Pythagorean Triples - Speech or Presentation Example

Pythagorean Triples A Pythagorean triple refer to a set of three positive integers a, b, c such that the sum of the squares of two smaller integers of them equal to the square of the third largest integer i.e. if c > a, b then a2 + b2 = c2, c2 – a2= b2, or, c2 – b2 = a2. Pythagorean triples are part of Pythagorean Theorem which says that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides 1. Each triple set has its many multiples which will be regarded as the same triple set i.e. the set (a, b, c) and its any multiple k, ka, kb, kc, will count as one set.

We are required to build the Pythagorean triples and to verify them through the Pythagorean Theorem. The Pythagorean triples (a; b; c) can be built through following method given positive integer x and y: 1a = 2xy b = x2 – y2 c = x2 + y2Using this formula, we can build many Pythagorean triples. Let’s take:x = 4; y = 3, then a = 2 (4) (3) = 24; b = (4)2 – (3)2 = 7; c = (4)2 + (3)2 = 25. The triple set is (7; 24; 25). It can be verified using the theorem: c2 = 72 + 242 = 49 + 576 = 625.

So, c = 25.x = 6; y = 5, then a = 2 (6) (5) = 60; b = (6)2 – (5)2 = 11; c = (6)2 + (5)2 = 61. The triple set is (11; 60; 61). It can be verified using the theorem: c2 = 112 + 602 = 3721. So, c = 61.x = 5; y = 4, then a = 2 (5) (4) = 40; b = (5)2 – (4)2 = 9; c = (5)2 + (4)2 = 41. The triple set is (9; 40; 41). It can be verified using the theorem: c2 = 92 + 402 = 1681. So, c = 41.x = 5 y = 2, then a = 2 (5) (2) = 20; b = (5)2 – (2)2 = 21; c = (5)2 + (2)2 = 29. The triple set is (20; 21; 29).

It can be verified using the theorem: c2 = 202 + 212 = 841. So, c = 29.x = 4; y = 1, then a = 2 (4) (1) = 8; b = (4)2 – (1)2 = 15; c = (4)2 + (1)2 = 17. The triple set is (8; 15; 17). It can be verified using the theorem: c2 = 82 + 152 = 64 + 225 = 289. So, c = 17.x = 8; y = 1, then a = 2 (8) (1) = 16; b = (8)2 – (1)2 = 63; c = (8)2 + (1)2 = 65. The triple set is (16; 63; 65). It can be verified using the theorem: c2 = 162 + 632 = 4225. So, c = 65.x = 6; y = 1, then a = 2 (6) (1) = 12; b = (6)2 – (1)2 = 35; c = (6)2 + (1)2 = 37.

The triple set is (12; 35; 37). It can be verified using the theorem: c2 = 122 + 352 = 1369. So, c = 37.That’s how the Pythagorean triples (integers) are generated and then can be verified through the theorem equation. To generate the triples, the given formula has been chosen among many other available formulas because comparing to other methods it is more generally applicable to many sets of integers that can build the vast range of Pythagorean triples. The Pythagorean Theorem can itself be applicable to different geometrical applications in many practical scenarios where we are required to set appropriate physical dimensions of different objects reflecting, or somehow, using the shapes of right triangles.