Exercise 14 Page 514

We are given the set of three numbers, and want to determine whether they form a Pythagorean triple. Therefore, we have to check if a triangle with the given lengths of sides is a right triangle. We will need to use the Converse of the Pythagorean Theorem.

Converse of the Pythagorean Theorem

If the sides of a triangle have lengths $a,$ $b,$ and $c,$ and $c^2=a^2+b^2,$ then the triangle is a right triangle.

This tells us that we can use the Pythagorean Theorem in reverse to test if a triangle is right. In general, the hypotenuse $c$ has the greatest value. Let's substitute $a=1,$ $b=2,$ and $c=3$ into $a^2+b^2=c^2,$ and see if they produce a true statement. The values produce a false statement, so the described triangle is not a right triangle and, as a result, the given set of numbers is not a Pythagorean triple.