Question

Determine whether and are parallel, perpendicular, or neither. w(3, 4), x(5, 7), y(8, 2), z(6, –1)

a. perpendicular
b. parallel
c. neither

Answer 1

The lines formed by the points w(3, 4), x(5, 7) and y(8, 2), z(6, –1) are neither parallel nor perpendicular, as their slopes are not equal, and the product of slopes is not -1. Option C is correct.

To determine whether the two lines formed by the points (w, x) and (y, z) are parallel, perpendicular, or neither, we can calculate the slopes.

The slope (m) of a line passing through two points
`\((x_1, y_1)\)` and
`\((x_2, y_2)\)` is given by:

`\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]`

Let's calculate the slopes of the two lines:

For line wx:

`\[ m_(wx) = \frac{{7 - 4}}{{5 - 3}} = (3)/(2) \]`

For line yz:

`\[ m_(yz) = \frac{{(-1) - 2}}{{6 - 8}} = (-3)/(2) \]`

Lines are perpendicular if the product of their slopes is -1, and they are parallel if their slopes are equal.

`\[ m_(wx) \cdot m_(yz) = (3)/(2) \cdot \left((-3)/(2)\right) = -(9)/(4) \\eq -1 \]`

Since the product of the slopes is not -1, the lines are not perpendicular. Also, since the slopes are not equal, the lines are not parallel.

Therefore, the correct answer is:

c. neither