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Difficulty: MediumGeometric Figures on the Coordinate Plane

An isosceles trapezoid ABCDABCD has vertices A(−4,−1)A(-4, -1), B(6,−1)B(6, -1), and C(3,3)C(3, 3) in the standard (x,y)(x, y) coordinate plane. If the base ABAB is parallel to the x-axis, which of the following represents the coordinates of vertex DD?

  1. A
    (−7,3)(-7, 3)
  2. B
    (−5,3)(-5, 3)
  3. C
    (−3,3)(-3, 3)
  4. (−1,3)(-1, 3)Answer
  5. E
    (1,3)(1, 3)

Answer

(−1,3)(-1, 3)
The correct answer is the coordinate pair representing (−1,3)(-1, 3). Since the bases are parallel to the x-axis, they are horizontal, meaning DD must share the same y-coordinate as CC, which is 3. The vertical line of symmetry passes through the midpoint of the segment ABAB, which is at x=1x = 1. The vertex C(3,3)C(3, 3) is 2 units to the right of this line of symmetry, so the vertex DD must be 2 units to the left of the line of symmetry, giving an x-coordinate of −1-1.

Step-by-Step Solution

1
Determine the orientation of the trapezoid and the y-coordinate of the missing vertex.
The base ABAB lies on the horizontal line y=−1y = -1. Because the bases of a trapezoid are parallel, the second base CDCD must also be horizontal and lie on the line y=3y = 3. Therefore, the y-coordinate of vertex DD is 33.
Since the trapezoid's bases are parallel to the x-axis, they are horizontal lines, meaning vertices on the same base share the same y-coordinate.
2
Find the equation of the line of symmetry of the isosceles trapezoid.
The midpoint of the base ABAB has an x-coordinate of −4+62=1\frac{-4 + 6}{2} = 1. The vertical line of symmetry is x=1x = 1.
An isosceles trapezoid is symmetric with respect to the perpendicular bisector of its bases. For horizontal bases, this is a vertical line passing through the midpoint of the base.
3
Use the line of symmetry to find the x-coordinate of vertex DD.
The x-coordinate of C(3,3)C(3, 3) is 3, which is 3−1=23 - 1 = 2 units to the right of the line of symmetry x=1x = 1. Thus, the x-coordinate of vertex DD must be 2 units to the left of the line of symmetry: 1−2=−11 - 2 = -1. Combining this with the y-coordinate gives D(−1,3)D(-1, 3).
Symmetry requires that corresponding vertices on the opposite sides of the line of symmetry are equidistant from it.

Key Concept

Using symmetry and coordinate geometry properties of an isosceles trapezoid to determine the coordinates of a missing vertex.
Estimated Time:1m 30s
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Topics

Geometric Figures on the Coordinate Plane

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