# Explainer: The Pythagorean Theorem

In this explainer, we will learn how to use the Pythagorean theorem to find the length of the hypotenuse or a leg of a right triangle and its area.

The Pythagorean theorem gives a relationship between the sides of any right triangle.

Before we start, let’s remember what a right triangle is and how to recognize its hypotenuse.

### Definition: Useful Language

A right triangle is a triangle that has one right angle and always one longest side. This longest side is always the side that is opposite the right angle, while the other sides, called the legs, form the right angle. The longest side is called the hypotenuse.

Let’s start by considering an isosceles right triangle, , shown in the figure. Squares have been added to each side of .

As is isosceles, we see that the squares drawn at the legs are made each of two s. And we see also that four s fit in the bigger square. We deduce from this that the area of the bigger square, , is equal to the sum of the area of the two other squares, and . We can write .

Let and be the sides of the legs of the triangle (so, in this special case, ) and be the side of the hypotenuse. Therefore, , , and , and by substituting these into the equation, we find

This result can be generalized to any right triangle: this is the Pythagorean 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.

If the hypotenuse is labeled “” and the two shorter sides are labeled “” and “,” then the Pythagorean theorem states that

There are many proofs of the Pythagorean theorem. We are going to look at one of them.

Let’s consider a square of length and another square of length that are placed in two opposite corners of a square of length as shown in the diagram below.

By expanding , we can find the area of the two little squares (shaded in blue and green) and of the yellow rectangles. We find . This can be found as well by considering that the big square of length is made of a square of area , another square of area , and two rectangles of area .

Now, the blue square and the green square are removed from the big square, and the yellow rectangles are split along one of their diagonals, creating four congruent right triangles. They are then placed in the corners of the big square, as shown in the figure.

As the four yellow triangles are congruent, the four sides of the white shape at the center of the big square are of equal length. (This is the hypotenuse of the yellow right triangles.) Also, the angle of the white shape and the two non-right angles of the right form a straight line. As the measures of the two non-right angles of a right triangle add up to , the angle of the white shape is . Therefore, the white shape is a square.

Let be the length of the white square’s side (and of the hypotenuse of the yellow triangles). Using the fact that the big square is made of the white square and the four yellow right triangles, we find that the area of the big square is , that is, .

Since the big squares in both diagrams are congruent (with side ), we find that , and so

This is a geometric proof of the Pythagorean theorem.

Now that we know the Pythagorean theorem, let’s see how to use it to solve some questions.

### Example 1: Using the Pythagorean Theorem to Find an Unknown Side of a Right Triangle

Find in the right triangle shown.

1. Identify the hypotenuse and the legs in the right triangle. The right angle is . The legs form the right angle, so they are the sides and . The hypotenuse is the side opposite ; it is .
2. With and the legs of the right triangle and the hypotenuse, the Pythagorean theorem states that .
3. Substitute , , and with the actual values given on the diagram into the equation:
4. Solve for :

As is a length, it is positive; so

### Example 2: Using the Pythagorean Theorem to Find an Unknown Side of a Right Triangle

Given that is a right angle triangle, where and , determine the length of .

1. Identify the hypotenuse and the legs in the right triangle. The right angle is . The legs form the right angle, so they are the sides and . The hypotenuse is the side opposite ; it is .
2. With and the legs of the right triangle and the hypotenuse, the Pythagorean theorem states that .
3. Let be . Substitute , , and with their actual values into the equation:
4. Solve for :
By subtracting 64 from each side of the equation,

As is a length, it is positive; so

Let’s summarize how to use the Pythagorean theorem to find an unknown side of a right triangle.

### How to Use the Pythagorean Theorem to Find an Unknown Side of a Right Triangle

1. Identify the hypotenuse and the legs in the right triangle.
2. With and the legs of the right triangle and the hypotenuse, write the Pythagorean theorem: .
3. Substitute , , and with their actual values, using for the unknown side, into the above equation.
4. Solve for .

The following example is a slightly more complex question where we need to use the Pythagorean theorem.

### Example 3: Using the Pythagorean Theorem in Multistep Problems

Find the perimeter of .

1. Identify the question: find the perimeter of .
2. Identify what is given:
1. We have , and .
2. is a right triangle at , and is a right triangle at .
3. Identify a solving strategy:
1. We want to find to be able to calculate the perimeter of . is the hypotenuse of , but we do not know . However, is the hypotenuse of where we know both and . Therefore, we will apply the Pythagorean theorem first in triangle to find and then in triangle to find .
2. To answer the question, we will work out .
4. Implement the strategy:
1. In triangle , is the hypotenuse. The Pythagorean theorem states that in a right triangle where and are the legs and is the hypotenuse. If we substitute the real values for and and let , we get As is a length, it is positive; so
2. In , is the hypotenuse. The Pythagorean theorem states that in a right triangle where and are the legs and is the hypotenuse. If we substitute the real values for and and let , we get As is a length, it is positive; so
3. We work out the perimeter of :

The perimeter of is 172 cm.