Pythagorean triples are given by the formulas x2 - y2, 2xy, and x2 + y2. use the formulas for the pythagorean triples to prove why it is not possible for a right triangle to have legs with lengths of 16 and an odd number. show all of your work for full credit.

Let  be an integer. Suppose there is a triangle with legs of length 16 and . Then by the Pythagorean theorem, the length of the hypotenuse should be



The formulas for Pythagorean triples say that if the legs are integers, then so must be the hypotenuse, because if  and  are integers, then so are , , and .

However,  is not a perfect square trinomial, which means for any integer , the length of the hypotenuse is not an integer, so such a triangle doesn't exist.