Mathwords logoReference LibraryMathwords

Perpendicular Slopes

Perpendicular slopes describe the relationship between the slopes of two lines that meet at a right angle. If one line has slope mm, the perpendicular line has slope −1m-\frac{1}{m}, which is the negative reciprocal.

Two non-vertical lines in a plane are perpendicular if and only if the product of their slopes equals −1-1. That is, if line ℓ1\ell_1 has slope m1m_1 and line ℓ2\ell_2 has slope m2m_2, the lines are perpendicular when m1⋅m2=−1m_1 \cdot m_2 = -1. Equivalently, m2=−1m1m_2 = -\frac{1}{m_1}. This relationship does not apply when one of the lines is vertical (undefined slope) and the other is horizontal (slope of zero), though those lines are still perpendicular to each other.

Key Formula

m1⋅m2=−1m_1 \cdot m_2 = -1
Where:
  • m1m_1 = the slope of the first line
  • m2m_2 = the slope of the second line, perpendicular to the first

Worked Example

Problem: A line passes through the points (1, 2) and (3, 8). Find the slope of any line perpendicular to it.
Step 1: Find the slope of the given line using the slope formula.
m1=8−23−1=62=3m_1 = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3
Step 2: Take the negative reciprocal to find the perpendicular slope. Flip the fraction and change the sign.
m2=−1m1=−13m_2 = -\frac{1}{m_1} = -\frac{1}{3}
Step 3: Verify by checking that the product of the two slopes equals −1-1.
3×(−13)=−1✓3 \times \left(-\frac{1}{3}\right) = -1 \checkmark
Answer: The slope of any line perpendicular to the given line is −13-\frac{1}{3}.

Visualization

Why It Matters

Perpendicular slopes show up whenever you need to construct right angles using equations — for example, finding the shortest distance from a point to a line, or determining whether streets on a city grid actually meet at 90°. In geometry proofs, verifying that two slopes multiply to −1-1 is a standard way to prove lines are perpendicular without measuring angles directly.

Common Mistakes

Mistake: Only flipping the fraction without changing the sign (or only changing the sign without flipping).
Correction: You must do both operations. The negative reciprocal of 23\frac{2}{3} is −32-\frac{3}{2}, not −23-\frac{2}{3} and not 32\frac{3}{2}.
Mistake: Applying the negative reciprocal rule to a vertical and horizontal line pair.
Correction: A vertical line has an undefined slope, so you cannot compute −1m-\frac{1}{m}. Vertical and horizontal lines are perpendicular by definition — just recognize the special case.

Related Terms