Heather is given the coordinates of the vertices of quadrilateral
ABCD.
ABCD.
She is asked to prove whether the quadrilateral is a parallelogram. Her proof follows the diagram.
Given:
Quadrilateral
ABCD
ABCD
has vertices at
A(−8,−1),
A(−8,−1),
B(−1,−7),
B(−1,−7),
C(2,2),
C(2,2),
and
D(−5,8).
D(−5,8).
Prove:
Quadrilateral
ABCD
ABCD
is a parallelogram
Well, let's take a closer look at this quadrilateral ABCD.
To prove that it is a parallelogram, we need to show that both pairs of opposite sides are parallel.
Let's start with side AB and side CD. The slope of the line passing through points A and B can be found using the formula (y2 - y1) / (x2 - x1). Plugging in the given coordinates, we get:
(y2 - y1) / (x2 - x1)
= (-7 - (-1)) / (-1 - (-8))
= (-7 + 1) / (-1 + 8)
= -6 / 7
Now let's find the slope of the line passing through points C and D:
(y2 - y1) / (x2 - x1)
= (8 - 2) / (-5 - 2)
= (8 - 2) / (-5 - 2)
= 6 / (-7)
= -6 / 7
Hey, hey! Look at that! The slopes of side AB and side CD are equal (-6/7), which means they are parallel to each other.
Now let's move on to side BC and side DA. We'll do a similar calculation:
Slope of BC:
= (2 - (-7)) / (2 - (-1))
= (2 + 7) / (2 + 1)
= 9 / 3
= 3
Slope of DA:
= (-1 - 8) / (-8 - (-5))
= (-1 - 8) / (-8 + 5)
= -9 / -3
= 3
Oh, snap! The slopes of side BC and side DA are equal (3), meaning they are parallel as well.
So we've shown that both pairs of opposite sides are parallel, which means that quadrilateral ABCD is indeed a parallelogram.
Congratulations, Heather! You've proved it! Now go celebrate with some geometrically shaped balloons.
To prove that quadrilateral ABCD is a parallelogram, we need to show that both pairs of opposite sides are parallel.
Step 1: Find the slopes of the two pairs of opposite sides.
The formula to find the slope between two points (x₁, y₁) and (x₂, y₂) is given by:
m = (y₂ - y₁)/(x₂ - x₁)
Slope of side AB:
m_AB = (-7 - (-1))/(-1 - (-8))
= -6/7
Slope of side CD:
m_CD = (2 - 8)/(2 - (-5))
= -6/7
Slope of side BC:
m_BC = (2 - (-7))/(2 - (-1))
= 9/3
= 3
Slope of side AD:
m_AD = (-1 - 8)/(-8 - (-5))
= -9/-3
= 3
Step 2: Compare the slopes of opposite sides.
If the opposite sides have the same slope, they are parallel.
Since m_AB = m_CD = -6/7 and m_BC = m_AD = 3, we can conclude that both pairs of opposite sides have equal slopes.
Step 3: Therefore, quadrilateral ABCD is a parallelogram.
Note: In this proof, we have shown that the opposite sides of the quadrilateral are parallel. If we also prove that the opposite angles are equal, then it would be a complete proof that ABCD is a parallelogram.
To prove whether quadrilateral ABCD is a parallelogram, we can use the slope criteria for parallelograms. The slope of one side of a parallelogram is equal to the slope of the opposite side.
First, let's calculate the slopes of the sides:
1. Side AB:
The formula for slope is given by: m = (y2 - y1) / (x2 - x1)
Using the coordinates (x1, y1) = (-8, -1) and (x2, y2) = (-1, -7):
The slope of AB is mAB = (-7 - (-1)) / (-1 - (-8)) = -6 / 7
2. Side BC:
Using the coordinates (x1, y1) = (-1, -7) and (x2, y2) = (2, 2):
The slope of BC is mBC = (2 - (-7)) / (2 - (-1)) = 9 / 3 = 3
3. Side CD:
Using the coordinates (x1, y1) = (2, 2) and (x2, y2) = (-5, 8):
The slope of CD is mCD = (8 - 2) / (-5 - 2) = 6 / -7 = -6/7
4. Side DA:
Using the coordinates (x1, y1) = (-5, 8) and (x2, y2) = (-8, -1):
The slope of DA is mDA = (-1 - 8) / (-8 - (-5)) = -9 / -3 = 3
Now, we can compare the slopes:
mAB = -6/7, mBC = 3, mCD = -6/7, mDA = 3
We observe that mAB = mCD, and mBC = mDA, which means that the opposite sides of the quadrilateral have equal slopes. Therefore, we can conclude that quadrilateral ABCD is a parallelogram based on the slope criteria.
Note: In addition to the slope criteria, there are other methods to prove that a quadrilateral is a parallelogram, such as showing that opposite sides are parallel or that opposite angles are congruent.
check each side length
there should be two pairs of equal length
check each side's slope
there should be two pairs that are the same