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Determine whether lines are perpendicular using slope

Perpendicularity in the coordinate plane is determined by the relationship between slopes: for two nonvertical lines, their slopes are negative reciprocals, so m1m2=−1m_1m_2=-1; a horizontal line and a vertical line are also perpendicular, even though the vertical line’s slope is undefined. The reasoning includes finding slopes from graphs, ordered pairs, or linear equations and distinguishes negative reciprocals from merely opposite slopes; angle-measure formulas and vector-based generalizations are beyond this treatment.

Detailed Explanation: Determine whether lines are perpendicular using slope

To determine whether two nonvertical lines are perpendicular:

  1. Find the slope of each line.
  2. Check whether the slopes are negative reciprocals.
  3. Equivalently, multiply the slopes. If
m1m2=−1,m_1m_2=-1,

the lines are perpendicular.

Remember: negative reciprocals have signs that are opposite and fractions that are flipped. For example, the negative reciprocal of 23\frac{2}{3} is −32-\frac{3}{2}, not −23-\frac{2}{3}.

Example: Are the lines

y=23x−1y=\frac{2}{3}x-1

and

y=−32x+4y=-\frac{3}{2}x+4

perpendicular?

The equations are in slope-intercept form, y=mx+by=mx+b, so the slopes are the coefficients of xx:

m1=23,m2=−32.m_1=\frac{2}{3}, \qquad m_2=-\frac{3}{2}.

Now multiply the slopes:

m1m2=23(−32)=−1.m_1m_2=\frac{2}{3}\left(-\frac{3}{2}\right)=-1.

Because the product of the slopes is −1-1, the lines are perpendicular.

A horizontal line and a vertical line are also perpendicular. A vertical line does not have a defined slope, so use this special case instead of multiplying slopes.

Learn by doing: Determine whether lines are perpendicular using slope

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