The Correct Answer and Explanation is:
Correct Answer: Rectangle
Explanation:
The geometric figure shown on the coordinate plane is a rectangle. This conclusion is reached by methodically analyzing the properties of the figure using its vertex coordinates.
First, we identify the coordinates of the four vertices from the graph. The vertices are located at the points (-4, 0), (2, 8), (6, 5), and (0, -3).
To classify the quadrilateral, we must examine its sides. We can do this by calculating the slope of each line segment. The slope (m) is found using the formula m = (y2 – y1) / (x2 – x1).
- The slope of the side connecting (-4, 0) and (2, 8) is (8 – 0) / (2 – (-4)) = 8 / 6, which simplifies to 4/3.
- The slope of the side connecting (0, -3) and (6, 5) is (5 – (-3)) / (6 – 0) = 8 / 6, which also simplifies to 4/3. Since these slopes are equal, this pair of opposite sides is parallel.
Next, we calculate the slopes of the other two sides.
- The slope of the side connecting (2, 8) and (6, 5) is (5 – 8) / (6 – 2) = -3 / 4.
- The slope of the side connecting (-4, 0) and (0, -3) is (-3 – 0) / (0 – (-4)) = -3 / 4. This pair of opposite sides is also parallel.
Because both pairs of opposite sides are parallel, the figure is confirmed to be a parallelogram. To determine if it is a more specific type of parallelogram, like a rectangle, we check the angles. Perpendicular lines, which form right angles, have slopes that are negative reciprocals of each other. Let’s check the slopes of two adjacent sides, such as 4/3 and -3/4. Multiplying these slopes gives (4/3) * (-3/4) = -1. This confirms the adjacent sides are perpendicular and form a 90-degree angle. A parallelogram with four right angles is a rectangle.
Finally, to ensure it is not a square, we can compare the lengths of adjacent sides. The side with a slope of 4/3 has a length of 10 units, while the adjacent side with a slope of -3/4 has a length of 5 units. Since the adjacent sides are not equal, the figure is not a square. Thus, the most specific classification is a rectangle.
