Pythagorean Triples Using the Following Expressions

Pythagorean three-folds using the future(a) expressions Takeshia Palmer ABO1220D 06/11/2012 Michael Hammoud Pythagorean triples ar sets of tercet integers that represent the sides of a flop triangle. Some of the roughly well-known primitive Pythagorean triples be (3, 4, 5), (5, 12, 13) and (8, 15, 17). (Primitive means that you potty 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(prenominal) 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 mold: (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 review: (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 carbon 400 = 400 A remarkable fact is that in that respect are continuously many primitive Pythagorean triples. besides 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, stray it on our website: BestEssayCheap.com

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