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Discovering Pythagorean Triples

The Pythagorean Theorem is certainly one of the most famous theorems in all of mathematics. Mathematicians and lay people alike have studied it for centuries, and people have proved it in many different ways. (Even President James Garfield was credited with a new, original proof.) So without further ado, here it is:

The Pythagorean Theorem: The sum of the squares of the legs (the two shortest sides) of a right triangle is equal to the square of the hypotenuse (the longest side).

If you pick any old numbers for two of the sides of a right triangle, the third side usually ends up being irrational — you know, the square root of something. For example, if the legs are 5 and 8, the hypotenuse ends up being the square root of 89, or approximately 9.43398 . . . (the decimal goes on forever without repeating). And if you pick whole numbers for the hypotenuse and one of the legs, the other leg usually winds up being the square root of something.

When this doesn't happen — namely, when all three sides are whole numbers — you have a Pythagorean triple.

Pythagorean Triple: A Pythagorean triple (like 3-4-5) is a set of three whole numbers that work in the Pythagorean Theorem and can thus be used for the three sides of a right triangle.

The four smallest Pythagorean triple triangles are the 3-4-5 triangle, the 5-12-13 triangle, the 7-24-25 triangle, and the 8-15-17 triangle — but infinitely more of them exist. If you're interested, one simple way to find more of them is to take any odd number, say 11, and square it — that's 121. The two consecutive numbers that add up to 121 (60 and 61) give you the two other numbers (to go with 11). So another Pythagorean triple is 11-60-61.

A family of right triangles is associated with each Pythagorean triple. For example, the 5 : 12 : 13 family consists of the 5-12-13 triangle and all other triangles of the same shape that you'd get by shrinking or blowing up the 5-12-13 triangle. Just multiply the length of each side by the same number. For example, multiply each side by 0.5 and you get a 2.5-6-6.5 triangle. Or you can quadruple each side and get a 20-48-52 triangle.

Understanding the Pythagorean triple families of triangles is important because they come up in so many right triangle problems.

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