Prove that A(–2, 0), B(0, 9), and C(2, 0) and D(0, –9) are the vertices of a parallelogram.
A. What are the properties of a parallelogram involving slope?
B. What must we KNOW (regarding slopes and sides) to confirm the shape is a parallelogram?
C. Do so to prove these are the vertices of a parallelogram (showing your work).
A. The properties of a parallelogram involving slope are:
- Opposite sides are parallel
- Opposite sides have the same length
- Opposite angles are congruent
B. To confirm that the shape is a parallelogram, we need to know:
1. The slope of one side is equal to the slope of its opposite side
2. The lengths of opposite sides are equal
3. The opposite angles are congruent
C. To prove that A(-2, 0), B(0, 9), C(2, 0), and D(0, -9) are the vertices of a parallelogram:
1. Find the slopes of the sides of the quadrilateral:
Slope of AB = (9 - 0) / (0 - (-2)) = 9/2
Slope of BC = (0 - 9) / (2 - 0) = -9/2
Slope of CD = (-9 - 0) / (0 - 2) = 9/2
Slope of DA = (0 - (-9)) / (-2 - 0) = -9/2
Since the slopes of opposite sides are equal (9/2 and -9/2), it satisfies the first property.
2. Calculate the lengths of opposite sides:
Length of AB = √[(0 - 9)^2 + (-2 - 0)^2] = √(81 + 4) = √85
Length of BC = √[(2 - 0)^2 + (0 - 9)^2] = √(4 + 81) = √85
Length of CD = √[(0 - (-9))^2 + (2 - 0)^2] = √(81 + 4) = √85
Length of DA = √[(-2 - 0)^2 + (0 - (-9))^2] = √(4 + 81) = √85
Since the lengths of opposite sides are equal (√85), it satisfies the second property.
3. Check the angles:
Angle A = angle BAC = angle BDC = angle D
Angle C = angle BCA = angle BDA = angle A
Since the opposite angles are congruent, it satisfies the third property.
Therefore, based on the properties of a parallelogram, we can confirm that the quadrilateral with vertices A(-2, 0), B(0, 9), C(2, 0), and D(0, -9) is indeed a parallelogram.
A. Properties of a parallelogram involving slope:
1. Opposite sides are parallel: If the slopes of two lines are equal, then they are parallel.
2. Opposite sides are congruent: If the lengths of two sides are equal, then they are congruent.
3. Diagonals bisect each other: The midpoint of one diagonal is also the midpoint of the other diagonal.
B. To confirm that the given shape is a parallelogram, we must know the following:
1. Slopes: The slopes of opposite sides need to be equal to prove they are parallel.
2. Side lengths: The lengths of opposite sides need to be equal to prove they are congruent.
3. Diagonal lengths: The diagonals need to have the same length to prove they bisect each other.
C. To prove that A(-2, 0), B(0, 9), C(2, 0), and D(0, -9) are the vertices of a parallelogram, we need to check the properties mentioned above.
Step 1: Find the slopes of AB, BC, CD, and DA.
The slope formula is: slope = (change in y) / (change in x).
Slope of AB = (9 - 0) / (0 - (-2)) = 9 / 2 = 4.5
Slope of BC = (0 - 9) / (2 - 0) = -9 / 2 = -4.5
Slope of CD = (-9 - 0) / (0 - 2) = -9 / (-2) = 4.5
Slope of DA = (0 - (-9)) / (-2 - 0) = 9 / (-2) = -4.5
Since the slopes of AB and CD are equal (4.5) and the slopes of BC and DA are equal (-4.5), we have proven that opposite sides are parallel.
Step 2: Find the lengths of AB, BC, CD, and DA.
The distance formula is: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2).
Length of AB = sqrt((0 - (-2))^2 + (9 - 0)^2) = sqrt(2^2 + 9^2) = sqrt(4 + 81) = sqrt(85)
Length of BC = sqrt((2 - 0)^2 + (0 - 9)^2) = sqrt(2^2 + (-9)^2) = sqrt(4 + 81) = sqrt(85)
Length of CD = sqrt((0 - 2)^2 + (-9 - 0)^2) = sqrt((-2)^2 + (-9)^2) = sqrt(4 + 81) = sqrt(85)
Length of DA = sqrt((-2 - 0)^2 + (0 - (-9))^2) = sqrt((-2)^2 + 9^2) = sqrt(4 + 81) = sqrt(85)
Since AB = CD and BC = DA, we have proven that opposite sides are congruent.
Step 3: Find the midpoint of AC and BD.
The midpoint formula is: midpoint = ((x1 + x2) / 2, (y1 + y2) / 2).
Midpoint of AC = ((-2 + 2) / 2, (0 + 0) / 2) = (0, 0)
Midpoint of BD = ((0 + 0) / 2, (9 + (-9)) / 2) = (0, 0)
Since the midpoint of AC is the same as the midpoint of BD, we have proven that the diagonals bisect each other.
Based on the properties verified in Steps 1-3, we can conclude that A(-2, 0), B(0, 9), C(2, 0), and D(0, -9) are the vertices of a parallelogram.
length of AB = √(2^2+9^2) = √85
slope of AB = 9/2
Now do the other sides, and you must have a pair with the same slope (parallel) and length.