Perpendicular Lines
Simple Explanation
Two lines are perpendicular if they meet at a right angle (90°). This happens exactly when one line's slope is the negative reciprocal of the other's — multiply the two slopes together and you always get −1.
Why Do We Need It?
Perpendicularity shows up anywhere right angles matter — building corners, road intersections, and squared-off designs.
Formula
Slope Formula
m = (y₂ − y₁) / (x₂ − x₁)
How steep a line is — the change in y for each unit change in x.
- m
- — the slope (gradient) of the line
- x₁, y₁
- — the coordinates of the first point on the line
- x₂, y₂
- — the coordinates of the second point on the line
When to use it: Whenever you need to measure or compare how steep a line is, or check whether two lines are parallel or perpendicular.
Worked Example
Find a perpendicular slope
A line has slope 2/3. What is the slope of a line perpendicular to it?
Why Does This Work?
Multiplying the two slopes must give −1: (2/3) × (−3/2) = −1, confirming the negative reciprocal rule — this specific product is what forces the two lines to meet at exactly 90°.
Real-Life Example
Squared wall corners
Builders check that two walls meet at a true right angle before finishing a room.
Two perpendicular walls have slopes that are negative reciprocals of each other, ensuring the corner is a perfect 90°.
Practice
A line has slope 5. What is the slope of a line perpendicular to it? (Give a decimal.)
EasyWhich slope is perpendicular to a line with slope -4?
MediumCommon mistake
Only flipping the fraction (finding the reciprocal) without also negating it — perpendicular slopes need both steps, not just one.
Quick Review
- Perpendicular slopes multiply to −1: m₁ × m₂ = −1.
- Equivalently, m₂ is the negative reciprocal of m₁.
- A horizontal line (slope 0) is perpendicular to a vertical line (undefined slope) — a special case the multiplication rule does not directly cover.