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
Slope AB = (8-5)/(1+5) = 3/6 = 1/2
Slope CB = (2-8)/(4-1) = -6/3 = -2
so clearly AB and CB are perpendicular, slopes are negative reciprocals
of each other.
looks like it could be ....
find slope CD and AD, then draw your conclusion
To determine if the quadrilateral ABCD is a rectangle, we need to check if all four sides are perpendicular to each other.
Step 1: Find the slopes of the four sides.
Slope of AB:
mAB = (y2 - y1) / (x2 - x1)
mAB = (8 - 5) / (1 - (-5))
mAB = 3 / 6
mAB = 1/2
Slope of BC:
mBC = (y2 - y1) / (x2 - x1)
mBC = (2 - 8) / (4 - 1)
mBC = -6 / 3
mBC = -2
Slope of CD:
mCD = (y2 - y1) / (x2 - x1)
mCD = (-2 - 2) / (-2 - 4)
mCD = -4 / -6
mCD = 2/3
Slope of DA:
mDA = (y2 - y1) / (x2 - x1)
mDA = (5 - (-2)) / (-5 - (-2))
mDA = 7 / -3
mDA = -7/3
Step 2: Check if opposite sides have slopes that are negative reciprocals of each other.
AB and CD:
mAB * mCD = (1/2) * (2/3) = 1/3 ≠ -1
So, AB and CD are not perpendicular.
BC and DA:
mBC * mDA = (-2) * (-7/3) = 14/3 ≠ -1
So, BC and DA are not perpendicular.
Since none of the opposite sides have slopes that are negative reciprocals of each other, the quadrilateral ABCD is not a rectangle.
To determine if a quadrilateral is a rectangle, we can use the slopes of its sides.
First, we need to find the slopes of all four sides of the quadrilateral using the formula:
Slope (m) = (change in y)/(change in x)
Let's calculate the slopes for each pair of consecutive vertices:
1. AB:
Slope of AB = (y2 - y1) / (x2 - x1)
= (8 - 5) / (1 - (-5))
= 3 / 6
= 1/2
2. BC:
Slope of BC = (y2 - y1) / (x2 - x1)
= (2 - 8) / (4 - 1)
= -6 / 3
= -2
3. CD:
Slope of CD = (y2 - y1) / (x2 - x1)
= (-2 - 2) / (-2 - 4)
= -4 / -6
= 2/3
4. DA:
Slope of DA = (y2 - y1) / (x2 - x1)
= (5 - (-2)) / (-5 - (-2))
= 7 / -3
= -7/3
Now, let's compare the slopes of the opposite sides:
AB is parallel to CD, so their slopes should be equal. But 1/2 is not equal to 2/3.
BC is parallel to DA, so their slopes should be equal. But -2 is not equal to -7/3.
Since the slopes of the opposite sides are not equal, we can conclude that the quadrilateral is not a rectangle.
To summarize, by comparing the slopes of the quadrilateral's sides, we found that the opposite sides' slopes are not equal. Therefore, the given quadrilateral with vertices A(-5,5), B(1,8), C(4,2), and D(-2,-2) is not a rectangle.