"Proof of the Pythagorean Theorem "

What is the Pythagorean Theorem?

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.