Question

Use the distance and slope formulas to justify your answer to the question. Determine whether WXYZ given W(0, 8), X(6, 10), Y(-1, -1), Z(-7, -3) is a parallelogram.

Answer 1

WXYZ is not a parallelogram. Although the slopes of WX and YZ are equal, indicating parallelism, the lengths of the opposite sides are not equal, confirming that WXYZ is a trapezoid.

Step-by-step explanation:

To determine whether WXYZ is a parallelogram, we can analyze the slopes of its sides. A parallelogram has opposite sides that are both parallel and equal in length. Let's calculate the slopes of WX and YZ to check for parallelism. The slope (m) is given by the formula:

`\[m = (y_2 - y_1)/(x_2 - x_1)\]`

For WX:

`\[m_(WX) = (10 - 8)/(6 - 0) = (2)/(6) = (1)/(3)\]`

For YZ:

`\[m_(YZ) = ((-3) - (-1))/((-7) - (-1)) = (-2)/(-6) = (1)/(3)\]`

Both slopes are equal
`(\(m_(WX) = m_(YZ)\))`, indicating that WX and YZ are parallel. Now, let's check the slopes of WY and XZ:

For WY:

`\[m_(WY) = ((-1) - 8)/((-1) - 0) = (-9)/(-1) = 9\]`

For XZ:

`\[m_(XZ) = ((-3) - 10)/((-7) - 6) = (-13)/(-13) = 1\]`

The slopes
`\(m_(WY)\) and \(m_(XZ)\)` are not equal, indicating that WY and XZ are not parallel. However, since WX and YZ are parallel, and WY and XZ are not parallel, WXYZ is a trapezoid.

To confirm that WXYZ is a parallelogram, we also need to check if the opposite sides have equal lengths. Using the distance formula:

`\[d = √((x_2 - x_1)^2 + (y_2 - y_1)^2)\]`

The distances between opposite sides are:

`\[d_(WX) = √((6 - 0)^2 + (10 - 8)^2) = √(36 + 4) = √(40)\]`

`\[d_(YZ) = √((-7 - (-1))^2 + ((-3) - (-1))^2) = √(36 + 4) = √(40)\]`

`\[d_(WY) = √(((-1) - 0)^2 + ((-1) - 8)^2) = √(1 + 81) = √(82)\]`

`\[d_(XZ) = √(((-7) - 6)^2 + ((-3) - 10)^2) = √(169 + 169) = √(338)\]`

Since
`\(d_(WX) = d_(YZ)\)` and
`\(d_(WY) \\eq d_(XZ)\)`, the opposite sides of WXYZ are not equal in length. Therefore, WXYZ is a trapezoid, not a parallelogram.