The question appears to be about a cube $ABCDEFGH$, with the given length $AC = 2\sqrt{6}$ cm. The problem asks for the distance from point $A$ to point $G$.

Solution Outline:

In a cube, $A$ and $G$ are opposite vertices. The diagonal distance between two opposite vertices of a cube can be found using the formula: $d = \sqrt{3} \times \text{side length}$ Where $d$ is the space diagonal (distance from $A$ to $G$) and "side length" refers to the length of one edge of the cube.

We are given that $AC = 2\sqrt{6}$. Since $AC$ is the diagonal across one face of the cube, we can use the formula for the face diagonal: $AC = \sqrt{2} \times \text{side length}$

Step-by-step Solution:

1. From $AC = 2\sqrt{6}$, solve for the side length of the cube: $2\sqrt{6} = \sqrt{2} \times \text{side length}$ $\text{side length} = \frac{2\sqrt{6}}{\sqrt{2}} = 2\sqrt{3} \, \text{cm}$

2. Now, calculate the distance from $A$ to $G$ using the space diagonal formula: $AG = \sqrt{3} \times \text{side length} = \sqrt{3} \times 2\sqrt{3} = 6 \, \text{cm}$

So, the distance from point $A$ to point $G$ is $6$ cm, which matches option (e).

Let me know if you'd like more details or have any questions!

Expanding the Topic:

1. How do you calculate the length of a face diagonal in a cube?
2. What is the formula for the volume of a cube given its side length?
3. How can you calculate the surface area of a cube?
4. How would the formula for the space diagonal change for a rectangular prism?
5. What is the relationship between the side length and the diagonals in a cube?

Tip: In a cube, every diagonal (face or space) is related to the side length through square roots, making the diagonal formulas easy to derive!