Math Week 4

Pythagorean Triples

July 30, 2012

Pythagorean Triples

In the projects section in mathematics in our world it says the numbers 3, 4, and 5 are called Pythagorean triples since 32+41=52 (Bluman, 2005, p.522). The numbers 52+122=132 are also Pythagorean triples. The projects section also asks to find other Pythagorean triples. In fact, there is a set of formulas that will generate an infinite number of Pythagorean triples. A Pythagorean is a set of positive integers a, b, and c (Bluman, 2005, p.522). A right triangle exists with legs a, b, and hypotenuse c. A Pythagorean triple is a triple of positive integers where a2+b2=c2. This is an example of a right triangle where the sides are positive integers. In the following project I will find five Pythagorean triples and complete the mathematical steps.

Pick two positive integers, m and n, with m less than n. Then pick the three numbers from the Pythagorean triple can be calculated from:

n²-m²

2mn

n² + m²

1) m=3, n=4

n²-m²= (4)²-(3)²=16-9=7

2mn=2(3) (4) =24

n² + m² = (4)² + (3)² = 16 + 9 = 25

Triple: 7, 24, 25

Check:

(7)²+ (24)²= (25)²

49+576=625

625 = 625

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

Check:

(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

Check:

(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

Check:

(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

Triple:12,16,20

Check:

(12)²+(16)²=(20)²

144+256=400

400 = 400

In conclusion, I have selected two positive integers, m and n, with m less than n. In doing so I was able to select five different Pythagorean triples 7,24,25 , 6,8,10 , 9,40,41, 11,60,61, 12,16,20, and complete all the work required....