If lengths of sides AB, BC and AC of a right angle triangle are in a GP, what is the ratio between AC and AB?

Question

If lengths of sides AB, BC and AC of a right angle triangle are in a geometric progression, what is the ratio between AC and AB?

Answer 1

Thanks for the query.

One way to proceed as below -

let the sides AB, BC & CA be a/r, a and a*r respectively where r is the c.r.
hence, ratio of CA and AB is (a*r) / (a/r) = r^2
Now,
As per Pythagoras' theorem

(a/r)^2 + a^2 = (a*r)^2
cancelling a^2 we get,
(1/r)^2 + 1 = r^2 or, r^4 - r^2 - 1 = 0
solving for r^2 we get,
r^2 = (1/2) +/ - sqrt (5) / 2
Ignoring the - sqrt (5) / 2
we get the ratio as (1/2) + sqrt(5)/2