What is the most precise name for quadrilateral ABCD with vertices A(-2, 4), B(3, 4), C(6, 0), and D(1, 0)?
A. rhombus
B. square
C. parallelogram
D. rectangle
The Correct Answer and Explanation is :
iturn0image1Quadrilateral ABCD, with vertices A(-2, 4), B(3, 4), C(6, 0), and D(1, 0), is best classified as a rectangle. Here’s a detailed explanation:
Step 1: Plotting the Vertices
First, plot the given vertices on the Cartesian plane:
- A(-2, 4)
- B(3, 4)
- C(6, 0)
- D(1, 0)
Connecting these points in order forms quadrilateral ABCD.
Step 2: Calculating the Slopes of the Sides
To determine the nature of the quadrilateral, calculate the slopes of its sides:
- Slope of AB:
[ \text{Slope of AB} = \frac{4 – 4}{3 – (-2)} = \frac{0}{5} = 0 ]
This indicates that line AB is horizontal. - Slope of BC:
[ \text{Slope of BC} = \frac{0 – 4}{6 – 3} = \frac{-4}{3} ]
This line has a negative slope, indicating it’s neither horizontal nor vertical. - Slope of CD:
[ \text{Slope of CD} = \frac{0 – 0}{1 – 6} = \frac{0}{-5} = 0 ]
This indicates that line CD is horizontal. - Slope of DA:
[ \text{Slope of DA} = \frac{4 – 0}{-2 – 1} = \frac{4}{-3} ]
This line has a negative slope, similar to BC.
Step 3: Analyzing the Slopes
The slopes of opposite sides are equal:
- Slope of AB = Slope of CD = 0 (both horizontal)
- Slope of BC = Slope of DA = -4/3
This confirms that opposite sides are parallel, a defining property of parallelograms.
Step 4: Checking for Right Angles
To verify if the quadrilateral is a rectangle, check for perpendicular adjacent sides by calculating the product of their slopes:
- Product of slopes of AB and BC:
[ 0 \times \left(-\frac{4}{3}\right) = 0 ]
Since the product is zero, lines AB and BC are perpendicular, confirming a right angle at vertex B. - Product of slopes of BC and CD:
[ \left(-\frac{4}{3}\right) \times 0 = 0 ]
This indicates a right angle at vertex C. - Product of slopes of CD and DA:
[ 0 \times \left(-\frac{4}{3}\right) = 0 ]
This indicates a right angle at vertex D. - Product of slopes of DA and AB:
[ \left(-\frac{4}{3}\right) \times 0 = 0 ]
This indicates a right angle at vertex A.
Since all adjacent sides are perpendicular, quadrilateral ABCD has four right angles, confirming it is a rectangle.
Conclusion
Based on the calculations and analyses, quadrilateral ABCD is a rectangle. Therefore, the correct answer is D.
Visual Representation
Below is a graphical representation of quadrilateral ABCD on the Cartesian plane:
Note: The image above is sourced from a related problem with similar vertex coordinates.