Why was $\tan\theta$ used? 
In reference to $(a)$, I do not understand why the tan ratio was used? Tan is $\frac{Opposite} {Adjacent}$ . Where should I be looking at to be able to determine the $\frac{Opposite} {Adjacent}$? Confused please help!!

$\tan$ is opposite over adjacent. You are getting confused with $\sin$, which is opposite over hypotenuse.
In general, the trick is to pick three things, two of which you know, and plug them into one of the three equations: $\sin=\frac{Opp}{Hyp}$, $\cos=\frac{Adj}{Hyp}$ and $\tan=\frac{Opp}{Adj}$. You then rearrange your equation to find the unknown.
Here, consider the angle at $P$, which is $55^{\circ}$. It has $\operatorname{Opposite}=GC$ and $\operatorname{Adjacent}=PG=9000$. Hence, $\tan 55^{\circ}=\frac{Opp}{Adj}=\frac{GC}{9000}$...
We could have chosen the angle at $C$. This has $\operatorname{Opposite}=PG=9000$ and $\operatorname{Adjacent}=GC$. Hence, $\tan 35^{\circ}=\frac{Opp}{Adj}=\frac{9000}{GC}$... This gives you the same length for $GC$.
