Question

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.

Answer 1

Let
`k` be an integer. Suppose there is a triangle with legs of length 16 and
`2k+1`. Then by the Pythagorean theorem, the length of the hypotenuse should be


`√(16^2+(2k+1)^2)=√(4k^2+4k+257)`

The formulas for Pythagorean triples say that if the legs are integers, then so must be the hypotenuse, because if
`x=16` and
`y=2k+1` are integers, then so are
`x^2-y^2`,
`2xy`, and
`x^2+y^2`.

However,
`4k^2+4k+257` is not a perfect square trinomial, which means for any integer
`k`, the length of the hypotenuse is not an integer, so such a triangle doesn't exist.