Why the Formulas Work · Superchallenge problem

A $45$-$45$-$90$ triangle $ABC$ and a $30$-$60$-$90$ triangle $ABD$ share the hypotenuse $\overline{AB}$, which has length $4$. The right angles are at $C$ and $D$, $\angle DAB = 30^\circ$, and $C$ and $D$ lie on opposite sides of line $AB$. Find $CD$. Give an exact answer.