Given a cube ABCDEFGH with a side length of 8 root 6 cm. Determine the distance from point A to C and D to F.

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.

Learn how to calculate the distances between points A to C and D to F in a cube with a side length of 8√6 cm. This solution applies key geometric concepts, including the Pythagorean theorem, to find the face diagonal and space diagonal of the cube. Ideal for students in Grades 9-10.

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