Quite recently, I received this comment on my Pythagoras’ Theorem post:

If I know the hypotenuse to be 20 inches what is formula that would tell me the must different combinations on the other (2) sides. They have to be whole numbers.

Thanks.

- james Creagh

If all three sides of a right-squared triangle are integers, then it is called a Pythagorean Triple. There are many ways to generate such triangles, but today I would write a short explanation to the question proposed above.

It is known for a fact that the simplest Pythagorean Triple is made up of sides with units 3, 4 and 5.

3² + 4² = 5²

This is written as (3,4,5). Multiples of this triangle can be generated using the formula:

(3n)² + (4n)² = (5n)²

So, you will have a combination of Pythagorean Triple as follows:

1 -> (3, 4, 5)
2 -> (6, 8, 10)
3 -> (9, 12, 15)
4 -> (12, 16, 20)
5 -> (15, 20, 25)
and so forth …

So, to answer the question above, the answer is highlighted in maroon above. For a hypotenuse of 20, the two sides would have to be 12 and 16. Hope this answers you, james Creagh.

This method would only be true for multiples of the triple (3, 4, 5). In short, this method can only be used for hypotenuses that are multiple of fives.

There are other kinds of triples, such as (5, 12, 13) and (7, 24, 25). In the coming posts, I would explore other methods to generate triples that are not multiples of fives.