From the image, the problem involves a cuboid $ABCD.EFGH$ with the dimensions $AB = 8 \, \text{cm}$, $BC = 6 \, \text{cm}$, and $AE = AG = \text{Height}$. The task is to calculate two distances:

Problem A:

Determine the distance from point $A$ to the center of the face $EFGH$.

Problem B:

Determine the distance from point $A$ to the plane $EFGH$.

Let's first solve these.

Step-by-Step Solution:

Given:

• $AB = 8 \, \text{cm}$
• $BC = 6 \, \text{cm}$
• $AE = AG = \text{height}$

Let's denote the height as $h$ for now (it should be calculated based on further context).

Solution for A: Distance from $A$ to the center of face $EFGH$.

1. Find the coordinates:

   • Coordinates of $A$ are $A(0,0,0)$.
   • Face $EFGH$ lies on the top of the cuboid, and the center of this face would be at the point: $\text{Center of } EFGH = \left( \frac{AB}{2}, \frac{BC}{2}, h \right) = \left( \frac{8}{2}, \frac{6}{2}, h \right) = (4, 3, h)$

2. Calculate the distance: Using the distance formula between point $A(0, 0, 0)$ and the center $(4, 3, h)$: $\text{Distance} = \sqrt{(4 - 0)^2 + (3 - 0)^2 + (h - 0)^2} = \sqrt{16 + 9 + h^2} = \sqrt{25 + h^2}$

So, the distance from point $A$ to the center of the face $EFGH$ is $\sqrt{25 + h^2}$.

Solution for B: Distance from $A$ to the plane $EFGH$.

The plane $EFGH$ is parallel to the $xy$-plane and is at a height $h$. So the distance from point $A$ to the plane is simply the height $AE$, which is $h$.

Further Steps:

In order to fully solve these, we need the value of the height $AE$. If provided, we can substitute it into the solutions.

Would you like to ask any questions or need additional details on this?

Here are some related questions that can help extend your understanding:

1. How do you derive the distance formula between two points in 3D space?
2. What is the significance of finding the distance to a plane in geometry?
3. How does the position of a plane affect the distance between a point and the plane?
4. Can we find the height $AE$ using other geometric properties or measurements in the cuboid?
5. How would the distances change if the cuboid were a different shape, such as a square-based prism?

Tip: When calculating distances in 3D geometry, always clearly define the coordinates of points and use the distance formula carefully to avoid mistakes.