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.
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.
To determine if the quadrilateral is a rectangle, we need to check if its opposite sides are parallel and if its adjacent sides are perpendicular. We can use slope to help us with this.
Step 1: Find the slopes of the sides.
The slope between two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
Slope of side AB:
A(-5, 5) and B(1, 8)
slope_AB = (8 - 5) / (1 - (-5))
= 3 / 6
= 1/2
Slope of side BC:
B(1, 8) and C(4, 2)
slope_BC = (2 - 8) / (4 - 1)
= -6 / 3
= -2
Slope of side CD:
C(4, 2), and D(-2, -2)
slope_CD = (-2 - 2) / (-2 - 4)
= -4 / -6
= 2/3
Slope of side DA:
D(-2, -2) and A(-5, 5)
slope_DA = (5 + 2) / (-5 + 2)
= 7 / -3
= -7/3
Step 2: Check for parallel sides.
To determine if opposite sides are parallel, we need to compare their slopes. If the slopes are equal, then the sides are parallel.
slope_AB = 1/2
slope_CD = 2/3
The slopes of sides AB and CD are not equal. Therefore, sides AB and CD are not parallel.
slope_BC = -2
slope_DA = -7/3
The slopes of sides BC and DA are not equal. Therefore, sides BC and DA are not parallel.
Step 3: Check for perpendicular sides.
To determine if adjacent sides are perpendicular, we need to check if the product of their slopes is -1.
slope_AB * slope_BC = (1/2) * (-2) = -1
slope_BC * slope_CD = (-2) * (2/3) ≈ -1
slope_CD * slope_DA = (2/3) * (-7/3) ≈ -1
slope_DA * slope_AB = (-7/3) * (1/2) ≈ -1
The product of the slopes of all adjacent sides is approximately -1.
Conclusion: Since the opposite sides are not parallel and the adjacent sides are perpendicular, the quadrilateral is a rectangle.
Note: We have shown that the opposite sides are not parallel and the adjacent sides are perpendicular, which is sufficient to prove that the quadrilateral is a rectangle.
To determine if a quadrilateral is a rectangle using slopes, we need to make use of two properties:
1. Opposite sides of a rectangle are parallel.
2. Adjacent sides of a rectangle are perpendicular.
Let's first find the slopes of the opposite sides of the quadrilateral.
The slope of side AB:
m₁ = (y₂ - y₁) / (x₂ - x₁)
= (8 - 5) / (1 - (-5))
= 3 / 6
= 1/2
The slope of side CD:
m₂ = (y₄ - y₃) / (x₄ - x₃)
= (-2 - 2) / (-2 - 4)
= -4 / (-6)
= 2/3
Since the slopes of AB (1/2) and CD (2/3) are not equal, it means that opposite sides AB and CD are not parallel. Therefore, the quadrilateral is not a rectangle.
However, let's also check if the adjacent sides are perpendicular because a quadrilateral could still be a rectangle even if opposite sides are not parallel.
The slope of side BC:
m₃ = (y₄ - y₃) / (x₄ - x₃)
= (2 - 8) / (4 - 1)
= -6 / 3
= -2
The slope of side DA:
m₄ = (y₂ - y₁) / (x₂ - x₁)
= (8 - 5) / (1 - (-5))
= 3 / 6
= 1/2
Since the product of slopes m₃⋅m₄ = (-2)⋅(1/2) ≠ -1, the adjacent sides BC and DA are not perpendicular. Hence, our conclusion remains the same: the quadrilateral is not a rectangle.