To solve this problem, we need to break it down into several steps.

Step 1: Find the Side Length of the Cube

The surface area of a cube with side length $s$ is given by: $6s^2 = 96 \text{ cm}^2$ Dividing both sides by 6: $s^2 = 16 \Rightarrow s = 4 \text{ cm}$

Step 2: Determine the Lengths of $AB$ and $CD$

Since $A$ and $B$ are midpoints of two adjacent edges, each of these edges has length $s = 4$ cm, so the length of $AB$ is half of the face diagonal of the cube.

The face diagonal $AC$ on one face of the cube is: $AC = s\sqrt{2} = 4\sqrt{2} \text{ cm}$ Since $AB$ is half of this face diagonal: $AB = \frac{4\sqrt{2}}{2} = 2\sqrt{2} \text{ cm}$ Similarly, $CD$ is also $2\sqrt{2}$ cm since it is parallel to $AB$ and located on a parallel face of the cube.

Step 3: Calculate the Area of Quadrilateral $ABCD$

The quadrilateral $ABCD$ forms a parallelogram with $AB \parallel CD$ and $AD \parallel BC$.

The height of this parallelogram (the distance between $AB$ and $CD$) is the side length $s = 4$ cm.

Thus, the area of the parallelogram $ABCD$ is: $\text{Area} = \text{base} \times \text{height} = (2\sqrt{2}) \times 4 = 8\sqrt{2} \approx 18 \text{ cm}^2$

Conclusion

The area of quadrilateral $ABCD$ is approximately: $\boxed{18 \text{ cm}^2}$

So, the correct answer is D) 18 cm².

Would you like further details on any step?

Here are five related questions for further exploration:

1. How would you calculate the face diagonal if only the surface area were given?
2. What if $A$ and $B$ were not midpoints but quarter points on the edges—how would the area change?
3. How does the area of a parallelogram depend on the angle between the sides?
4. How can you find the volume of the cube given only the side length?
5. What are other properties of a parallelogram that can help in calculating its area?

Tip: For any square or cube, knowing just one dimension (side length or surface area) allows you to calculate all other geometric properties.