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):

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   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.