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

We should already have learned about the Pythagorean theorem.

### Theorem: The Pythagorean Theorem

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides (called the legs).

Write for the length of the hypotenuse and and for the lengths of the legs. The Pythagorean theorem can be expressed algebraically as

In view of this, it makes sense to ask the following question: Does this mean that a triangle with side lengths satisfying the equation is necessarily a right triangle?

Let us assume that has side lengths , , and and satisfies . Then, let be a right triangle with side lengths , , and and a right angle at .

By applying the Pythagorean theorem to , we find

Now, we know that . Furthermore, for the given and , the sum has a single value. Therefore, we must have . Since and are lengths, we can take positive square roots of both sides of the last equation, from which it follows that .

Since a triangle is defined uniquely by the lengths of its three sides, we conclude that and are congruent. Consequently, must have a right angle at .

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

### Theorem: The Converse of the 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 .

A corollary of this result is that if , the triangle does not have a right angle at .

Thus, the converse of the Pythagorean theorem allows us to determine whether a given triangle has a right angle.

### How To: Determining Whether a Given Triangle Has a Right Angle

To find whether a given triangle has a right angle, we can do the following:

- Identify the longest side of the triangle.
- Calculate the square of the longest side and the sum of the squares of the other two side lengths.
- If the two quantities are equal, the triangle has a right angle. If not, the triangle has no right angle.

Let us start with an example to check our understanding of the converse of the Pythagorean theorem.

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

What can the converse of the Pythagorean theorem be used for?

- Demonstrating that a triangle is equilateral
- Demonstrating that a triangle has a right angle
- Finding the angles in a triangle
- Finding lengths in an equilateral triangle
- Demonstrating that a triangle is an isosceles triangle

### Answer

First, recall the Pythagorean theorem, which states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides (called the legs). Writing for the length of the hypotenuse and and for the lengths of the legs, we can express this theorem algebraically as .

Note that the Pythagorean theorem applies to right triangles only; it shows that the presence of a right angle in a triangle implies the relationship between its side lengths.

The converse of the Pythagorean theorem works in the opposite direction. It starts from a relationship between the side lengths of a triangle and uses this to prove that the triangle has a right angle. In other words, write for the length of the longest side of a triangle and and for the lengths of the two shorter sides; the converse of the Pythagorean theorem tells us that if , then the triangle has a right angle.

Reviewing the five possible answer choices, we conclude that the correct one is B: the converse of the Pythagorean theorem can be used for demonstrating that a triangle has a right angle.

Next, we will use the converse of the Pythagorean theorem to test whether a given triangle is a right triangle.

### Example 2: Identifying the Type of a Triangle by Applying the Converse of the Pythagorean Theorem

Is this triangle a right triangle?

- Yes
- No

### Answer

First, we recall the converse of the Pythagorean theorem. Write for the length of the longest side of a triangle and and for the lengths of the two shorter sides. The converse of the Pythagorean theorem tells us that if , then the triangle has a right angle.

A corollary of this result is that if , the triangle does not have a right angle.

From the diagram, the given triangle has side lengths of 13 cm, 12 cm, and 5 cm, so the longest side has length . Squaring this value gives

The two shorter sides have lengths and . The sum of the squares of these side lengths is

Both calculations give the same answer, so and the converse of the Pythagorean theorem implies this is a right triangle. Therefore, the correct answer is βYes,β which is A.

We will now look at an example without a diagram.

### Example 3: Identifying the Type of a Triangle by Applying the Converse of the Pythagorean Theorem

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

- No
- Yes

### Answer

Write for the length of the longest side of a triangle and and for the lengths of the two shorter sides. The converse of the Pythagorean theorem tells us that if , then the triangle has a right angle. A corollary of this result is that if , then the triangle does not have a right angle.

We start by identifying the longest side length of the triangle, which is . Squaring this value gives

The shorter two sides have lengths and . The sum of the squares of these side lengths is

Comparing these values, clearly , so . We conclude that the given lengths cannot form a right triangle, so the correct answer is βNo,β which is A.

The converse of the Pythagorean theorem can also be applied to solve geometric problems of greater complexity, sometimes in conjunction with the Pythagorean theorem itself.

### Example 4: Applying the Pythagorean Theorem and Its Converse to Determine Whether an Angle Is Right

Is a right triangle at ?

- No
- Yes

### Answer

In this question, we are given two of the three side lengths of , namely, and , and need to work out if it is a right triangle at .

From the diagram, we also know that is a right triangle at and are given and , the lengths of its two shorter sides. Furthermore, the missing side length of is , which is exactly half of , the missing length of the hypotenuse of .

Therefore, as is a right triangle, we can apply the Pythagorean theorem to find the length of its hypotenuse. Once we have this answer, we can halve it to provide the missing side length of . Finally, we can apply the converse of the Pythagorean theorem in to check whether it is a right triangle at .

Starting with , we write for , the length of the hypotenuse. The Pythagorean theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. Hence, we can substitute the known values into the equation given by the Pythagorean theorem, , as follows:

As is a length, it is positive, so

Next, we calculate the missing side length of by halving this value to get . We now have all three side lengths of : 19.1 cm, 15.6 cm, and 10.25 cm.

Writing for the length of the longest side of a triangle and and for the lengths of the two shorter sides, we recall the converse of the Pythagorean theorem, which tells us that if , then the triangle has a right angle. A corollary of this result is that if , then the triangle does not have a right angle.

The longest side of has length , and squaring this value gives

The other two side lengths are and . The sum of the squares of these lengths is

Comparing these values, clearly , so .

By the converse of the Pythagorean theorem, we deduce that does not have a right angle at . This means the correct answer is βNo,β which is A.

In our final example, we show how we can apply the converse of the Pythagorean theorem to help us find an area.

### Example 5: Applying the Converse of the Pythagorean Theorem to Find the Area of a Triangle

A triangle has sides of lengths 36.4, 27.3, and 45.5. What is its area?

### Answer

Here, we are given the three side lengths of a triangle. Writing for the length of the longest side and and for the lengths of the two shorter sides, we recall the converse of the Pythagorean theorem, which tells us that if , then the triangle has a right angle.

If we can prove that the given triangle is a right triangle, it will be straightforward to work out its area, for the following reason. Remember that the legs of a right triangle form a right angle. If the lengths of the legs are and , then would represent the area of a rectangle with side lengths and . Therefore, the quantity , which is half of this area, represents the area of the corresponding right triangle. This is shown in the diagram below.

We start by identifying the longest side of the triangle, which is . Squaring this value gives

The other two side lengths are and , and the sum of the squares of these lengths is

Both calculations give the same answer, so , and the converse of the Pythagorean theorem implies this is a right triangle.

Lastly, we can calculate the area of this right triangle. The two shorter sides (the legs) must form a right angle, and they have lengths 36.4 and 27.3. We, therefore, obtain the area by multiplying these values and then dividing the result by 2:

Thus, we have found that the area of the triangle is 496.86 square units.

Let us finish by recapping some key concepts from this explainer.

### key Points

- The converse of the Pythagorean theorem states that 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 . - A corollary of this result is that if , then the triangle does not have a right angle at .