In this explainer, we will learn how to use the converse of the Pythagorean theorem to determine whether a triangle is a right triangle.

You should have already learned about the Pythagorean theorem.

The Pythagorean theorem states that, in a right triangle, the area of a square on the hypotenuse of length is equal to the sum of the areas of the squares on the legs of lengths and . Does this mean that a triangle where the lengths of the sides verify the equation is necessarily a right triangle?

Let us assume that is of side lengths , , and , with . Let be a right triangle of side lengths , , and , with a right angle at .

By applying the Pythagorean theorem in , we find

We know that . For the given and , has only one value. It follows that .

Since a triangle is uniquely defined by its three sidesβ lengths, we conclude that and are congruent. Consequently, has a right angle at .

This is a proof of the converse of the Pythagorean theorem.

### The Converse of Pythagorean Theorem

If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

If , then the triangle has a right angle at .

The converse of the Pythagorean theorem allows you to determine whether a given triangle has a right angle or not.

### How To: Determining Whether a Given Triangle Has a Right Angle or Not

If we are given the three sides of a triangle and want to determine whether the triangle has a right angle or not, the first step is to identify the longest side. Then, we need to check whether the square of the longest side is equal to the sum of the squares of the other two sides or not.

If yes: the triangle has a right angle opposite its longest side.

If not: the triangle has no right angle.

### Example 1: Using the Converse of the Pythagorean Theorem

Can the lengths 7.9 cm, 8.1 cm, and 5.3 cm form a right triangle?

### Answer

- Identify the longest side of the triangle: 8.1 cm.
- Square it: .
- Calculate the sum of the squares of the other two sides:
- Compare both numbers: .
- Conclude: the given lengths cannot form a right triangle.

### Example 2: Using the Converse of the Pythagorean Theorem

Is a right triangle at ?

### Answer

- Identify the question: does have a right angle at ?
- Identify what is given: and . is the hypotenuse of , and and are given.
- Identify a solving strategy: to answer the question, we will need to apply the converse of the Pythagorean theorem in , but for this, we need first to find . As it is the hypotenuse of and we know the other two sides, we will apply the Pythagorean theorem in first.
- Implement the solving strategy:
- We apply the Pythagorean theorem in . Let be the length of , the hypotenuse of . The legs are 17.6 cm and 21 cm. By substituting these into the equation given by the Pythagorean theorem, , we get As is a length, it is positive, and so .
- We apply the converse of the Pythagorean theorem in .
- Identify the longest side of the triangle: 44 cm.
- Square it: .
- Calculate the sum of the squares of the other two sides:
- Compare both numbers: .
- Conclude: has no right angle at .