The lengths of the sides of a right-angled triangle form a geometric sequence.

Find the two possible values for the common ratio of the sides.

Use Pythagoras' Theorem

Let the three sides be $a,ar,a{r}^2.$

The two cases to consider are when

• $r>1$ (so the sequence is increasing) and $a{r}^2$ is the hypotenuse (as it is the longest side)

• $0<r<1$ and $a$ is the hypotenuse.