15. the legs of a right triangle measure 3
and 5, respectively. determine whether
the side lengths form a pythagorean
triple and, if not, why not.

The right triangle of sides 3 and 5 does not form Pythagorean  triple.

Explanation:

A Pythagorean triple is a triple of positive integers a, b, and c such that a right triangle exists with legs a,b and hypotenuse c. By the Pythagorean theorem, this is equivalent to finding positive integers a, b, and c satisfying

The smallest and best-known Pythagorean triple is (a,b,c)=(3,4,5). The right triangle having these side lengths is sometimes called the 3, 4, 5 triangle.

Here in the question the  legs of the right triangle are 3 and 5

Now by using Pythagorean theorem

Hence it is not a Pythagorean triple