9. Point C is the center of the regular hexagon shown above. Which of the following expressions represents the area of this hexagon?
Answer: B
The area of the regular hexagon can be represented by the expression 6xy.
The area of the regular hexagon is given by the formula \( \frac{3\sqrt{3}}{2} s^2 \), where \( s \) is the length of a side. In this case, the expression 6xy corresponds to the derived area based on the side lengths represented by \( x \) and \( y \).
A) 12xy
This option overestimates the area of the hexagon. The expression 12xy suggests that the area is double the correct area derived from the standard formula for the area of a regular hexagon, which indicates a misunderstanding of the geometric properties involved.
B) 6xy
This option correctly represents the area of the regular hexagon. The expression aligns with the calculations derived from its geometric properties, where the product of \( x \) and \( y \) appropriately accounts for the dimensions of the hexagon.
C) 3xy
This option underestimates the area of the hexagon. The expression 3xy does not encompass the full area covered by the hexagon's sides and angles, leading to an inaccurate representation of its total area.
D) xy
This option significantly underestimates the area of the hexagon. The expression xy suggests a very limited area, which fails to account for the complexity and size of the hexagon's geometry, resulting in a gross miscalculation.
Conclusion
The expression 6xy accurately reflects the area of the regular hexagon based on its geometric properties and the appropriate formula. All other options either overestimate or underestimate the area, demonstrating a lack of understanding of how to calculate the area of a regular hexagon correctly. Thus, option B is the only viable choice for representing the hexagon's area.