What is the name for a triangle's longest side, opposite the right angle?

The hypotenuse, from Greek meaning stretched under. The relationship named after Pythagoras was known and used in Babylon over a thousand years before him - the tablet Plimpton 322 lists sets of numbers that satisfy it - and in India and China independently. What the Greeks contributed was proof rather than observation.

That distinction is the beginning of mathematics as a discipline. A surveyor who knows a 3-4-5 triangle gives a right angle has a useful fact; a proof establishes it holds for every right triangle that could ever exist, which is a different kind of claim entirely and is why Euclid's Elements remained a textbook for two thousand years.

The theorem also produced the first known mathematical crisis. Apply it to a square of side one and the diagonal is the square root of two, which cannot be written as a fraction - and the Pythagoreans, whose philosophy held that everything is number meaning whole-number ratio, had proved the existence of a quantity their worldview could not contain. The story that they drowned the man who revealed it is almost certainly legend, and the discomfort it records was real.