Mathematics and Statistics · Ch 5 — Locus and Straight Line
Parallel and Perpendicular Lines
Parallel and Perpendicular Lines
The slope captures a line's direction, so questions about two lines being parallel or perpendicular reduce to simple statements about their slopes.
Parallel lines point in the same direction, so they make the same angle with the x-axis and therefore have equal slopes:
Condition for Parallel Lines
Two non-vertical lines with slopes and are parallel if and only if .
A quick consequence for the general form: any line parallel to can be written as — the coefficients of and stay the same and only the constant changes. This makes "find the line through a given point parallel to a given line" a one-step substitution.
Perpendicular lines cross at a right angle. If one has slope and the other slope , then:
Condition for Perpendicular Lines
Two lines with slopes and are perpendicular if and only if , equivalently .
In words, the slope of a perpendicular line is the negative reciprocal of the original slope. For the general form, any line perpendicular to can be written as (the coefficients of and are swapped and one sign is changed), which is often the fastest route. …
. Equivalently, a line parallel to is $ …
; the slope of a perpendicular line is the negative reciprocal of the original. A line perpendicular to $ax+b …