Pythagoras Theorem

What is the Pythagorean Theorem?

Pythagoras (puh thag or us) was a brilliant Greek philosopher and mathematician, born in the sixth century, B.C. He and his followers tried to explain everything with numbers.

He was able to demonstrate that there is a special relationship among the sides of a right triangle. A right triangle is one that has two sides that form a right angle. The side opposite the right triangle is called the hypotenuse and the two shorter sides are called the legs. Here's a right triangle:

Notice that the sides of the triangle in the middle are 3, 4, and 5 units. The square formed by the leg of length 3 is 9. You could also say 3² = 9. The square formed by the leg of length 4 is 16. You could also say 4² = 16.

The sum of these two squares, 3² and 4², is equal to the square of the hypotenuse which is 25 or 5². Or, 3³ + 4² = 5².

Don't worry if you have to reread the previous few sentences to understand!

This relationship is only true for right triangles.

Here's what the Pythagorean (puh thag or ee un) Theorem says:

the sum of the squares of the lengths of the legs of any right triangle is equal to the square of the length of the hypotenuse.

This is often stated as a² + b² = c²

where a and b are legs and c is the hypotenuse.

Therefore if you know the length of any two sides of a right triangle you can determine the 3rd side.

Remember: the hypotenuse is always the LONGEST side in a right triangle.

Pythagoras Theorem in 2 Min:

Pythagoras Theorem Video:

Example

In the triangle below, what is the length of side a?

Solution:

You know from the Pythagorean Theorem that sum of the squares of the length of the two legs is equal to the square of the length of the hypotenuse. In this case we know the length of ONE leg and the hypotenuse and have to find the other leg. We know side b = 7cm and side c = 8cm but we don't know side a.

So you know that a² + 7² = 8² which is the same as a² + 49 = 64.

To isolate your variable, subtract 49 from each side of the equation.

a² + 49 - 49 = 64 - 49

If you know what a² is you'll need to find its square root in order to find the value of a.

a² = 15 so a = √15

Use your calculator to determine the √15 = 3.87.

So the length of side a is 3.87cm.

Is it a right triangle?

The Pythagorean Theorem can also be used to determine whether or not a triangle is a right triangle. In the image below you can see the area of the squares on each of the three sides. Is the angle (?) a 90⁰ angle?

Pythagorean Triple

A set of 3 whole numbers that satisfies the Pythagorean Theorem is called a Pythagorean triple.

Example: 3² + 4² = 5² OR 9+16=25

Another example of a Pythagorean Triple is 7-12-13 because:

5² + 12² = 13² OR 25 + 144 = 169