# Pythagorean Theorem Algebra Proof

## What is the Pythagorean Theorem?

You can learn all about the Pythagorean theorem, but here is a quick summary:

The Pythagorean theorem says that, *in a right triangle,* the square of **a** (which is a×a, and is written **a ^{2}**) plus the square of

**b**(

**b**) is equal to the square of

^{2}**c**(

**c**):

^{2}a^{2} + b^{2} = c^{2}

## Proof of the Pythagorean Theorem using Algebra

We can show that **a ^{2} + b^{2} = c^{2}** using Algebra

Take a look at this diagram ... it has that "abc" triangle in it (four of them actually):

## Area of Whole Square

It is a big square, with each side having a length of **a+b**, so the **total area** is:

A = (a+b)(a+b)

## Area of The Pieces

Now let's add up the areas of all the smaller pieces:

^{2}

^{2}+ 2ab

## Both Areas Must Be Equal

The area of the **large square** is equal to the area of the **tilted square and the 4 triangles**. This can be written as:

(a+b)(a+b) = c^{2} + 2ab

NOW, let us rearrange this to see if we can get the Pythagorean theorem:

^{2}+ 2ab

^{2}+ 2ab + b

^{2}= c

^{2}+ 2ab

^{2}+ b

^{2}= c

^{2}

DONE!

Now we can see why the Pythagorean theorem works ... and it is actually a **proof** of the theorem.

This proof came from China over 2000 years ago!

There are **many** more proofs of the Pythagorean theorem, but this one works neatly.