Mathematically similar problem/golden ratio

Nov 2009
from he rectangle ABCD a square is cut off to leave rectangle BCEF. Rectangle BCEF is similar to ABCD. Find x and hence state the ratio of the sides of rectangle ABCD.

Diagram attached.

Because BCEF is a square does that not mean that x=1 due to FE being part of the rectangle, where it states that side is 1. Therefore because it is a square, that side must equal with x, which equals 1.

However this is not correct because the answer is the golden ratio, what did I do wrong?

furthermore the thing I normally do does not work for this as shown below
\(\displaystyle \frac{1}{1}=\frac{1}{x}\)

P.S. used bb codes for image

Please help :(