ABCD is a rectangle with length and breadth in the ratio α : 1. It is divided into a square APQD and a second rectangle PBCQ, as shown. Show that the length and breadth of rectangle PBCQ are also in the ratio α : 1.
Also this question could possibly be wrong so please let me know if it is.
I tried proving it but I am stuck and I have been trying such a simple problem for hours still can't solve it.
Since your title mentions the golden mean a.k.a the golden ratio, I am going to assume that your $\alpha$ is the golden ratio i.e. a root of
$\alpha^2 = \alpha + 1$
(even though you do not state this in your question).
Rectangle $PBCQ$ has side with length $1$ and $\alpha-1$ so the ratio of its sides is
$1:\alpha-1 = \alpha:\alpha^2-\alpha$
I'll let you complete the final step.
Of course, if $\alpha$ is not the golden ratio then the claim in the question is false. For example, if $\alpha=2$ then $PBCQ$ is a square and the ratio of its sides is $1:1$.