Geometry

This problem is actually simple because it is a 30-60-90 triangle.

Because we know that $CD = 8\sqrt3$, $AD$ must equal $\sqrt2\times 8\sqrt3 = 8\sqrt6$

We also know that $BD$ must equal to ${8\sqrt3}\over\sqrt2$, which can be rewritten as $4\sqrt6$. 

This means that $\color{brown}\boxed{AB=12\sqrt6}$

Note that triangles  ACD ,  CBD  are  similar  and each are 30-60-90 right triangles

And since  AD is opposite a 60° angle it =  sqrt 3 * CD =  sqrt 3 * 8 sqrt 3 =  24

And, by similar triangles, we have the following relationship

DB / DC  = DC / AD

DC^2 = DB * AD

(8sqrt 3)^2  = DB * 24

256 / 24  = DB  = 32 / 3  (cm)

I misread the question... DB should be $4 \sqrt6$, because $\triangle{CDB}$ is a 30-60-90.