To determine if a quadrilateral is a rectangle, we need to check if its opposite sides are parallel and if its adjacent sides are perpendicular.
First, let's calculate the slopes of the sides of the quadrilateral:
Slope of AB = (y2 - y1) / (x2 - x1) (Using points A(-5, 5) and B(1, 8)):
Slope of AB = (8 - 5) / (1 - (-5)) = 3 / 6 = 1/2
Slope of BC = (y2 - y1) / (x2 - x1) (Using points B(1, 8) and C(4, 2)):
Slope of BC = (2 - 8) / (4 - 1) = -6 / 3 = -2
Slope of CD = (y2 - y1) / (x2 - x1) (Using points C(4, 2) and D(-2, -2)):
Slope of CD = (-2 - 2) / (-2 - 4) = -4 / -6 = 2/3
Slope of DA = (y2 - y1) / (x2 - x1) (Using points D(-2, -2) and A(-5, 5)):
Slope of DA = (5 - (-2)) / (-5 - (-2)) = 7 / -3 = -7/3
Now, let's analyze the slopes:
Opposite sides AB and CD have slopes 1/2 and 2/3, respectively. Since these slopes are different, AB and CD are not parallel.
Opposite sides BC and DA have slopes -2 and -7/3, respectively. Since these slopes are different, BC and DA are not parallel.
The adjacent sides AB and BC have slopes 1/2 and -2, respectively. These slopes are not negative reciprocals of each other, so AB and BC are not perpendicular.
The adjacent sides BC and CD have slopes -2 and 2/3, respectively. These slopes are not negative reciprocals of each other, so BC and CD are not perpendicular.
Therefore, since the opposite sides are not parallel and the adjacent sides are not perpendicular, the quadrilateral ABCD is not a rectangle.
A quadrilateral has vertices at A(−5, 5), B(1, 8), C(4, 2), and D(−2, −2). Use slope to determine if the quadrilateral is a rectangle. Show your work.
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