Pythagorean Triples
Pythagorean Triples To begin you must understand the Pythagoras theorem is an equation of a2 + b2 = c2. This simply means that the sum of the areas of the two squares formed along the two small sides of a right angled triangle equals the area of the square formed along the longest. Let a, b, and c be the three sides of a right angled triangle. To define, a right angled triangle is a triangle in which any one of the angles is equal to 90 degrees. The longest side of the right angled triangle is called the 'hypotenuse '. Once you have this basic understanding you can apply the understanding that if a, b, and c are positive integers, they are called Pythagorean Triples. Our textbook explains that “the numbers 3, 4, and 5 are called Pythagorean triples since 32 + 42 = 52” (Bluman, pg. 522) and that there is a set of formulas that will generate an infinite number of Pythagorean triples. Here are some examples: | | | 3,4,5 Triangle 5,12,13 Triangle 9,40,41 Triangle | | | 32 + 42 = 52 52 + 122 = 132 92 + 402 = 412(Ganesh, 2010.) | | | The set of Pythagorean Triples is endless. Let n be any integer greater than 1 as in this example which is a set of Pythagorean triple, is true because: (3n)2 + (4n)2 = (5n)2 Some other Pythagorean Triples are (5,12,13); (7,24,25); (8,15,17); (9,40,41); (17,144,145); and (25,312,313). | | | To summarize, it is fairly easy to show that no matter what numbers you use for n you can use this method to show that there are infinitely many Pythagorean Triples. Since the Pythagorean Theorem states that every right triangle has side lengths satisfying the formula a2 + b2 = c2; the Pythagorean triples describe the three integer side lengths of a right triangle.
References:
Bluman, A. G. (2005). Mathematics in Our World (1st Ed.) Ashford University, Custom Edition. New York: McGraw-Hill.
Ganesh, J. (2010). Pythagorean Triples
References:
Bluman, A. G. (2005). Mathematics in Our World (1st Ed.) Ashford University, Custom Edition. New York: McGraw-Hill.
Ganesh, J. (2010). Pythagorean Triples
References: Bluman, A. G. (2005). Mathematics in Our World (1st Ed.) Ashford University, Custom Edition. New York: McGraw-Hill. Ganesh, J. (2010). Pythagorean Triples –Advanced. Retrieved on June 27, 2010 from http://www.mathsisfun.com/numbers/pythagorean-triples.html Examples and Calculations: