The Pythagorean Triples

MAT 126

Instructor

March, 15, 2011

The Pythagorean Triples

The Pythagorean Theorem is a theory that was used by ancient civilizations and focuses on the relationships of all right triangles: that is, the sum of the squares of the legs of the triangle equals the area to the square of the hypotenuse. Triangles are characterized by the measures of their angles. A right has a 900 angle (Bluman, 2011). A Pythagorean Triple includes sets of three numbers, where the smallest set include the integers:3,4, and 5. Furthermore, if a and b are the legs of a right triangle, and c is the hypotenuse, then a formula such as a2 + b2=c2 satisfies what is known as the Pythagorean Triples.

There are many real life applications for the Pythagorean Theorem. For instance, as carpenters build frames for roofs, they must calculate the measurements of diagonal beams to support the roofs. The Pythagorean Theorem is used to find the proper dimensions of missing measurements. The formula is used to calculate many measurements including widths of rivers, facts at scenes of motor vehicle accidents, and heights of buildings.

To answer the problem raised in Chapter 10, Project 4 in our text, Mathematics in our world (Bluman, 2005), we must find a set formula that generates an undefined number of Pythagorean triples. They must be verified in the Pythagorean Theorem equation.

A known formula generates an infinite number of Pythagorean triples: a = (2mn); b= (m^2 - n^2), and c= (m^2 + n^2); m and n are positive integers and m > n. Five Pythagorean triples are included in the table listed below:

a b c

m

n

2mn

m2 – n2

m2 + n2

1 4 1 2(4*1) 8 16- 12 15 16+1 17

2 7 4 2(7*4) 56 49-16 33 47+16 65

3 8 2 2(8*2) 32 64-4 60 64+4 68

4 10 4 2(10*4) 80 100-16 84 100+16 116

5 23 6 2(23*6) 276 529-36 493 529+36 565

The following verifies each of the above Pythagorean triples, per the...