[bandrami]

For "most" right triangles, yes, C/A is irrational. In fact the triangles for which C/A is rational are vanishingly rare (though Pythagoras proved many important things about them[1])

But before Pythagoras, it was still an open question if for any two reals A, B there might be a rational Q such that QA = B. Whereas we now know that for "most" reals there is no such Q, thanks to Pythagoras.

1: https://en.wikipedia.org/wiki/Pythagorean_triple