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Difficulty: MediumGeometric Figures on the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, a parallelogram has vertices at P(−2,−1)P(-2, -1), Q(4,−1)Q(4, -1), R(6,4)R(6, 4), and S(0,4)S(0, 4). What is the area of parallelogram PQRSPQRS, in square units?

  1. A
    15
  2. B
    18
  3. C
    24
  4. 30Answer
  5. E
    40

Answer

30
The correct answer is 30. The area of a parallelogram is determined by multiplying its base by its perpendicular height. The base segment PQPQ is horizontal and extends from x=−2x = -2 to x=4x = 4, giving a length of 4−(−2)=64 - (-2) = 6 units. The height is the vertical distance between the line containing the base PQPQ (y=−1y = -1) and the line containing the opposite side SRSR (y=4y = 4). This distance is 4−(−1)=54 - (-1) = 5 units. Multiplying the base of 6 units by the height of 5 units yields an area of 30 square units.

Step-by-Step Solution

1
Identify the base of the parallelogram by calculating the length of the horizontal side PQPQ.
PQ=4−(−2)=6PQ = 4 - (-2) = 6 units
The segment PQPQ lies on the horizontal line y=−1y = -1, so its length is the difference between the x-coordinates of its endpoints.
2
Identify the height of the parallelogram by calculating the vertical distance between the parallel horizontal sides PQPQ (on y=−1y = -1) and SRSR (on y=4y = 4).
height=4−(−1)=5height = 4 - (-1) = 5 units
The height of a parallelogram is the perpendicular distance between its parallel bases.
3
Calculate the area of the parallelogram using the formula Area=base×heightArea = \text{base} \times \text{height}.
Area=6×5=30Area = 6 \times 5 = 30 square units
Multiplying the base length by the vertical height gives the total area of the parallelogram.

Key Concept

Finding the area of a parallelogram on the coordinate plane using base and height calculations from coordinates.
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Topics

Geometric Figures on the Coordinate Plane

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