Quick answer: Distinct nonvertical parallel lines have equal slopes. Nonvertical perpendicular lines with defined slopes satisfy m₁m₂ = −1. A horizontal line is perpendicular to a vertical line, handled separately.
Worked example
Find a line through (1, 2) perpendicular to y = 2x + 3.
- The given slope is 2.
- The perpendicular slope is −1/2.
- Use y − 2 = −(x − 1)/2.
- Multiply by 2 and rearrange: x + 2y = 5.
Check the result
At (1, 2), 1 + 4 = 5. The slope product is 2(−1/2) = −1.
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Common mistake to avoid
Equal slopes can also mean the same line, so distinguish coincident from distinct parallel lines. Never calculate a reciprocal of slope zero.
Try it yourself
1. Find a parallel to y = 3x + 1 through (0, −2). 2. What is perpendicular to y = 4 through (2, 0)?
Answers and reasoning
1. y = 3x − 2. 2. The vertical line x = 2.
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