Two nonvertical lines are perpendicular if and only if which relationship between their slopes holds?

Two nonvertical lines are perpendicular if and only if which relationship between their slopes holds?

• A. One line is vertical
• B. Sum of slopes is 0
• C. Slopes are equal
• D. Product of slopes is -1

Answer: (D)

The main idea is that perpendicular lines have slopes that are negative reciprocals. If a line makes an angle θ with the positive x-axis, its slope is m = tan θ. For two lines to be perpendicular, their angles differ by 90 degrees, so the second slope is tan(θ + 90°) = -cot θ = -1/tan θ = -1/m. Therefore the slopes multiply to -1. Because both lines are nonvertical, both slopes are defined, and this negative reciprocal relationship precisely captures perpendicularity. That’s why the product of the slopes being -1 is the correct criterion. If the sum of the slopes were zero, that would just mean the slopes are negatives of each other, which does not guarantee perpendicularity in general. Equal slopes mean the lines are parallel, not perpendicular. And if one line were vertical, its slope wouldn’t be defined, so the product criterion wouldn’t apply in the same way.

The main idea is that perpendicular lines have slopes that are negative reciprocals. If a line makes an angle θ with the positive x-axis, its slope is m = tan θ. For two lines to be perpendicular, their angles differ by 90 degrees, so the second slope is tan(θ + 90°) = -cot θ = -1/tan θ = -1/m. Therefore the slopes multiply to -1.

Because both lines are nonvertical, both slopes are defined, and this negative reciprocal relationship precisely captures perpendicularity. That’s why the product of the slopes being -1 is the correct criterion.

If the sum of the slopes were zero, that would just mean the slopes are negatives of each other, which does not guarantee perpendicularity in general. Equal slopes mean the lines are parallel, not perpendicular. And if one line were vertical, its slope wouldn’t be defined, so the product criterion wouldn’t apply in the same way.