Length of Segment in Triangle with Right Angles

In triangle $ABC$, angle $A$ measures $30^{ ext{o}}$ and side $BC$ measures $10$ units. If $AB$ is extended to a point $D$ such that it is perpendicular to $AC$, calculate the length of segment $AD$.

Use the Law of Sines and the properties of right triangles to solve for $AD$. Remember to consider the relationship between the angles and sides.