Is the following set of numbers a perfect triple? 13, 15, 18

Why are they, or why not

That depends upon how you define a "perfect triple".

It usually means a trio of integers like 3, 4, and 5, that satisfy
a^2 + b^2 = c^2

7, 15 and 18 are such a triple, but 13, 15 and 18 are not.