 Lesson Explainer: The Pythagorean Theorem | Nagwa Lesson Explainer: The Pythagorean Theorem | Nagwa

# Lesson Explainer: The Pythagorean Theorem Mathematics • 8th Grade

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: Right Triangle and Hypotenuse

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 each made of two s, and we also see that four s fit in the bigger square. We deduce from this that area of the bigger square, , is equal to the sum of the area of the two other squares, and . We can write this as .

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

This result can be generalized to any right triangle, and this is the essence of 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.

Writing for the length of the hypotenuse and and for the lengths of the legs, we can express the Pythagorean theorem algebraically as

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 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 diagnoals, 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 lengths. (They are the hypotenuses of the yellow right triangles.) Also, the angle of the white shape and the two non-right angles of the right triangle from a straight line. As the measure of the two non-right angles ofa right triangle add up to , the angle of the white shape is . Therefore, the white shape isa square.

Let be the length of the white square’s side (and of the hypotenuses 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 triangles, we find that the area ofthe big square is ; that is, .

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

This is ageometric proof of the Pythagorean theorem.

Now that we know the Pythagorean theorem, let’s look at an example.

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

Find in the right triangle shown.

We are given a right triangle and must start by identifying its hypotenuse and legs. The right angle is , and the legs form the right angle, so they are the sides and . The hypotenuse is the side opposite , which is therefore .

From the diagram, we have been given the lengths of the two legs and need to work out , the length of the hypotenuse. Writing and for the lengths of the legs and for the length of the hypotenuse, we recall the Pythagorean theorem, which states that

Substituting for , , and with the actual values given in the diagram, we get

We must now solve this equation for . Simplifying the left-hand side, we have

To find the value of , we take the square roots of both sides, remembering that is positive because it is a length. Thus,

In the first example, we were asked to find the length of the hypotenuse of a right triangle. Now, let’s see what to do when we are asked to find the length of one of the legs.

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

Find in the right triangle shown.

We are given a right triangle and must start by identifying its hypotenuse and legs. The right angle is , and the legs form the right angle, so they are the sides and . The hypotenuse is the side opposite , which is therefore .

From the diagram, we have been given the length of the hypotenuse and one leg, and we need to work out , the length of the other leg, . Writing and for the lengths of the legs and for the length of the hypotenuse, we recall the Pythagorean theorem, which states that

Substituting for , , and with the actual values given on the diagram, we get

To solve for , we start by expanding the square numbers:

Then, we subtract 225 from both sides, which gives us

Finally, to find , we take the square roots of both sides, remembering that is positive because it is a length. Thus,

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

### How To: Using the Pythagorean Theorem to Find an Unknown Side of a Right Triangle

1. Identify the hypotenuse and the legs of the right triangle.
2. With and as the legs of the right triangle and as 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 .

Once we have learned how to find the length of the hypotenuse or a leg, we can also use the Pythagorean theorem to answer geometric questions expressed as word problems. Here is an example of this type.

### Example 3: Finding the Diagonal of a Rectangle Using the Pythagorean Theorem

Determine the diagonal length of the rectangle whose length is 48 cm and width is 20 cm.

Here, we are given the description of a rectangle and need to find its diagonal length. It helps to start by drawing a sketch of the situation.

The rectangle has length 48 cm and width 20 cm. Therefore, its diagonal length, which we have labeled as cm, will be the length of the hypotenuse of a right triangle with legs of length 48 cm and 20 cm.

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 (the legs). Writing for the length of the hypotenuse and and for the lengths of the legs, this can be expressed algebraically as

Substituting for , , and with the values from the diagram, we have

To solve this equation for , we start by writing on the left-hand side and simplifying the squares:

Then, we take the square roots of both sides, remembering that is positive because it is a length. Hence, we have

The dimensions of the rectangle are given in centimetres, so the diagonal length will also be in centimetres. We conclude that a rectangle of length 48 cm and width 20 cm has a diagonal length of 52 cm.

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

### Example 4: Applying the Pythagorean Theorem to Solve More Complex Problems

Find the perimeter of .

In this question, we need to find the perimeter of , which is a quadrilateral made up of two right triangles, and .

From the diagram, is a right triangle at , and is a right triangle at . We also know three of the four side lengths of the quadrilateral, namely , , and . To calculate the perimeter of , we need to find its missing side length, .

Note that is the hypotenuse of , but we do not know . However, is the hypotenuse of , where we know both and .

Now, recall the Pythagorean theorem, which states that in a right triangle where and are the lengths of the legs and is the length of the hypotenuse, we have

Therefore, we will apply the Pythagorean theorem first in triangle to find and then in triangle to find .

In triangle , is the length of the hypotenuse, which we denote by . Substituting for all three side lengths in the Pythagorean theorem and then simplifying, we get

As is a length, it is positive, so taking the square roots of both sides gives us

In triangle , is the length of the hypotenuse, which we denote by . Since we now know the lengths of both legs, we can substitute them into the Pythagorean theorem and then simplify to get

As is a length, it is positive, so taking the square roots of both sides gives us

Finally, we can work out the perimeter of quadrilateral by summing its four side lengths:

All lengths are given in centimetres, so the perimeter of is 172 cm.

The Pythagorean theorem can also be applied to help find the area of a right triangle as follows. When given the lengths of the hypotenuse and one leg, we can always use the Pythagorean theorem to work out the length of the other leg. Note that 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.

We will finish with an example that requires this step.

### Example 5: Applying the Pythagorean Theorem to Solve More Complex Problems

In the trapezoid below, and . Find its area.

Here, we are given a trapezoid and must use information from the question to work out more details of its properties before finding its area.

The fact that is perpendicular to implies that is a right triangle with its right angle at .

Similarly, since both and are perpendicular to , then they must be parallel. When combined with the fact that is parallel to (and hence to ), this implies that is a rectangle.

Therefore, the area of the trapezoid will be the sum of the areas of right triangle and rectangle .

First, consider . We know that the hypotenuse has length . In addition, we can work out the length of the leg because .

Now, recall the Pythagorean theorem, which states that, in a right triangle where and are the lengths of the legs and is the length of the hypotenuse, we have

As we know two side lengths of the right triangle , we can apply the Pythagorean theorem to find the missing length of leg . Writing for this length and substituting for , , and , we have

To solve for , we start by expanding the square numbers:

Then, we subtract 81 from both sides, which gives us

To find , we take the square roots of both sides, remembering that is positive because it is a length. Thus,

Since we now know the lengths of the legs of right triangle are 9 cm and 12 cm, we can work out its area by multiplying these values and dividing by 2. Therefore,

Secondly, consider rectangle . Notice that its width is given by . Moreover, we also know its height because it is the same as the missing length of leg of right triangle that we calculated above, which is 12 cm. Therefore,

Finally, the area of the trapezoid is the sum of these two areas: . Since the lengths are given in centimetres then this area will be in square centimetres. The area of the trapezoid is 126 cm2.

Let’s finish by recapping some key concepts from this explainer.

### Key Points

• 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).
• Writing for the length of the hypotenuse, and and for the lengths of the legs, we can express the Pythagorean theorem algebraically as
• We can use the Pythagorean theorem to find the length of the hypotenuse or a leg of a right triangle and to solve more complex geometric problems involving areas and perimeters of right triangles.