The Pythagorean theorem is one of the most important equations in mathematics as it allows us to use algebra in geometry. This theorem tells us that, in a right triangle, the square of the hypotenuse is equal to the sum of the square of the legs. With the Pythagorean theorem, we can solve a large number of geometry problems. In fact, this theorem is useful in several areas of Engineering and even in Architecture.

Here, we will learn how to solve these types of problems.

## How to solve problems with the Pythagorean theorem?

To solve problems with the Pythagorean theorem, we have to carefully read the type of problem we have and determine what we want to find. We can use the Pythagorean theorem when we have the length of two sides of a right triangle and we want to find the length of the third side.

Depending on the side we want to find, we have two different cases:

- We have the lengths of two legs and we want to find the length of the hypotenuse.
- We have the lengths of one leg and the hypotenuse and we want to find the length of the other leg.

In the triangle below, the hypotenuse is side *c* (the side opposite the right angle) and the legs are sides *a* and *b*:

The Pythagorean theorem is:

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

Therefore, if we know the lengths of the two legs, we simply plug the values into the equation to get the length of the hypotenuse.

However, if we want to find the length of a leg, we can use one of the variations of the Pythagorean theorem:

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

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

## Pythagorean theorem – Problems with answers

The Pythagorean theorem and its variations are used to solve some application problems of the Pythagorean theorem. Try to solve the problems yourself before looking at the solution.

**PROBLEM 1**

Given the right triangle below, find the value of X.

##### Solution

In this case, we have the lengths of the legs and we want to find the length of the hypotenuse. Therefore, we recognize the following values:

- a=6
- b=8

Now, we substitute these values into the Pythagorean theorem and solve:

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

$latex {{c}^2}={{6}^2}+{{8}^2}$

$latex {{c}^2}=36+64$

$latex {{c}^2}=100$

$latex c=\sqrt{100}$

$latex c=10$

The length of *X* is 10.

**PROBLEM 2**

If we have the following lengths, what is the value of Y?

##### Solution

Here, we have the length of one leg and the length of the hypotenuse and we want to find the length of the other leg. Thus, we recognize the following:

- a=12
- c=13

We use one of the variations of the Pythagorean theorem and solve:

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

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

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

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

$latex b=5$

The length of *b* is 5.

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**PROBLEM 3**

A right triangle has legs of lengths 8 m and 11 m, what is the length of its hypotenuse?

##### Solution

The problem gives us the lengths of the legs of the triangle:

- a=8
- b=11

Therefore, we apply the Pythagorean theorem with these values and solve:

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

$latex {{c}^2}={{8}^2}+{{11}^2}$

$latex {{c}^2}=64+121$

$latex {{c}^2}=185$

$latex c=13.6$

The hypotenuse has a length of 13.6 m.

**PROBLEM 4**

Two brothers go for a walk at the same time, one of them going south and the other going west. After half an hour, the brother who went south has walked 700 meters and the brother who went west has walked 850 meters. What is the shortest distance between the two at that time?

##### Solution

To solve the problem more easily, we are going to use the following diagram:

We can see that the south and west directions form a right angle. Also, the shortest distance between two points is a straight line joining the points. Therefore, we can use the Pythagorean theorem to find the length of that line:

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

$latex {{c}^2}={{700}^2}+{{850}^2}$

$latex {{c}^2}=490000+722500$

$latex {{c}^2}=1212500$

$latex c=1101.1$

Therefore, the shortest distance between the siblings is 1101.1 meters.

**PROBLEM 5**

The square of the hypotenuse of an isosceles right triangle is equal to 128 m². What is the length of one of the legs?

##### Solution

We know the square of the hypotenuse. Also, we know that the legs of an isosceles right triangle have the same length. Therefore, we can illustrate this problem with the following diagram:

Now, we apply the Pythagorean theorem with this information and we have:

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

$latex 128={{a}^2}+{{a}^2}$

$latex 128=2{{a}^2}$

$latex {{a}^2}=\frac{128}{2}$

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

$latex a=8$

Therefore, the length of each leg is equal to 8 m.

## Pythagorean theorem – Practice problems

Use the Pythagorean theorem and its variations to solve the following problems. If you need help with this, use the solved problems above.

## See also

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

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