PYTHAGOREAN THEOREM

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!

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