Pythagorean Theorem Algebra Proof

What is the Pythagorean Theorem?

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

triangle abc

The Pythagorean theorem says that, in a right triangle, the square of a (which is a×a, and is written 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):

Squares and Triangles

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:
c2
Each of the four triangles has an area of:
ab2
So all 4 of them together is:
4ab2 = 2ab
The tilted square and the 4 triangles together is:
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 Pythagorean 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 theorem.

This proof came from China over 2000 years ago!

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