Pythagoras... proved in an awesome way

Yeah, yeah, for both of the people who follow this blog, here is something pretty cool. Most people who do maths up to year 8 have seen pythagoras' theorem, but they have no idea how we know it works. Well here's one way of proving that it works. You can actually do this with paper, but because im so kind im going to do it on paint for you.

Ok so start with a piece of paper. Fold it into 4 so that you can cut a triangle off one of the corners. What happens is that you get 4 equally sized right angles triangles. Now arrange them into a square as shown below.

What you get is a neat little square inside a larger one. If we label one of the outside edges of the identical green triangles 'a' and one 'b', we can see that we get a big square of side length a+b. Also, if we label the hypotenuse (the longest side of the triangle) c, we can see that the area of the blue square in the middle is c²

Now, draw a square around the big square with a pen. What this proof relies on is that, no matter how much you move the green triangles around, the area inside the big square you drew with a pen will remain the same. If you're having trouble visualising it, get out the pen and paper and do it yourself, it really helps.

Ok so no matter how much you move the green triangles, the area inside the big square remains the same. Also, the blue area will remain the same (no matter how much you move the green triangles, the remaining area will remain constant). This means that i can rearrange the triangles. One way to do so is shown above.

Remember that before, the blue area was c². Now, the area is made up by 2 triangles of side length a and b, so the blue area is a²+b². Since the area of the blue parts remains constant, this means that the old area=the new area, or, written more nicely in a larger font and centred for your reading pleasure...

a²+b²=c²

Well, when i was shown this, I thought it was pretty cool. I mean, finally, a simple non-rigorous proof of something in maths! Of course, there's much more there, but at this stage im only just scratching the surface of it. I'll post some more when I have some more :)