# What Does the Pythagorean Theorem State?

Formula

### PythagoreanTheorem

$\begin{array}{llll}\hfill {\text{leg}}_{1}^{2}+{\text{leg}}_{2}^{2}& ={\text{hypotenuse}}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {a}^{2}+{b}^{2}& ={c}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Pythagoras lived from about 572 to 497 BC. He was born on the Greek island of Samos in the Aegean Sea. The equation you see above is named after Pythagoras, although it was already known among the Babylonians 1000 years before his time.

In the figure above, you see a right triangle where a square is drawn from each side of the triangle. This figure illustrates what the theorem is really saying.

In the formula box above it says “leg one squared plus leg two squared is equal to the hypotenuse squared”. You know from previous lessons that when you raise a length to the power of 2 (that is, when you square it), you will find the area of a square. The side of that square is the length that was squared.

Therefore, the Pythagorean theorem says that if you take the area of the square of one leg and add that to the area of the square of the second leg, their combined area is equally as large as the area of the square of the hypotenuse (the longest side length). When you add the sizes of the areas, you get this formula:

 ${\text{leg}}_{1}^{2}+{\text{leg}}_{2}^{2}={\text{hypotenuse}}^{2}$

If you know three sides of a triangle, you can use this formula, and see whether it’s fulfilled, to check if the triangle is a right triangle. You can also use the Pythagorean theorem to find an unknown length of a side, given the lengths of the two other sides in a right triangle.

When we work with the Pythagorean theorem, we often write the answer as just a positive solution, even though we’re working with powers of 2, which often have positive and negative solutions. Why do we not include the negative solution?

Since we are talking about the length of the sides of a triangle, it makes no sense to say that it is, say, “minus four meters long”. In such a case, where would it even be? Would it have disappeared into a warp hole, into another dimension? Since warp holes only exist in science fiction and video games, this makes no sense in reality. So we only use the positive answer when we talk about actual units of measure.

A Pythagorean number triple is a collection of three integers—or whole numbers—that satisfy the Pythagorean theorem, such as $\left(3,4,5\right)$, $\left(5,12,13\right)$ and $\left(8,15,17\right)$.