Question

ABCD is a rectangle where points C and D have coordinates (−2, 0) and (2, 0), respectively. If the area of the rectangle is 24, what is the best way to describe the equation of the line AB?

• A. y = 3
• B. y = 6
• C. y = −3
• D. y = −6

Hint:

For rectangles symmetric about the x-axis, the height determines the vertical location of lines parallel to the base. Use the area formula to calculate dimensions efficiently.

Answer 1

The Correct Option is A

Understanding the Geometry of the Rectangle:

\(C\) and \(D\) are on the x-axis with coordinates \((-2, 0)\) and \((2, 0)\). The length of side \(CD\) (the base of the rectangle) is:

\[ \text{Length of } CD = |2 - (-2)| = 4 \]

Determine the Height of the Rectangle: The area of a rectangle is given by:

\[ \text{Area} = \text{Base} \times \text{Height} \]

Substituting the known values:

\[ 24 = 4 \times \text{Height} \]

\[ \text{Height} = \frac{24}{4} = 6 \]

Finding the Line \(AB\):

Since the height is perpendicular to \(CD\) and the rectangle is symmetric about the x-axis, the lines \(AB\) and \(CD\) are parallel. The line \(AB\) lies at a height of \(\frac{6}{2} = 3\) units above the x-axis. The equation of \(AB\) is:

\[ y = 3 \]

Thus, the best description of the equation of \(AB\) is \(y = 3\).