# Pythagorean Theorem!

### Because irrational numbers DO exist...

## History

A long long time ago, not in a galaxy far away, in ancient Greece, a mathematician named Pythagoras lived. Pythagoras lived from about 569-500 B.C.E. In his lifetime, he created a group of followers called the Brotherhood of Pythagoras. They were devoted to studying mathematics. Pythagoras and his group of followers believed that "Numbers ruled the universe." One of the mathematical ideas Pythagoras came up with is the Pythagorean theorem. He noticed that there was a special relationship between the sides of a right triangle and came up with a formula to describe that relationship.

## The theorem.

The Pythagorean theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. In other words, for a right triangle with legs a,b and hypotenuse c,

Pythagoras in 2 minutes 2

## So what do we need in order to use this theorem?

## A right triangle. The only time we can use the Pythagorean theorem is with right triangles. The reason this occurs is because there is a unique relationship between the three sides of the triangle. If the relationship was not equal, what kind of triangle do you think we would have? Why? | ## The measure of two sides. With our formula, we have to know the lengths of two of the sides in order to find out the third side. If we know two of the sides, we can substitute the known values into the formula in order to solve for the unknown variable. We can use any two sides, so we could know he value of both legs or we could know the value of one leg and the hypotenuse. | ## In some cases, if we know the angles, life gets easy. In some cases, we are given a very special right triangle. Such triangles are 30-60-90 triangles (shown above) and isosceles right triangles (where the angle measures are 45-45-90). For these special cases, there is a pattern and we only need to know one of the sides. |

## A right triangle.

The only time we can use the Pythagorean theorem is with right triangles. The reason this occurs is because there is a unique relationship between the three sides of the triangle. If the relationship was not equal, what kind of triangle do you think we would have? Why?

## The measure of two sides.

With our formula, we have to know the lengths of two of the sides in order to find out the third side. If we know two of the sides, we can substitute the known values into the formula in order to solve for the unknown variable. We can use any two sides, so we could know he value of both legs or we could know the value of one leg and the hypotenuse.

## In some cases, if we know the angles, life gets easy.

In some cases, we are given a very special right triangle. Such triangles are 30-60-90 triangles (shown above) and isosceles right triangles (where the angle measures are 45-45-90). For these special cases, there is a pattern and we only need to know one of the sides.

## Practice makes perfect! Do a few examples on your own!

## Common Questions

**How can I prove the Pythagorean theorem?**

There are MANY ways to prove this theorem. Here is a website that shows the various ways to prove this fundamental theorem.

http://www.cut-the-knot.org/pythagoras/index.shtml

**What if I'm only given one side of the triangle?**

Unless you have a special triangle (30-60-90 or isosceles) you will not be able to use the Pythagorean theorem.

**Can I use this for all triangles?**

No. Only right triangles.

## Now put it all together!

Colin Dodds - Pythagorean Theorem (Math Song)