Roger is an interior designer and one morning he sees that his ladder is making a shadow with respect to the floor in the backyard as indicated in the diagram from below. He's curious and found that the angles formed between the corners are the following: $\frac{\angle CAB}{2}=\frac{\angle ADC}{2}=\frac{\angle ABC}{3}=\frac{\angle ACD}{8}$. He also notices that $AB=CD$ and $AC=8$. Find the length of the segment AD which is seen by Roger.

$\begin{array}{ll}

1.&16\,cm\\

2.&20\,cm\\

3.&15\,cm\\

4.&21\,cm\\

\end{array}$

I'm not sure exactly what sort of theorem or identity should be used here. I attempted different way to tackle this problem. Can someone help me?.

I'm slow at seeing things in space, so an answer which would help me the most is one which can be detailed the most as possible. This problem belongs to a section of euclidean geometry. So I appreciate that an answer could follow an euclidean geometry approach and a trigonometrical one as well so I can compare both methods and see which one is quicker. So can someone help me?.