As shown in the figure, consider a rectangular stone block with a vertex $A$ and a face containing point $A$. Let the midpoints of the edges of this face be $B , E , F , D$ respectively. Another face of the rectangular block containing point $B$ has its edge midpoints as $B , C , H , G$ respectively. Given that $\overline { B C } = 8$ and $\overline { B D } = \overline { D C } = 9$. The stone block is now cut to remove eight corners, such that the cutting plane for each corner passes through the midpoints of the three adjacent edges of that corner. Find the length of $\overline { A D }$ and the volume of tetrahedron $A B C D$, and find the height from vertex $A$ to the base plane when $\triangle B C D$ is the base of the tetrahedron. (Volume of pyramid $= \frac { \text{Base area} \times \text{Height} } { 3 }$) (Non-multiple choice question, 8 points)
As shown in the figure, consider a rectangular stone block with a vertex $A$ and a face containing point $A$. Let the midpoints of the edges of this face be $B , E , F , D$ respectively. Another face of the rectangular block containing point $B$ has its edge midpoints as $B , C , H , G$ respectively. Given that $\overline { B C } = 8$ and $\overline { B D } = \overline { D C } = 9$. The stone block is now cut to remove eight corners, such that the cutting plane for each corner passes through the midpoints of the three adjacent edges of that corner.
Find the length of $\overline { A D }$ and the volume of tetrahedron $A B C D$, and find the height from vertex $A$ to the base plane when $\triangle B C D$ is the base of the tetrahedron. (Volume of pyramid $= \frac { \text{Base area} \times \text{Height} } { 3 }$) (Non-multiple choice question, 8 points)