Pythagorean triples are sets of three numbers which represent measurements for the lengths of the sides of right triangles. They stem from Pythagoras Theorem which was developed to find the missing length of one side of a right triangle when the other lengths are known. Even though several civilizations already had the knowledge of the triples, it was not until one thousand years later when Pythagoras proved the theorem.

Pythagoras Theorem states that the squares of both sides of a right triangle are equal to the square of its hypotenuse. Therefore, if you were to imagine that each side of a right triangle formed squares, the squares formed on the legs would be equal in area to the square formed on the hypotenuse.

The formula used in this theorem is a²+b²=c². The theorem in this form is helpful in finding a missing side length when the other two lengths are known. There are a few professions for which this knowledge comes in handy. In ancient times, the formula was used to build strong foundations and the pyramids. Today, architects use it for the same purposes. Contractors and other developers also use it to layout floor plans and erect walls and roofs.

There is another side to this mathematical method. That is, finding new lengths for the sides to design different shaped and scaled right triangles. To do this, a series formulas is necessary. First, use the terms m and n, where both are always integers and m is always valued at 1 less than n. Next, calculate each side (a, b, c) in terms of m and n. To find sets of Pythagorean triples, a=m+n; b=2mn; and c=m²+n². Other than the basic 3²+4²=5² that all geometry students encounter at least once in their educational journey, others can be found using this series of formulas. Some other primitive sets include the following: 7, 12, and 25; 27, 364, and 365; and 35, 612, and 613.

Using this series of formulas will produce primitive Pythagorean triples. Prime sets, or primitive sets, can then be scaled by multiplying the lengths of each side by the desired number. For example, the triple 3, 4, 5 can be multiplied by 3 to get 9, 12, and 15. Using these triples, one can create equivalent right triangles of differing sizes.

References:

Bluman, A.G. (2005) Mathematics in our world. McGraw-Hill Companies: New York.

Lindauer, H. (1998). The Pythagorean theorem. Retrieved May 30, 2009 from http://www.ms.uky.edu/~lee/ma502/pythag/pythag.htm.

Ganesh, J. (2006). Pythagorean triples – advanced. Retrieved May 30, 2009 from http://www.mathsisfun.com/numbers/pythagorean-triples.html.