For a tetrahedron ABCD with an equilateral triangle BCD of side length 12 as one face, let H be the foot of the perpendicular from vertex A to plane BCD. The point H lies inside triangle BCD. The area of triangle CDH is 3 times the area of triangle BCH, the area of triangle DBH is 2 times the area of triangle BCH, and $\overline { \mathrm { AH } } = 3$. Let M be the midpoint of segment BD, and let Q be the foot of the perpendicular from point A to segment CM. What is the length of segment AQ? [4 points] (1) $\sqrt { 11 }$ (2) $2 \sqrt { 3 }$ (3) $\sqrt { 13 }$ (4) $\sqrt { 14 }$ (5) $\sqrt { 15 }$
For a tetrahedron ABCD with an equilateral triangle BCD of side length 12 as one face, let H be the foot of the perpendicular from vertex A to plane BCD. The point H lies inside triangle BCD. The area of triangle CDH is 3 times the area of triangle BCH, the area of triangle DBH is 2 times the area of triangle BCH, and $\overline { \mathrm { AH } } = 3$. Let M be the midpoint of segment BD, and let Q be the foot of the perpendicular from point A to segment CM. What is the length of segment AQ? [4 points]\\
(1) $\sqrt { 11 }$\\
(2) $2 \sqrt { 3 }$\\
(3) $\sqrt { 13 }$\\
(4) $\sqrt { 14 }$\\
(5) $\sqrt { 15 }$