Answer
$\bf{C}$
Work Step by Step
We are given that the $\triangle ABD$ is a $30^{\circ}-60^{\circ}-90^{\circ}$
This means:
$AB=\sqrt 3BD$
Here, $BC=1 \ cm$, so we have $AB=\sqrt 3 \ cm$
Also, $\triangle ABC$ is a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle. That is, $BC=\sqrt 3 AB$
Here, $BC=CD+1 \ cm$ and $AB=\sqrt 3 \ cm$, so we have $AB=\sqrt 3 \ cm$
$CD+1=(\sqrt 3)(\sqrt 3) \\ CD=2 \ cm$
Therefore, the correct option is $\bf{C}$.
