A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length 
 $\textbf{(A)}\ 3\sqrt3\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 6\sqrt3\qquad\textbf{(D)}\ 12\sqrt3\qquad\textbf{(E)}\ \text{ none of these}$
Draw a diagram as shown.  Using the [Pythagorean Theorem](https://artofproblemsolving.com/wiki/index.php/Pythagorean_Theorem) (or by using 30-60-90 triangles), half of the chord length is $6\sqrt{3}$, so the chord’s length is $\boxed{\textbf{(D) } 12\sqrt{3}}$.  One can also find the length directly by using the [Law of Cosines](https://artofproblemsolving.com/wiki/index.php/Law_of_Cosines).