Question

In the rectangular coordinate plane, point A has coordinates (-4, 0), point B has coordinates (0, 4), point C has coordinates (4, 0), and point D has coordinates (0, -4). What is the area of quadrilateral ABCD?

• A. 8
• B. 16
• C. 24
• D. 32
• E. 64

Hint:

When given coordinates, a quick sketch on a coordinate plane can be very helpful to visualize the shape. For quadrilaterals with vertices on the axes, the diagonals are often easy to find and the area formula \( \frac{1}{2} d_1 d_2 \) is very efficient.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question asks for the area of a quadrilateral defined by four points on a coordinate plane. We need to identify the shape of the quadrilateral and use the appropriate area formula.
Step 2: Key Formula or Approach:
1. Plot the points to visualize the shape. 2. The given quadrilateral is a rhombus (or a square, which is a special rhombus). 3. The area of a rhombus can be calculated using its diagonals: \( \text{Area} = \frac{1}{2} d_1 d_2 \), where \(d_1\) and \(d_2\) are the lengths of the diagonals.
Step 3: Detailed Explanation:
Let's identify the vertices of the quadrilateral ABCD: A = (-4, 0) B = (0, 4) C = (4, 0) D = (0, -4) Let's find the lengths of the diagonals. The diagonals connect opposite vertices. Diagonal 1 (AC): This is the horizontal distance between A(-4, 0) and C(4, 0). \[ d_1 = |4 - (-4)| = |4 + 4| = 8 \] Diagonal 2 (BD): This is the vertical distance between B(0, 4) and D(0, -4). \[ d_2 = |4 - (-4)| = |4 + 4| = 8 \] The shape is a rhombus with equal diagonals, which means it is a square. Now we can calculate the area using the formula for a rhombus: \[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \] \[ \text{Area} = \frac{1}{2} \times 8 \times 8 = \frac{1}{2} \times 64 = 32 \] Alternatively, we can see the quadrilateral is composed of four identical right-angled triangles in each quadrant. For example, the triangle in the first quadrant has vertices (0,0), (4,0), and (0,4). Its area is \( \frac{1}{2} \times 4 \times 4 = 8 \). The total area is \( 4 \times 8 = 32 \).
Step 4: Final Answer:
The area of the quadrilateral ABCD is 32.