# What is the difference between the Pythagorean Theorem and Pythagorean Triples?

Apr 1, 2018

The theorem is a statement of fact about the sides of a right-angled tri9angle, and the triples are set of three exact values which are valid for the theorem.

#### Explanation:

The theorem of Pythagoras is the statement that there is a specific relationship between the sides of a right-angled triangle.

ie: ${a}^{2} = {b}^{2} + {c}^{2}$

In finding the length of a side, the last step involves finding a square root which is often an irrational number.

For example, if the shorter sides are $6 \mathmr{and} 9$ cm, then the hypotenuse will be:

${c}^{2} = {6}^{2} + {9}^{2} = 117$

$c = \sqrt{117} = 10.8166538 \ldots \ldots \ldots$

This theorem ALWAYS works, but the answers can be rational or irrational.

In some triangles, the sides work out to be exact answers. For example if the shorter sides are $3 \mathmr{and} 4$ cm, then the hypotenuse is:
${c}^{2} = {3}^{2} + {4}^{2} = 25$
$c = \sqrt{25} = 5$

The ratio $3 : 4 : 5$ is known as a Pythagorean Triple ... meaning a set of three values which works for Pythagoras' Theorem.

Some of the common triples are:

$3 : 4 : 5$
$5 : 12 : 13$
$7 : 24 : 25$
$8 : 15 : 17$
$9 : 40 : 41$
$11 : 60 : 61$

Notice that their multiples also work, so from $3 : 4 : 5$ we can get:
$6 : 8 : 10$
$9 : 12 : 15$
$12 : 16 : 20$
$15 : 20 : 25$ ... and so on.