a parallelogram is a quadrilateral whose opposite sides are parallel. The vertices of the quadrilateral ABCD are A(2,0), B(-2,2),C(2,8), D(6,6). Determine whether or not ABCD is a Parallel.
To determine whether ABCD is a parallelogram, we need to compare the slopes of its opposite sides.
Let's start by finding the slopes of the two pairs of opposite sides:
1. Slope of AB:
Slope = (y₂ - y₁) / (x₂ - x₁)
= (2 - 0) / (-2 - 2)
= 2 / -4
= -1/2
2. Slope of CD:
Slope = (y₂ - y₁) / (x₂ - x₁)
= (6 - 8) / (6 - 2)
= -2 / 4
= -1/2
Now let's compare the slopes of both pairs of opposite sides. If the slopes are equal, then the opposite sides are parallel.
The slopes of AB and CD are both -1/2. Since the slopes are equal, we can conclude that the opposite sides AB and CD are parallel.
Therefore, ABCD is a parallelogram because its opposite sides AB and CD are parallel.
To determine whether or not the quadrilateral ABCD is a parallelogram, we need to check if the opposite sides are parallel.
Let's start by labeling the coordinates of the given points: A(2,0), B(-2,2), C(2,8), and D(6,6).
Using the formula for the slope of a line (m), we can calculate the slopes of the two pairs of opposite sides: AB and CD, and BC and AD.
The formula for calculating the slope (m) between two points (x₁, y₁) and (x₂, y₂) is given by: m = (y₂ - y₁) / (x₂ - x₁).
For side AB, the coordinates are A(2,0) and B(-2,2). Calculating the slope, we have:
m₁ = (2 - 0) / (-2 - 2) = 2 / -4 = -1/2.
For side CD, the coordinates are C(2,8) and D(6,6). Calculating the slope, we have:
m₂ = (6 - 8) / (6 - 2) = -2 / 4 = -1/2.
Since m₁ = m₂ = -1/2, the slopes of sides AB and CD are equal.
Now, let's calculate the slopes for sides BC and AD.
For side BC, the coordinates are B(-2,2) and C(2,8). Calculating the slope, we have:
m₃ = (8 - 2) / (2 - (-2)) = 6 / 4 = 3/2.
For side AD, the coordinates are A(2,0) and D(6,6). Calculating the slope, we have:
m₄ = (6 - 0) / (6 - 2) = 6 / 4 = 3/2.
Since m₃ = m₄ = 3/2, the slopes of sides BC and AD are equal.
Now, compare the slopes for the opposite sides: AB and CD have slopes of -1/2, and BC and AD have slopes of 3/2.
Since the opposite sides AB and CD have the same slope (-1/2), and BC and AD have the same slope (3/2), we can conclude that the opposite sides of ABCD are parallel.
Therefore, quadrilateral ABCD is a parallelogram.