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Pythagorean Triples (Triplets)Pythagorean Triples (Triplets) : In this section we consider some combinations of whole numbers that satisfy the Pythagorean Theorem. Knowing these combinations is not essential, but knowing some of them can save you appreciable time and effort.Definition : Any three whole numbers that satisfy the equation a ^{2} + b ^{2} = c ^{2} form a Pythagoras triplets. These four triangles are all members of the (3,4,5) family. For example, the triple (6,8,10) is ( 3 x 2, 4 x 2 , 5 x 2). Even though the last triangle, ( 3√3, 4√3 , 5√3), is a member of the (3,4,5) family, the measures of its sides are not a Pythagorean triplets because they are not whole numbers. Other common families are (5,12,13) , of which (15,36,39) is another member. (7,24,25), of which (14,48,50) is another member. (8,15,17), of which (4, 7.5,8.5) is another member. There are infinitely many families, including (9,40,41),(11,60,61), (20,21,29) and (12,35,37), but most are not used very often. Examples : 1) Find AB . Solution : By Pythagoras triplets method (10,24,?) belongs to the (5,12,13) family. 10 = 5 x 2 24 = 12 x 2 So, AB = 13 x 2 = 26 The following problems shows how a knowledge of Pythagoras triplets can be useful even in situations where their applicability is not immediately apparent. 1) Given : The right triangle shown Find : y The fraction may complicate our work, we may not wish to complete a long calculation to solve 4 ^{2} + (7½) ^{2} = y ^{2} . An alternative method is to find a more easily recognized member of the same family. We multiply each side by the denominator of the fraction ,2. 4 x 2 = 8 ; 7½ x 2 ; 2y Clearly the family is (8,15,17) ∴ 2y = 17 y = 17/2 y = 8½ Pythagorean Theorem • Introduction of Pythagorean Theorem • Converse of Pythagorean Theorem • Pythagorean Triples • Application of Pythagorean Theorem • Proof on Pythagorean Theorem
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