# Pythagorean Theorem

The **Pythagorean Theorem** states that for a right triangle with legs of length and and hypotenuse of length we have the relationship . This theorem has been know since antiquity and is a classic to prove; hundreds of proofs have been published and many can be demonstrated entirely visually(the book *The Pythagorean Proposition* alone consists of more than 370). The Pythagorean Theorem is one of the most frequently used theorems in geometry, and is one of the many tools in a good geometer's arsenal. A very large number of geometry problems can be solved by building right triangles and applying the Pythagorean Theorem.

This is generalized by the Pythagorean Inequality and the Law of Cosines.

## Contents

## Proofs

In these proofs, we will let be any right triangle with a right angle at .

### Proof 1

We use to denote the area of triangle .

Let be the perpendicular to side from .

Since are similar right triangles, and the areas of similar triangles are proportional to the squares of corresponding side lengths,

.

But since triangle is composed of triangles and , , so . ∎

### Proof 2

Consider a circle with center and radius . Since and are perpendicular, is tangent to . Let the line meet at and , as shown in the diagram:

Evidently, and . By considering the power of point with respect to , we see

. ∎

### Proof 3

and are squares.

. ∎

## Common Pythagorean Triples

A Pythagorean Triple is a set of 3 positive integers such that , i.e. the 3 numbers can be the lengths of the sides of a right triangle. Among these, the Primitive Pythagorean Triples, those in which the three numbers have no common divisor, are most interesting. A few of them are:

Also Pythagorean Triples can be created with the a Pythagorean triple by multiplying the lengths by any integer.
For example,
Note that (-1,0,1) and (3,4,5) are the only pythagoren triplets that consist of consecutive integers.

Also, if (a,b,c) are a pythagorean triplet it follows that (ka,kb,kc) will also form a pythagorean triplet for any constant k.

k can also be imaginary.

## Problems

### Introductory

### Sample Problem

Right triangle has legs of length and . Find the hypotenuse of .

#### Solution 1 (Bash)

.

#### Solution 2 (Using 3-4-5)

We see looks like the legs of a right triangle with a multiplication factor of 111. Thus .

### Another Problem

Right triangle has side lengths of and . Find the sum of all the possible hypotenuses.

#### Solution (Casework)

Case 1:

3 and 4 are the legs. Then 5 is the hypotenuse.

Case 2:

3 is a leg and 4 is the hypotenuse.

There are no more cases as the hypotenuse has to be greater than the leg.

This makes the sum .