My college algebra teacher was explaining how an isoceles right angle triangle will always have a hypotenuse of an irrational number, where 1 ^2 + 1^2 = square root of 2 = 1.414213562...etc. My observation and question to him was that, while I can kind of understand how a number like Pi is infinite, given that it's based on a circle, I do not see why a triangle with 2 equal sides have a hypotenuse of an irrational number. In my mind, such straight lines of known proportions would be the DEFINITION of rational. He really couldn't explain WHY this is the case.
Can anyone here offer an explanation, other than "that's just the way it is"? Thanks.