Pythagorean Triples
Pythagorean Triples
Ashley Walker
MAT126
Bridget Simmons
November 28, 2011
A Pythagorean triple is a triple of positive integers a, b, and c such that a right triangle exists with legs a, b, and hypotenuse c (Bluman, 2005). A Pythagorean triple is a triple of positive integers (a, b, c) where a2 + b2 = c2. A triple is simply a right triangle whose sides are positive integers. An easy way to generate Pythagorean triples is to multiply any known Pythagorean triple by an integer (any integer) (Vargas, 2008).
In project #4, pg. 522, (Mathematics in Our World) introduced some new information to add to my mathematics knowledge of numbers. The numbers 3, 4, and 5 are called Pythagorean triples since 32 + 42 = 52. The numbers 5, 12, and 13 are also Pythagorean triples since 52 + 122 = 132. Can you find any other Pythagorean triples? Actually, there is a set of formulas that will generate an infinite number of Pythagorean triples and write a brief report on the subject (Bluman, 2005).
When asked to find any other Pythagorean triples, I found 5, choosing 2 integers, m and n, with m less than n (Manuel, 2010). Three formulas I choose to form the Pythagorean triple, which can be calculated from: n2 – m2
2mn
n2 + m2 (Manuel, 2010).
Project 4, pg.522 1) m=3, n=4 n2– m2=(4)2–(3)2=16-9=7 VERIFICATION: (7)2+(24)2=(25)2
2mn=2(3)(4)=24 49+576=625 n2+m2=(4)2+(3)2=16+9=25 625=625
TRIPLE: 7, 24, 2 2) m=1, n=3 n2-m2=(3)2-(1)2=9-1=8 2mn=2(1)(3)=6 n2+m2=(3)2+(1)2=9+1=10 TRIPLE: 6,8,10
VERIFICATION: (6)2+(8)2=(10)2 36+64=100 100=100 3) m=4, n=5 n2-m2=(5)2=(4)2=25-16=9 2mn=2(4)(5)=40 n2+m2=(5)2+(4)2=25+16=41 TRIPLE: 9,40,41
VERIFICATION: (9)2=(40)2=(41)2 81+1600=1681 1681=1681 4) m=5, n=6 n2-m2=(6)2-(5)2=36-25=11 2mn=2(5)(6)=60 n2+m2=(6)2+(5)2=36+25=61 TRIPLE: 11,,60,61
VERIFICATION: (11)2=(60)2=(61)2 121+3600=3721 3721=3721 5) m=2, n=4 n2-m2=(4)2-(2)2=16-4=12 2mn=2(2)(4)=16
n2+m2=(4)2+(2)2=16+4=20
Ashley Walker
MAT126
Bridget Simmons
November 28, 2011
A Pythagorean triple is a triple of positive integers a, b, and c such that a right triangle exists with legs a, b, and hypotenuse c (Bluman, 2005). A Pythagorean triple is a triple of positive integers (a, b, c) where a2 + b2 = c2. A triple is simply a right triangle whose sides are positive integers. An easy way to generate Pythagorean triples is to multiply any known Pythagorean triple by an integer (any integer) (Vargas, 2008).
In project #4, pg. 522, (Mathematics in Our World) introduced some new information to add to my mathematics knowledge of numbers. The numbers 3, 4, and 5 are called Pythagorean triples since 32 + 42 = 52. The numbers 5, 12, and 13 are also Pythagorean triples since 52 + 122 = 132. Can you find any other Pythagorean triples? Actually, there is a set of formulas that will generate an infinite number of Pythagorean triples and write a brief report on the subject (Bluman, 2005).
When asked to find any other Pythagorean triples, I found 5, choosing 2 integers, m and n, with m less than n (Manuel, 2010). Three formulas I choose to form the Pythagorean triple, which can be calculated from: n2 – m2
2mn
n2 + m2 (Manuel, 2010).
Project 4, pg.522 1) m=3, n=4 n2– m2=(4)2–(3)2=16-9=7 VERIFICATION: (7)2+(24)2=(25)2
2mn=2(3)(4)=24 49+576=625 n2+m2=(4)2+(3)2=16+9=25 625=625
TRIPLE: 7, 24, 2 2) m=1, n=3 n2-m2=(3)2-(1)2=9-1=8 2mn=2(1)(3)=6 n2+m2=(3)2+(1)2=9+1=10 TRIPLE: 6,8,10
VERIFICATION: (6)2+(8)2=(10)2 36+64=100 100=100 3) m=4, n=5 n2-m2=(5)2=(4)2=25-16=9 2mn=2(4)(5)=40 n2+m2=(5)2+(4)2=25+16=41 TRIPLE: 9,40,41
VERIFICATION: (9)2=(40)2=(41)2 81+1600=1681 1681=1681 4) m=5, n=6 n2-m2=(6)2-(5)2=36-25=11 2mn=2(5)(6)=60 n2+m2=(6)2+(5)2=36+25=61 TRIPLE: 11,,60,61
VERIFICATION: (11)2=(60)2=(61)2 121+3600=3721 3721=3721 5) m=2, n=4 n2-m2=(4)2-(2)2=16-4=12 2mn=2(2)(4)=16
n2+m2=(4)2+(2)2=16+4=20
References: Bluman, A.G. (2005). Mathematics in Our World. New York: McGraw Hill Vargas, S.P. (2008). Pythagorean Triple. Retrieved November 28, 2011 from http:// www.mathworld.wolfram.com/pythagroean Manuel, K.P. (2010). Comment on Pythagorean Triples. Retrieved November 28, 2011 from http://www.solving-math-problems.com/pythagroean-triples-comments.html