Irrational numbers and Pythagoras Theorem

Question

Is it always true that if the right angled triangle with is also isosceles and having lengths that can be denoted in terms of a rational number, the length of its hypotenuse will always be an irrational number? Another way to look at it would be that the diagonal of a square is always irrational. Does this always hold true?

Answer 1

If the legs each have length $x$, then the hypotenuse has length $x\sqrt{2}$.

So if $x$ is rational, then the hypotenuse has irrational length. If $x$ is irrational, then the hypotenuse could have irrational or rational length.

For example:

If $x=5$, the hypotenuse has length $5\sqrt{2}$, which is irrational.

If $x=\sqrt{2}$, the hypotenuse has length $2$, which is rational.

If $x=\sqrt{3}$, the hypotenuse has length $\sqrt{6}$, which is irrational.

Answer 2

If the adjacent sides of a right triangle are sqrt(2) then the hypotenuse will be 2 which is rational. However if the side lengths are rational then a$^2$+b$^2$=c$^2$ so 2a$^2$=c$^2$ and c = $\sqrt{2a}$ which is irrational since $\sqrt{2}$ is irrational and a is rational.