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Pythagorean Triples

In this lesson, we will learn

Pythagorean Triples
Families of Pythagorean Triples
Pythagorean Triples and Right Triangles
Solving Problems using the Pythagorean Triples
How to generate Pythagorean Triples

Pythagorean Triples

Pythagorean triples are formed by positive integers a, b and c, such that a² + b² = c². We may write the triple as (a, b, c)

For example, the numbers 3, 4 and 5 form a Pythagorean Triple because 3² + 4² = 5². There are infinitely many Pythagorean triples.

Some examples are

( 3 , 4 , 5 )	( 5, 12, 13)	( 7, 24, 25)	( 8, 15, 17)
( 9, 40, 41)	(11, 60, 61)	(12, 35, 37)	(13, 84, 85)
(16, 63, 65)	(20, 21, 29)	(28, 45, 53)	(33, 56, 65)
(36, 77, 85)	(39, 80, 89)	(48, 55, 73)	(65, 72, 97)

Families of Pythagorean Triples

If we multiply each number of a Pythagorean triple by the same number, we form another Pythagorean triple.

For example, (6, 8, 10) is a family of the Pythagorean triple (3, 4, 5) because it can be obtained by 2 × 3 = 6, 2 × 4 = 8, 2 × 5 = 10. If we multiply (3, 4, 5) by 3, we get another triple (9, 12, 15). This can be repeated with different multiples.

Similarly, the family for (5, 12, 13) would be (10, 24, 26), (15, 36, 39) and so on.

Pythagorean Triples and Right Triangles

When we make a triangle with sides whose lengths are the Pythagorean Triples, the triangle will form a right triangle. This follows from the converse of the Pythagorean Theorem.

5-12-13 triangles

Memorising some common Pythagorean triples can helpful.

For example, if we see that the sides of a triangle form a Pythagorean triple then we know that it is a right triangle.

The following video gives some examples of Pythagorean triples and right triangles.

Solving problems using the Pythagorean Triples

Pythagorean triples may also help us to find the missing side of a right triangle faster. If two sides of a right triangle form part of a triple then we can know the value of the third side without having to calculate using the Pythagorean theorem.

Example:

Find the value of x