If two lines are perpendicular, their slopes multiply to what value (provided both are defined)?

• A. -1
• B. 0
• C. 1
• D. Undefined

Answer: (A)

Slopes of perpendicular lines are negative reciprocals of each other, assuming neither line is vertical. If one line has slope m, it makes an angle θ with the x-axis, so m = tan θ. A line perpendicular to it makes angle θ + 90°, with slope tan(θ + 90°) = -cot θ = -1/tan θ = -1/m. Multiplying the two slopes gives m * (-1/m) = -1. Another way to see this is using the tangent of the angle between lines: tan(φ - θ) = (m2 - m1) / (1 + m1m2); for a 90° difference, the tangent is undefined, so the denominator must be zero, giving 1 + m1m2 = 0 and thus m1*m2 = -1. Therefore, for two non-vertical lines that are perpendicular, their slopes multiply to -1.