This theorem says that if we have a triangle inscribed in a circle with the diameter as the hypotenuse, the triangle will be a right triangle and will form a right angle at the vertex located at any point on the circumference.

Here, we will briefly review Thales’ theorem. Also, we will learn to solve problems with this theorem.

## Thales’ theorem revision

Thales’ theorem indicates that a triangle inscribed in a circle, where the hypotenuse corresponds to the diameter of the circle, is a right triangle. For example, in the following diagram, we have points A, B, and C located on the circumference that form an inscribed triangle.

Since segment AC corresponds to the diameter of the circle, the angle formed at vertex B is a 90° angle.

Therefore, in general terms, Thales’ theorem tells us that if we have three points A, B, and C located on the circumference of a circle, where the line AC is the diameter of the circle, then the angle ∠ABC is a right angle (90°).

To prove this theorem, we have to use a bisector to obtain two isosceles triangles and then use 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.

## Problems with answers of Thales’ theorem

The following problems are solved by applying Thales’ theorem. Each problem has its respective solution where you can look at the process used to arrive at the answer.

**PROBLEM 1**

Determine the measure of angle Z in the diagram below if segment AC is the diameter of the circle.

##### 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°+60°+Z=180°

150°+Z=180°

Z=180°-150°

Z=30°

The measure of angle Z is 30°.

**PROBLEM 2**

If O represents the center of the circle, what is the measure of angle *a*?

##### Solution

If point O is the center of the circle, we know that segment XZ represents the diameter of the circle. Therefore, we can apply Thales’ theorem. Using the theorem, we know that angle Y is right, that is, 90°.

Now, we use the sum of interior angles of a triangle to find the measure of angle *a*:

90°+40°+*a*=180°

130°+*a*=180°

*a*=180°-130°

*a*=50°

The measure of angle *a* is 50°.

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**PROBLEM 3**

If segment XY is the diameter of the circle, what is its length?

##### Solution

Applying Thales’ theorem, we find that triangle XYZ is a right triangle. That is, angle Z is a right angle. Therefore, we find the length of XY by applying the Pythagorean theorem, where, XY is the hypotenuse of a right triangle:

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

$latex {{c}^2}={{6}^2}+{{9}^2}$

$latex {{c}^2}=36+81$

$latex {{c}^2}=117$

$latex c=10.82$

The length of the XY segment is 10.82 units.

**PROBLEM 4**

What is the length of AC if AB is the diameter of the circle?

##### Solution

Since AB is the diameter of the circle, we know that Thales’ theorem applies. This means that the triangle is a right triangle and we can use the Pythagorean theorem to find the length of the segment AC:

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

$latex {{a}^2}+{{8}^2}={{9}^2}$

$latex {{a}^2}+64=81$

$latex {{a}^2}=81-64$

$latex {{a}^2}=17$

$latex a=4.12$

The length of segment AC is 4.12 units.

**PROBLEM 5**

If XY is the diameter of the circle, what is the measure of angle *a*?

##### Solution

We are going to represent the center of the circle with O. Since the XY segment is the diameter, by Thales’ theorem, we know that the triangle is a right triangle.

Both internal triangles formed are isosceles since two of their sides correspond to the radii of the circle. Therefore, we can deduce that two interior angles are also equal:

∠O*YZ *= ∠*YZO* =60°

By Thales’ theorem, we also know that:

∠*XZY *=90°

Therefore, we have:

∠*a*= 90°-60°=30°

Thus, the measure of angle ∠*a* is 30°.

## Thales’ theorem – Problems to solve

Use Thales’ theorem to solve the following practical problems. Solve the problems and select your answer obtained. If you need help with this, you can look at the solved examples above.

## See also

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

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