Reading: Pythagorean Theorem

Learning Objective(s)

·Use the Pythagorean Theorem to find the unknown side of a right triangle.

·Solve application problems involving the Pythagorean Theorem.

A long time ago, a Greek mathematician named Pythagoras discovered an interesting property about right triangles: the sum of the squares of the lengths of each of the triangle's legs is the same as the square of the length of the triangle's hypotenuse. This property--which has many applications in science, art, engineering, and architecture--is now called the Pythagorean Theorem.

Let's take a look at how this theorem can help you learn more about the construction of triangles. And the best part--you don't even have to speak Greek to apply Pythagoras' discovery.

Pythagoras studied right triangles, and the relationships between the legs and the hypotenuse of a right triangle, before deriving his theory.

In the box above, you may have noticed the word "square,” as well as the small 2s to the top right of the letters in . To squarea number means to multiply it by itself. So, for example, to square the number 5 you multiply 5 • 5, and to square the number 12, you multiply 12 • 12. Some common squares are shown in the table below.

When you see the equation , you can think of this as "the length of side a times itself, plus the length of side b times itself is the same as the length of side c times itself.”

Let's try out all of the Pythagorean Theorem with an actual right triangle.

This theorem holds true for this right triangle--the sum of the squares of the lengths of both legs is the same as the square of the length of the hypotenuse. And, in fact, it holds true for all right triangles.

The Pythagorean Theorem can also be represented in terms of area. In any right triangle, the area of the square drawn from the hypotenuse is equal to the sum of the areas of the squares that are drawn from the two legs. You can see this illustrated below in the same 3-4-5 right triangle.

Note that the Pythagorean Theorem only works with right triangles.

You can use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle if you know the length of the triangle's other two sides, called the legs. Put another way, if you know the lengths of a and b, you can find c.

In the triangle above, you are given measures for legs a and b: 5 and 12, respectively. You can use the Pythagorean Theorem to find a value for the length of c, the hypotenuse.

Using the formula, you find that the length of c, the hypotenuse, is 13.

In this case, you did not know the value of c--you were given the square of the length of the hypotenuse, and had to figure it out from there. When you are given an equation like  and are asked to find the value of c, this is called finding the square root of a number. (Notice you found a number, c, whose square was 169.)

Finding a square root takes some practice, but it also takes knowledge of multiplication, division, and a little bit of trial and error. Look at the table below.

It is a good habit to become familiar with the squares of the numbers from 0‒10, as these arise frequently in mathematics. If you can remember those square numbers--or if you can use a calculator to find them--then finding many common square roots will be just a matter of recall.

You can use the same formula to find the length of a right triangle's leg if you are given measurements for the lengths of the hypotenuse and the other leg. Consider the example below.

The Pythagorean Theorem is perhaps one of the most useful formulas you will learn in mathematics because there are so many applications of it in real world settings. Architects and engineers use this formula extensively when building ramps, bridges, and buildings. Look at the following examples.

The Pythagorean Theorem states that in any right triangle, the sum of the squares of the lengths of the triangle's legs is the same as the square of the length of the triangle's hypotenuse. This theorem is represented by the formula . Put simply, if you know the lengths of two sides of a right triangle, you can apply the Pythagorean Theorem to find the length of the third side. Remember, this theorem only works for right triangles.

This reading is taken from the Developmental Math Open Program created by The NROC Project. It is available under a Creative Commons license.