Pythagorean triple

Question

do the side lengths 9,16,25 equal a Pythagorean triple?

Answer 1

No, they are not a Pythagorean triple.

The definition of a Pythagorean triple is:
a² + b² = c²
This is also the formula for the hypotenuse (c) of a right triangle with legs (a) and (b).

This looks like a trick question because a triangle with sides 3, 4 and 5 is widely recognized as a right triangle.
If you look at the given side lengths 9, 16 and 25, you see that this is 3², 4² and 5², which could be easily confused.
9 + 16 = 25, so 3, 4 and 5 do constitute a Pythagorean triple.

The given numbers don't fit the definition:
9² = 81
16² = 256
25² = 625

81 and 256 do not add up to equal 625.

Answer 2

No.

If they were, then they would be sides of a right triangle.

These are not sides of any triangle as 9 + 16 = 25.

However, the equation above can be rewritten as 3² + 4² = 5² so that (3,4,5) is a Pythagorean triple, and a very important one. In fact, the most commonly used Pythagorean triples in SAT questions on right triangles are (3,4,5) and (5,12,13).

In every Pythagorean triple, (a,b,c), at least one of a or b is even.

The triple of positive integers (a,b,c) is Pythagorean with b even if and only if there are positive integers m, n with m < n such that

a = m² - n²

b = 2mn

c = m² + n²