Pythagorean TriplesDate: 10/07/97 at 05:35:01 From: Al Subject: Pythagorean triples What is the definition of a Pythagorean Triple? We haven't covered it (or not under that name). Thanks, Al Date: 10/07/97 at 08:33:36 From: Doctor Jerry Subject: Re: Pythagorean triples Hi Al, A set {x,y,z} of positive integers is a Pythagorean Triple if the square of one of them is equal to the sum of the squares of the other two. So, {3,4,5} is a Pythagorean Triple. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 10/08/97 at 23:15:13 From: Doctor Chita Subject: Re: Pythagorean triples Dear Al: I think to answer your question, we should first review the Pythagorean theorem and its converse. The Pythagorean theorem says that if you have a right triangle with sides a, b, and c, and c is the hypotenuse, then a^2 + b^2 = c^2. You can then use this theorem to find the length of one side if you know the lengths of the other two sides. The converse of the Pythagorean theorem says that if you have a triangle whose sides are a, b, and c, and c is the longest side, and if a^2 + b^2 = c^2, then the triangle is a right triangle. A Pythagorean triple is a set of three integers that satisfy the Pythagorean theorem. That is, if you substitute the given sides of a triangle a, b, and c (with c the longest side) into the equation a^2 + b^2 = c^2, the equation will be true. It's an application of the converse of the Pythagorean theorem. For example, 3, 4, and 5 is the simplest triple because 3^2 + 4^2 = 5^2. Therefore, a triangle whose sides are 3, 4, and 5 is a right triangle. The numbers 2, 3, and 4 are not a Pythagorean triple because 2^2 + 3^2 does not equal 4^2. There is a way to generate sets of Pythagorean triples using algebraic expressions. Let a and b be two positive integers with a > b. Then 2ab, a^2 - b^2 represent the legs and a^2 + b^2 represents the hypotenuse of a right triangle. For example, if a is 2 and b is 1, then 2ab = 4, a^2 - b^2 = 3, and a^2 + b^2 = 5. By choosing different values for a and b, you can produce many triples. Some will be multiples of others. For example, if a = 3 and b = 1, then the triple is 6, 8, and 10. This triple represents the sides of a right triangle that is similar to a 3-4-5 triangle with sides that are twice as long. Play with different values of a and b and see how many triples you can come up with. How many are relatively prime - that is, how many are not multiplies of simpler triples? If you want some algebra practice, use the Pythagorean theorem to prove that 2ab, a^2 - b^2, and a^2 + b^2 are the sides of a right triangle. Have fun! -Doctor Chita, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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