In simple terms the Pythagorean Theorem states that the "square" of the hyponeneuse of a right triangle is equal to the sum of the two legs "squared".

To understand this better:

Make any right triange. Measure each side. Draw a square from each leg of the triange using the leg as one side of your square. Do the same with the hyponteneuse. In the center of each square write its area measurement. (it will be the the leg or hyponteneuse's side measurement times itself, or squared). Now add the two legs area measurement together. That number will add up to be the same number as the area of the square made by the hypoteneuse.

I would not be surprised if this is how the ancient Greek's discovered the relationship of the sides!

We'll start from the fact that if we'll construct a square whose one side lies on the hypotenuse of a right angle triangle, we'll notice that the area of this square is equal to the sum of areas of the squares whose one side lies on each leg of this triangle.

If we'll note the length of hypotenuse as`` z, the area of the square is `z^(2)`.

If the lengths of the legs are x and y, therefore the areas of the squares are `x^(2)` and `y^(2)` .

Therefore, according to Pyhtagorean theorem, we'llĀ get:

`x^(2)` + `y^(2)` = `z^(2)`

this is a very vague question, if you are looking for the history of the pythagorean theorem, then perhaps wikipedia might be useful to you.

have referenced a link to the wikipedia page on pythagorean theorem