Question

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

Answer 1

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 and 9` cm, then the hypotenuse will be:

`c^2 = 6^2 + 9^2 = 117`

`c = sqrt117 = 10.8166538.........`

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 and 4` cm, then the hypotenuse is:
`c^2 = 3^2+4^2 = 25`
`c = sqrt25 = 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.