As shown in the figure, in triangle ABC with $\overline { \mathrm { AB } } = 5 , \overline { \mathrm { AC } } = 2 \sqrt { 5 }$, let D be the foot of the perpendicular from vertex A to segment BC. For point E that divides segment AD internally in the ratio $3 : 1$, we have $\overline { \mathrm { EC } } = \sqrt { 5 }$. If $\angle \mathrm { ABD } = \alpha , \angle \mathrm { DCE } = \beta$, what is the value of $\cos ( \alpha - \beta )$? [4 points] (1) $\frac { \sqrt { 5 } } { 5 }$ (2) $\frac { \sqrt { 5 } } { 4 }$ (3) $\frac { 3 \sqrt { 5 } } { 10 }$ (4) $\frac { 7 \sqrt { 5 } } { 20 }$ (5) $\frac { 2 \sqrt { 5 } } { 5 }$
As shown in the figure, in triangle ABC with $\overline { \mathrm { AB } } = 5 , \overline { \mathrm { AC } } = 2 \sqrt { 5 }$, let D be the foot of the perpendicular from vertex A to segment BC.
For point E that divides segment AD internally in the ratio $3 : 1$, we have $\overline { \mathrm { EC } } = \sqrt { 5 }$. If $\angle \mathrm { ABD } = \alpha , \angle \mathrm { DCE } = \beta$, what is the value of $\cos ( \alpha - \beta )$? [4 points]\\
(1) $\frac { \sqrt { 5 } } { 5 }$\\
(2) $\frac { \sqrt { 5 } } { 4 }$\\
(3) $\frac { 3 \sqrt { 5 } } { 10 }$\\
(4) $\frac { 7 \sqrt { 5 } } { 20 }$\\
(5) $\frac { 2 \sqrt { 5 } } { 5 }$