Pythagorean Triples Using the Following Expressions

Pythagorean triples using the following expressions
Takeshia Palmer
ABO1220D
06/11/2012
Michael Hammoud



Pythagorean triples ar sets of tercet integers that represent the sides of a right triangle. Some of the some well-known primitive Pythagorean triples are (3, 4, 5), (5, 12, 13) and (8, 15, 17). (Primitive means that you tail endnot drainage area each number by a common factor, i.e. the GCD = 1.)You provide verify that these give the sides of a right triangle by using the Pythagorean Theorem:
a² + b² = c²,
You can have Pythagorean triples using the following expressions:
Pick two positive integers, m and n, with m less than n.




Then the three numbers that form the Pythagorean triple can be calculated from:
n² - m²
2mn
n² + m²
Examples:
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
decide:
(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 = four hundred
400 = 400
A remarkable fact is that there are continuously many primitive Pythagorean triples. But how can you generate them all? It turns out there are two soft methods for creating new Pythagorean triangles.




References
Bluman, A. G. (2011). Mathematics in our world (1st ed. Ashford University...If you wish to get a full essay, order it on our website: Orderessay



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