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Pythagorean Triples

Date: 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!   

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!   
Associated Topics:
High School Geometry
High School Number Theory
High School Triangles and Other Polygons

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