生成这个题目的讲解---**Extraction Content:**
**Question 19:**
**(Points:** 12 分)
**Question Stem:**
如图, 直三棱柱 $ABC - A_1B_1C_1$ 的体积为 4, $\triangle A_1BC$ 的面积为 $2\sqrt{2}$.
**(1) Sub-question 1:**
求 $A$ 到平面 $A_1BC$ 的距离;
**(2) Sub-question 2:**
设 $D$ 为 $A_1C$ 的中点, $AA_1 = AB$, 平面 $A_1BC \perp$ 平面 $ABB_1A_1$, 求二面角 $A - BD - C$ 的正弦值.
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**Chart/Diagram Description:**
* **Type:** Three-dimensional geometric figure, specifically a right triangular prism.
* **Main Elements:**
* **Vertices:** Six vertices are labeled: A, B, C (forming the bottom triangular base) and A1, B1, C1 (forming the top triangular base, corresponding to A, B, C respectively). An additional point D is marked within the prism.
* **Edges:**
* **Bottom Base (triangle ABC):** Edges AB and BC are shown as solid lines. Edge AC is shown as a dashed line, indicating it is a hidden edge from this perspective.
* **Top Base (triangle A1B1C1):** Edges A1B1, B1C1, and C1A1 are all shown as solid lines.
* **Lateral Edges:** AA1, BB1, and CC1 are shown as solid vertical lines, connecting corresponding vertices of the two bases.
* **Internal Lines:**
* Solid lines are drawn connecting A1 to B and A1 to C, forming triangle A1BC, which is explicitly mentioned in the problem statement.
* Dashed lines are drawn from point D to A, B, and C (AD, BD, CD).
* A dashed line is also drawn from B1 to D (B1D).
* **Point D:** Point D is located on the edge A1C (implied by context or stated as midpoint of A1C in the problem description, which is "D 为 A1C 的中点"). In the diagram, it appears as an intersection point of internal dashed lines.
* **Relative Position:** The figure represents a prism, meaning the top and bottom bases are parallel and congruent, and the lateral faces are rectangles (or parallelograms, but specified as a "right triangular prism," implying rectangular lateral faces and lateral edges perpendicular to the bases). The perspective shows the prism from slightly above and to the right, with vertex B and edges AB, BB1, BC, A1B1, B1C1, C1A1, AA1, CC1 being visible, while AC is hidden.