# 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 states that, in a right triangle, the square of a (a2) plus the square of b (b2) is equal to the square of c (c2):

a2 + b2 = c2

## Proof of the Pythagorean Theorem using Algebra

We can show that a2 + b2 = c2 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:

 First, the smaller (tilted) square has an area of A = c2 And there are four triangles, each one has an area of A =½ab So all four of them combined is A = 4(½ab) = 2ab So, adding up the tilted square and the 4 triangles gives: A = c2+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) = c2+2ab

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

 Start with: (a+b)(a+b) = c2 + 2ab Expand (a+b)(a+b): a2 + 2ab + b2 = c2 + 2ab Subtract "2ab" from both sides: a2 + b2 = c2

DONE!

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

This proof came from China over 2000 years ago!

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