I'm a little confused about what a Pythagorean Triple is.
As long as your two legs are both whole numbers, isn't every one a triple?
A Pythagorean triple consists of 3 whole numbers a, b, and c such that
a^2 + b^2 = c^2 ---> that's the Pythagorean Theorem
e.g. a = 5, b=12, and c=13 is one of them
because 5^2 + 12^2 = 13^2
25 + 144 = 169
169 = 169
If you randomly pick any such 3 numbers, the vast majority will not fit that pattern.
e.g. a = 4 , b = 7 , c = 9
is 4^2 + 7^2 = 9^2 ? , not very likely
here is a little algorithm that will produce Pythagorean triples.
let m and n be any two whole numbers, where m > n
then
2mn
m^2 - n^2 , and
m^2 + n^2 will always produce such a triple
e.g. let m = 7, n = 4
2mn = 56
m^2-n^2 = 49-16 = 33
m^2 + n^2 = 49+16 = 65
is 56^2 + 33^2 = 65^2 ? , it sure is !
Great question! A Pythagorean Triple is a set of three positive whole numbers (or integers) a, b, and c, such that a^2 + b^2 = c^2. In other words, it's a set of three numbers that satisfy the Pythagorean Theorem.
Now, not every pair of whole numbers is a Pythagorean Triple. The condition a^2 + b^2 = c^2 has to be satisfied for it to be a valid triple. Let's take an example to clarify things.
Suppose we have two legs, a = 3 and b = 4, then using the Pythagorean theorem, we can check if it's a Pythagorean Triple:
a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25.
Now, if we take the square root of both sides, we have:
√(a^2 + b^2) = √25 --> c = 5.
Here, a = 3, b = 4, and c = 5 form a Pythagorean Triple since a^2 + b^2 = c^2 is satisfied.
So, while it's true that any pair of whole numbers can potentially form a Pythagorean Triple, not every pair will actually satisfy the condition a^2 + b^2 = c^2. Therefore, only specific combinations of whole numbers can form Pythagorean Triples.