Question

Determine whether the line through (3, 0) and (7, -2) are parallel, perpendicular, or neither to the line through (-1, 4) and (1, 5).

Answer 1

The line through points (3, 0) and (7, -2) has a slope of -1/2, while the line through points (-1, 4) and (1, 5) has a slope of 1/2. Since the slopes are neither equal nor negative reciprocals of each other, the lines are neither parallel nor perpendicular.

Step-by-step explanation:

To determine whether the lines through (3, 0) and (7, -2), and through (-1, 4) and (1, 5) are parallel, perpendicular, or neither, we need to compare their slopes. The slope is found by using the formula (change in y)/(change in x), or Δy/Δx.

For the first line, the slope, m1, is calculated as follows:

1. Calculate the change in y: -2 - 0 = -2.
2. Calculate the change in x: 7 - 3 = 4.
3. Divide the change in y by the change in x to find the slope: m1 = -2/4 = -1/2.

For the second line, the slope, m2, is calculated in the same manner:

1. Calculate the change in y: 5 - 4 = 1.
2. Calculate the change in x: 1 - (-1) = 2.
3. Divide the change in y by the change in x to find the slope: m2 = 1/2.

If two lines are parallel, their slopes are equal. If they are perpendicular, the product of their slopes is -1. In this case, since m1 = -1/2 and m2 = 1/2, the lines are neither parallel nor perpendicular because -1/2 × 1/2 is not -1, and -1/2 is not equal to 1/2.