Question

The coordinates of the vertices of quadrilateral ABCD are A(−3, 2), B(3, 6), C(8, 1), and D(3, −4). Select from the drop-down menus to correctly complete the sentence.

Quadrilateral ABCD is:
a) A parallelogram because both pairs of opposite sides are parallel.
b) A trapezoid because exactly one pair of opposite sides is parallel.
c) A trapezoid because both pairs of opposite sides are parallel.
d) Neither a parallelogram nor a trapezoid because no pairs of opposite sides are parallel.

Answer 1

To determine if quadrilateral ABCD is a parallelogram or a trapezoid, we need to examine the slopes of the opposite sides. The slope of AB = 2/3, BC = -1, CD = 1, and AD = -1. Since AB and CD have the same slope and BC and AD have the same slope, we can conclude that both pairs of opposite sides in quadrilateral ABCD are parallel.

Step-by-step explanation:

The coordinates of the vertices of quadrilateral ABCD are A(−3, 2), B(3, 6), C(8, 1), and D(3, −4). To determine if quadrilateral ABCD is a parallelogram or a trapezoid, we need to examine the slopes of the opposite sides. The slope of a line can be found using the formula m = (y2 - y1)/(x2 - x1). Let's calculate the slopes of AB, BC, CD, and AD.

The slope of AB = (6 - 2)/(3 - (-3)) = 4/6 = 2/3

The slope of BC = (1 - 6)/(8 - 3) = -5/5 = -1

The slope of CD = (-4 - 1)/(3 - 8) = -5/-5 = 1

The slope of AD = (2 - (-4))/(-3 - 3) = 6/-6 = -1

Since AB and CD have the same slope (2/3 and 1), and BC and AD have the same slope (-1 and -1), we can conclude that both pairs of opposite sides in quadrilateral ABCD are parallel. Therefore, the correct answer is: Quadrilateral ABCD is a parallelogram because both pairs of opposite sides are parallel.