To solve the problem, let's analyze the given cube $ABCDEFGH$ with a side length of $8\sqrt{6}$ cm. In a cube, points $A$ and $C$ are opposite vertices on the same face, while points $D$ and $F$ are opposite vertices on another face.

1. Distance from point A to C:

In a cube, the distance between two opposite vertices on the same face (like $A$ and $C$) is the diagonal of the face. The diagonal $AC$ of a square face in a cube can be calculated using the Pythagorean theorem:

$AC = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}$

Given that the side length $s = 8\sqrt{6}$ cm, we can substitute it into the formula:

$AC = 8\sqrt{6} \times \sqrt{2} = 8\sqrt{12} = 8 \times 2\sqrt{3} = 16\sqrt{3} \text{ cm}$

2. Distance from point D to F:

Points $D$ and $F$ are opposite vertices of the cube, meaning the distance $DF$ is the space diagonal of the cube. The space diagonal $DF$ of a cube can be calculated using the formula:

$DF = \sqrt{s^2 + s^2 + s^2} = \sqrt{3s^2} = s\sqrt{3}$

Using the side length $s = 8\sqrt{6}$ cm:

$DF = 8\sqrt{6} \times \sqrt{3} = 8\sqrt{18} = 8 \times 3\sqrt{2} = 24\sqrt{2} \text{ cm}$

Final Results:

• The distance from $A$ to $C$ is $16\sqrt{3}$ cm.
• The distance from $D$ to $F$ is $24\sqrt{2}$ cm.

Would you like any more details on this solution, or do you have any other questions?

Here are 5 related questions you might find interesting:

1. How do you calculate the surface area of a cube with side length $8\sqrt{6}$ cm?
2. What is the volume of the cube in this problem?
3. How would you find the distance between any two adjacent vertices in a cube?
4. How does the distance formula for space diagonals in a cube relate to 3D vectors?
5. Can you determine the length of a line segment from a vertex to the midpoint of the opposite edge in a cube?

Tip: For any regular geometric shape like a cube, knowing the formulas for face diagonals and space diagonals is crucial for solving distance-related problems efficiently.