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 the quadrilateral is a rectangle, we need to check if all four sides are perpendicular to each other.
The slope between two points (x₁, y₁) and (x₂, y₂) can be found using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Let's calculate the slopes for all four sides AB, BC, CD, and DA:
Slope of AB:
m₁ = (8 - 5) / (1 - (-5)) = 3 / 6 = 1 / 2
Slope of BC:
m₂ = (2 - 8) / (4 - 1) = -6 / 3 = -2
Slope of CD:
m₃ = (-2 - 2) / (-2 - 4) = -4 / (-6) = 2 / 3
Slope of DA:
m₄ = (5 - (-2)) / (-5 - (-2)) = 7 / (-3) = -7 / 3
Now, let's check if any of these slopes are reciprocals of each other (indicating perpendicular lines).
m₁ and m₃ are reciprocals: m₁ * m₃ = (1/2) * (2/3) = 1/3 ≠ -1
Therefore, the quadrilateral is not a rectangle because at least one pair of opposite sides is not perpendicular.
To determine if the quadrilateral is a rectangle, we will need to consider the slopes of the sides.
A rectangle is a quadrilateral with four right angles, meaning that opposite sides must be parallel and equal in length.
Let's calculate the slopes of the sides.
Slope of AB:
m(AB) = (y2 - y1)/(x2 - x1)
= (8 - 5)/(1 - (-5))
= 3/6
= 1/2
Slope of BC:
m(BC) = (y2 - y1)/(x2 - x1)
= (2 - 8)/(4 - 1)
= -6/3
= -2
Slope of CD:
m(CD) = (y2 - y1)/(x2 - x1)
= (-2 - 2)/(-2 - 4)
= -4/-6
= 2/3
Slope of AD:
m(AD) = (y2 - y1)/(x2 - x1)
= (5 + 2)/(-5 - (-2))
= 7/-3
= -7/3
Now, let's compare the slopes:
Since the slopes of opposite sides are not equal, the quadrilateral is not a rectangle.
To confirm, we can check if the opposite sides are parallel by comparing their slopes:
Opposite sides AB and CD have slopes 1/2 and 2/3, respectively. Since the slopes are not equal, the opposite sides are not parallel.
Therefore, based on the slope, we can conclude that the quadrilateral is not a rectangle.
To determine if the given quadrilateral is a rectangle, we can use the slopes of the sides of the quadrilateral.
First, let's find the slopes of the four sides of the quadrilateral.
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), B(1, 8)
slope_AB = (8 - 5) / (1 - (-5))
slope_AB = 3 / 6
slope_AB = 1/2
Slope of side BC:
B(1, 8), C(4, 2)
slope_BC = (2 - 8) / (4 - 1)
slope_BC = -6 / 3
slope_BC = -2
Slope of side CD:
C(4, 2), D(-2, -2)
slope_CD = (-2 - 2) / (-2 - 4)
slope_CD = -4 / -6
slope_CD = 2/3
Slope of side DA:
D(-2, -2), A(-5, 5)
slope_DA = (5 - (-2)) / (-5 - (-2))
slope_DA = 7 / (-5 + 2)
slope_DA = 7 / (-3)
slope_DA = -7/3
Now, let's check if the slopes indicate that the quadrilateral is a rectangle.
A rectangle has opposite sides that are parallel and equal in length. The slopes of the opposite sides should be negative reciprocals of each other. In other words, if the slope of one side is m, then the slope of the opposite side should be -1/m.
In this case, the slopes are:
slope_AB = 1/2
slope_BC = -2
slope_CD = 2/3
slope_DA = -7/3
-1/slope_AB = -1/(1/2) = -2
-1/slope_BC = -1/-2 = 1/2
-1/slope_CD = -1/(2/3) = -3/2
-1/slope_DA = -1/(-7/3) = 3/7
Since opposite sides have slopes that are negative reciprocals of each other, we can conclude that the given quadrilateral is indeed a rectangle.
Remember, to determine if a quadrilateral is a rectangle, you need to calculate the slopes of its sides and check if the slopes of opposite sides are negative reciprocals.