3
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ Assuming that the two sides are of positive rational length, your question is whether $\sqrt{2x^2}$ is always irrational for any positive rational $x$. the answer is of course yes as it is equal to $x\sqrt{2}$ $\endgroup$ – JMoravitz Nov 6 '16 at 3:04
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ I edited my question to include the length be denoted by a rational number. $\endgroup$ – naveen dankal Nov 6 '16 at 3:08
  • $\begingroup$ @naveendankal I included both cases in this answer - In the case where both legs have rational length $x$, the hypotenuse must have irrational length because $x\sqrt{2}$ is irrational. $\endgroup$ – MightyTyGuy Nov 6 '16 at 3:31
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ I edited my question to include the length be denoted by a rational number. $\endgroup$ – naveen dankal Nov 6 '16 at 3:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.