Pythagorean triples - Essay Example

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It is about a property of all right triangles showing a relationship between their sides given by the equation c2 = a2 + b2; that is, the square of the longest side is equal to the…
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PYTHAGOREAN TRIPLES The Pythagorean Theorem is probably one of the first principles learned in a Trigonometry It is about a property of all right triangles showing a relationship between their sides given by the equation c2 = a2 + b2; that is, the square of the longest side is equal to the sum of the squares of the remaining two sides (Knott 2009). Hence, given any two sides of a right triangle (triangle with a 90-degree angle) the unknown side can always be solved by algebraically manipulating the equation presented above.
A special case of such triangles is when all the three sides, a, b and c, are integers, thus generating a Pythagorean Triple (Bogomolny 2009). The most popular example of which is the 3-4-5 triangle, the triple which, according to Knott (2009), was known to the Babylonians since way back 5,000 years and was possibly used as a basis in making true right angles in ancient building construction.
Then again, the 3-4-5 triangle is just one of the infinitely many Pythagorean Triples, and mind you, there are various ways of generating such triples. One is, given two integers n and m, where n > m, then sides a, b and c are define as n2 - m2, 2nm and n2 + m2, respectively, following a simple proof (Bogomolny 2009):
Taking for instance the triple 8-15-17, which is generated by taking n = 4 and m = 1, then a = n2 - m2 = 4­­2 - 12 = 16 - 1 = 15; b = 2mn = 2(1)(4) = 8, and; c = n2 + m2 = 4­­2 + 12 = 16 + 1 = 17. Another example is 7-24-25, which can be verified using n = 4 and m = 3. Such triples are examples of Primitive Pythagorean Triples, or those triples that are not multiples of another and are found using the n-m formula (Knott 2009). Other Pythagorean Triples can be found using a variety of methods as presented by Bogomolny (2009) and Knott (2009), some of which are:
a) by generating multiples of Primitive Pythagorean Triples, ka, kb, kc: 2 * (3,4,5) = 6, 8, 10;
b) by taking a series of multiples 1, 11, 111, …: 3-4-5, 33-44-55, 333-444-555, …;
c) by Two-fractions method—choose any two fractions whose product is 2, add 2 to each fraction, then cross multiply, getting the two shorter sides of the triple: 4/2, 2/2 → 8/2, 6/2 → 16, 12 → 162 + 122 = 202, and;
d) by taking (m+1) for n, with m as powers of 10, hence simplifying the triples into 2m+1, 2m(m+1), 2m2+2m+1: 2(10) + 1 = 21, 2(10)(10+1) = 220, 2(10)2 + 2(10) + 1 = 221.
To sum it up, there are infinitely many Pythagorean Triples existing. But one thing is for sure, a variety of techniques are available that will serve useful in generating patterns among such triples. Hence, if you cannot list them all, be familiar of their patterns at least.
Bogomolny, A. (2009). Pythagorean Triples. Retrieved November 20, 2009, from Interactive Mathematics Miscellany and Puzzles Web site:
Knott, R. (2009). Pythagorean Triangles and Triples. Retrieved November 20, 2009, from The University of Surrey, Mathematics Web site: Read More
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