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Difficulty: MediumProperties of Quadrilaterals

Quadrilateral ABCDABCD has vertices A(−2,1)A(-2, 1), B(2,4)B(2, 4), C(5,0)C(5, 0), and D(1,−3)D(1, -3) in the standard (x,y)(x,y) coordinate plane. What is the area, in square units, of quadrilateral ABCDABCD?

  1. A
    12.512.5
  2. B
    1515
  3. C
    2020
  4. 2525Answer
  5. E
    5050

Answer

25
Using the distance formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, each side length of quadrilateral ABCDABCD is calculated to be 55 units. Evaluating the slopes of adjacent sides shows that the slope of ABAB is 34\frac{3}{4} and the slope of BCBC is −43-\frac{4}{3}. Since their product is −1-1, adjacent sides are perpendicular. A quadrilateral with four equal sides and right angles is a square, and its area is 52=255^2 = 25 square units.

Step-by-Step Solution

1
Calculate the length of side ABAB using the distance formula.
AB=(2−(−2))2+(4−1)2=42+32=25=5AB = \sqrt{(2 - (-2))^2 + (4 - 1)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5.
The distance formula finds the exact side length of the quadrilateral.
2
Calculate adjacent side length BCBC and check the slopes to determine the figure type.
BC=(5−2)2+(0−4)2=32+(−4)2=5BC = \sqrt{(5 - 2)^2 + (0 - 4)^2} = \sqrt{3^2 + (-4)^2} = 5. The slope of ABAB is 34\frac{3}{4} and the slope of BCBC is −43-\frac{4}{3}.
Because all sides are equal to 55 and adjacent slopes are negative reciprocals, the figure is a square.
3
Calculate the area of the square.
Area=side2=52=25\text{Area} = \text{side}^2 = 5^2 = 25.
The area of a square with side length ss is s2s^2.

Key Concept

Finding the area of a quadrilateral in the coordinate plane by verifying side lengths and right angles using distance and slope formulas.
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Topics

Properties of Quadrilaterals

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