The problem in the image is asking to calculate the length of segment $GC$ in a regular hexagon. We are given that the perimeter of the regular hexagon is 24 cm.

Step-by-step Solution:

1. Determine the side length of the hexagon:

   • The perimeter of a regular hexagon is the sum of the lengths of its six equal sides.
   • Let the side length of the hexagon be $s$.
   • Since the perimeter is 24 cm: $6s = 24 \quad \Rightarrow \quad s = \frac{24}{6} = 4 \text{ cm}$ The side length $s$ of the hexagon is 4 cm.

2. Identify segment $GC$:

   • In a regular hexagon, any diagonal connecting two non-adjacent vertices can be related to the side length.
   • Segment $GC$ connects two vertices that are separated by two other vertices, meaning it is a longer diagonal of the hexagon.
   • In a regular hexagon, the length of the diagonal connecting two opposite vertices (which is the longest diagonal) is twice the side length. The second longest diagonal, which is the case here for $GC$, spans two sides and its length is equal to $s \times \sqrt{3}$.

3. Calculate the length of diagonal $GC$:

   • The formula for this type of diagonal is $s \times \sqrt{3}$: $GC = 4 \times \sqrt{3} \text{ cm}$

Thus, the length of segment $GC$ is $4\sqrt{3}$ cm.

Final Answer:

The correct choice is C) $4\sqrt{3}$ cm.

Would you like further details, or do you have any questions?

Here are some related questions to expand your understanding:

1. What is the formula for the area of a regular hexagon given its side length?
2. How do you calculate the height of an equilateral triangle with a given side length?
3. How many diagonals does a regular hexagon have, and how do you calculate their lengths?
4. What is the perimeter of a regular hexagon if one of its diagonals is $12$ cm?
5. How would you derive the formula for the longest diagonal of a regular hexagon?

Tip: In a regular hexagon, all interior angles are 120 degrees, and it can be divided into 6 equilateral triangles for various geometric calculations.