Pythagorean Triplets

How to identify and create pythagorean triplets or triples

When the side lengths of a right triangle satisfy the pythagorean theorem, these three numbers are known as pythagorean triplets or triples.

    The most common examples of pythagorean triplets are
  • 3,4,5 triangles
    • a 3,4,5 triplet simply stands for a triangle that has a side of length 3, a side of length 4 and a side of length 5.
    • If a triangle has these side lengths, then it MUST be a right triangle.
  • 5, 12, 13 right triangles
  • 7, 24, 25 right triangles
  • 8, 15, 17 triangles

NOTE: the triplets above such as 3,4,5 represent the ratios of side lengths that satisfy the pythagorean theorem.

Therefore, you can create other triplets by multiplying any of these triplets by a number.

  • (3,4,5) ×2 = 8,6,10 and 8,6,10 is also a pythagorean triplet
  • (5,12,13) ×2 = 10,24,26 and 10,24,26 is also a pythagorean triplet

Any multiple of the ratios above represent the sides of a right triangle. In other words just as 3,4,5 represents the

The Babylonians came up with a general formula for creating pythagorean triplets. This formula represents triplets for any U and V:

Hypotenuse Larger leg Smaller Leg
U² + V ² U² − V² 2UV
As an example if you make u = 2 and V= 1, you will get the 3,4,5 triplets
4 +1 = 5 4 − 1 = 3 2(2)(1) = 4

Hints for creating unique triplets using the Babylonian formula:

  • U and V should not have any common factors
  • U and V should not both be ODD
  • U and V should not both be EVEN

Therefore, to create a unique triplet, chose an even and an odd number for U and K and make sure that the numbers do not have common factors

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