The problem in the image asks to find the distance from point $S$ to the line $CD$ in a cube with a side length of 8 cm.

Step-by-step Solution:

1. Identify the position of point $S$:
   In the diagram, point $S$ is the midpoint of the top edge $HG$ of the cube. Since the cube has a side length of 8 cm, the position of $S$ relative to the other vertices can be determined.

2. Coordinates of key points:
   Let's set the origin at point $A$ with the following coordinates:

   • $A(0, 0, 0)$
   • $B(8, 0, 0)$
   • $D(0, 8, 0)$
   • $C(8, 8, 0)$
   • $H(0, 8, 8)$
   • $G(8, 8, 8)$
   • Point $S$ is the midpoint of $HG$, so its coordinates are: $S\left( \frac{0 + 8}{2}, \frac{8 + 8}{2}, \frac{8 + 8}{2} \right) = S(4, 8, 8)$

3. Distance from point $S(4, 8, 8)$ to line $CD$:
   The line $CD$ lies on the plane $z = 0$ and is a vertical line between $C(8, 8, 0)$ and $D(0, 8, 0)$. The direction vector of this line is: $\overrightarrow{CD} = D(0, 8, 0) - C(8, 8, 0) = (-8, 0, 0)$

   To find the perpendicular distance from point $S(4, 8, 8)$ to the line $CD$, we use the formula for the distance between a point and a line in 3D space: $d = \frac{| \overrightarrow{CS} \times \overrightarrow{CD} |}{|\overrightarrow{CD}|}$ where $\overrightarrow{CS} = S(4, 8, 8) - C(8, 8, 0) = (-4, 0, 8)$.

4. Calculating the cross product $\overrightarrow{CS} \times \overrightarrow{CD}$:

   \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -4 & 0 & 8 \\ -8 & 0 & 0 \end{vmatrix} = \hat{i}(0 - 0) - \hat{j}(-32 - 0) + \hat{k}(0 - 0) = 32\hat{j}$$ So, $$\overrightarrow{CS} \times \overrightarrow{CD} = (0, 32, 0)$$.

5. Magnitude of the cross product: $| \overrightarrow{CS} \times \overrightarrow{CD} | = \sqrt{0^2 + 32^2 + 0^2} = 32$

6. Magnitude of the direction vector $\overrightarrow{CD}$: $|\overrightarrow{CD}| = \sqrt{(-8)^2 + 0^2 + 0^2} = 8$

7. Final distance: $d = \frac{32}{8} = 4 \text{ cm}$

Thus, the distance from point $S$ to line $CD$ is 4 cm.

Would you like more details on any of the steps or have additional questions?

Here are five related questions for further practice:

1. How would you find the distance from point $S$ to line $AB$?
2. What is the distance from point $H$ to line $CD$?
3. How can you calculate the distance between two skew lines in 3D?
4. What is the length of the diagonal $AC$ of the cube?
5. How do you find the angle between two lines in 3D?

Tip: When dealing with cubes, try to set up a coordinate system to simplify the calculation of distances and vectors.