The Pythagorean Theorem states that, given a right triangle with side lengths a, b, and c (where a and b are the legs of the triangle, and c is the hypotenuse),
Of all the math formulas you ought to learn, here is the one you absolutely MUST know for Mathcounts. Learn it. Love it. It will show up again, and again, and again, and again. If you see a right triangle anywhere, the Pythagorean Theorem should immediately spring to mind.
The main usefulness of the Pythagorean Theorem is that if you know the lengths of any two sides of a right triangle, you can find the third side. Make sure, however, that the triangle is a right triangle--not just any ol' triangle will do!
Pythagorean triples are sets of whole-number side lengths that satisfy the equation. For example, (3, 4, 5) is a Pythagorean triple because
32 + 42 = 52
and 3, 4, and 5 are all whole numbers.
It helps to know some common Pythagorean triples because Mathcounts problem-writers use them a lot:
Also note that you can "scale up" these Pythagorean triples by multiplying all three numbers by the same constant. The result will still be a Pythagorean triple. For example, if you double the numbers in (3, 4, 5), you will get a new Pythagorean triple: (6, 8, 10).