As shown in the figure, there is a right triangle ABC with $\overline { \mathrm { AB } } = 3$, $\overline { \mathrm { BC } } = 4$, and $\angle \mathrm { B } = \frac { \pi } { 2 }$. Let D be the point that divides segment AB internally in the ratio $2 : 1$, let E be the point where the circle centered at A with radius $\overline { \mathrm { AD } }$ meets segment AC, let F be the point where the line AB meets this circle other than D, and let G be a point on arc EF such that $\overline { \mathrm { CG } } = 2 \sqrt { 6 }$. When point H on the circle passing through the three points C, E, G satisfies $\angle \mathrm { HCG } = \angle \mathrm { BAC }$, what is the length of segment GH? [4 points] (1) $\frac { 6 \sqrt { 15 } } { 5 }$ (2) $\frac { 38 \sqrt { 10 } } { 25 }$ (3) $\frac { 14 \sqrt { 3 } } { 5 }$ (4) $\frac { 32 \sqrt { 15 } } { 25 }$ (5) $\frac { 8 \sqrt { 10 } } { 5 }$
As shown in the figure, there is a right triangle ABC with $\overline { \mathrm { AB } } = 3$, $\overline { \mathrm { BC } } = 4$, and $\angle \mathrm { B } = \frac { \pi } { 2 }$. Let D be the point that divides segment AB internally in the ratio $2 : 1$, let E be the point where the circle centered at A with radius $\overline { \mathrm { AD } }$ meets segment AC, let F be the point where the line AB meets this circle other than D, and let G be a point on arc EF such that $\overline { \mathrm { CG } } = 2 \sqrt { 6 }$. When point H on the circle passing through the three points C, E, G satisfies $\angle \mathrm { HCG } = \angle \mathrm { BAC }$, what is the length of segment GH? [4 points]\\
(1) $\frac { 6 \sqrt { 15 } } { 5 }$\\
(2) $\frac { 38 \sqrt { 10 } } { 25 }$\\
(3) $\frac { 14 \sqrt { 3 } } { 5 }$\\
(4) $\frac { 32 \sqrt { 15 } } { 25 }$\\
(5) $\frac { 8 \sqrt { 10 } } { 5 }$