The Pythagorean theorem can be applied to find the lengths of some of the sides of a right triangle. This theorem is one of the most important in mathematics and is one of the foundations of trigonometry. To use the theorem, we have to start by identifying the different lengths given. Then, we plug the lengths into the theorem formula and solve for the unknown.

Here, we will learn about the process that we can use to solve problems related to the Pythagorean theorem. Then, we will use this process to solve some practice problems.

## How to use the Pythagorean theorem?

To use the Pythagorean theorem, we have to start by recognizing the different values we have in a given problem and then apply the Pythagorean theorem and solve for the unknown.

Recall that the Pythagorean theorem tells us that the square of the hypotenuse in a right triangle is equal to the sum of the squares of the other two sides. The hypotenuse is the side that is opposite the right angle and the other two sides are called the legs. Therefore, we can consider the following triangle:

In this triangle, the Pythagorean theorem is equal to:

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

Therefore, we can use the following steps to apply the Pythagorean theorem:

**Step 1:** Identify the legs and the hypotenuse of the right triangle.

** Step 2:** Substitute the values into the Pythagorean theorem formula, remembering that “

*c*” is the hypotenuse.

** Step 3:** Solve for the unknown.

## Solved Pythagorean theorem application examples

The following examples are solved using the process of applying the Pythagorean theorem. Each exercise has its respective solution, where you can look at the process used to get the answer.

**EXAMPLE 1**

What is the length of X in the triangle below?

##### Solution

We use the steps above to solve this exercise:

**Step 1:** The legs are the sides a=3 and b=4. The hypotenuse is the side opposite the right angle, so the hypotenuse is the X.

**Step 2:** We substitute these values in the Pythagorean theorem:

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

$latex {{X}^2}={{3}^2}+{{4}^2}$

**Step 3:** Solving for the unknown, we have:

$latex {{X}^2}={{3}^2}+{{4}^2}$

$latex {{X}^2}=9+16$

$latex {{X}^2}=25$

$latex X=5$

The length of X is 5.

**EXAMPLE 2**

Determine the length of Y in the triangle below.

##### Solution

Using the steps given, we have:

**Step 1:** The hypotenuse is c=13 and one of the legs is equal to a=12. Therefore, *b* is the other leg.

**Step 2:** We use these values in the Pythagorean theorem:

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

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

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

**Step 3:** Now, we solve for *b*:

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

$latex {{b}^2}=169-144$

$latex {{b}^2}=25$

$latex b=5$

The length of the other leg is 5.

Start now: Explore our additional Mathematics resources

**EXAMPLE 3**

What is the value of the hypotenuse if the legs of a right triangle are a=9 and b=13?

##### Solution

We use the given steps:

**Step 1:** In this case, the question tells us directly that the legs are a=9 and b=13. Therefore, we have to find the hypotenuse, c.

**Step 2:** We use the Pythagorean theorem with this information:

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

$latex {{c}^2}={{9}^2}+{{13}^2}$

**Step 3:** We solve for*c*:

$latex {{c}^2}={{9}^2}+{{13}^2}$

$latex {{c}^2}=81+169$

$latex {{c}^2}=250$

$latex c=15.8$

The length of c is 15.8.

**EXAMPLE 4**

The side opposite the right angle in a right triangle is 20 and another side is 15. What is the length of the third side?

##### Solution

Again, we follow the following steps:

**Step 1:** We have that the side opposite the right angle measures 20, so this side is the hypotenuse. That means the side of 15 is one of the legs. Therefore, we have c=20 and a=15.

**Step 2:** We use these values in the Pythagorean theorem:

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

$latex {{20}^2}={{15}^2}+{{b}^2}$

**Step 3:** We solve for *b*:

$latex {{20}^2}={{15}^2}+{{b}^2}$

$latex 400=225+{{b}^2}$

$latex {{b}^2}=400-225$

$latex {{b}^2}=175$

$latex b=13.2$

The length of *b* is 13.2.

## Pythagorean theorem – Practice problems

Use what you have learned about how to apply the Pythagorean Theorem to solve the following practice problems. Select an answer and check it to see if you got the correct answer.

## See also

Interested in learning more about the Pythagorean theorem? Take a look at these pages:

### Learn mathematics with our additional resources in different topics

**LEARN MORE**