ACT® · Math: Geometry — Coordinate Geometry and Conic Sections
Parallel and Perpendicular Slopes
Parallel lines always have equal slopes. Perpendicular lines have slopes that are negative reciprocals of each other, meaning they multiply to -1.
Slope measures steepness as rise over run. If two lines never cross, they climb at exactly the same rate forever, so their slopes must be identical. That's the whole idea behind parallel lines: same slope, different starting point (different y-intercept), so they stay the same distance apart forever.
Perpendicular lines meet at a right angle, a quarter turn. Rotating a line 90 degrees swaps its rise and run, which flips the slope upside down, and the turn also sends one direction positive while the other goes negative, which flips the sign. So a slope of m becomes -1/m. Multiply the two together and you always get -1: that product is the real test of perpendicularity, not just 'flip the sign.' A slope of 3 and a slope of -3 look like opposites, but 3 times -3 is -9, not -1, so they are not perpendicular. The true perpendicular partner of 3 is -1/3.
Once you have the right slope, point-slope form, y - b = m(x - a), builds the actual equation through a given point. This sits right next to writing equations of lines in general: find the slope first from the relationship (parallel or perpendicular), then plug in a point. The one exception worth knowing: a horizontal line has slope 0, and nothing multiplied by 0 gives -1, so its perpendicular partner is a vertical line, which has no defined slope at all.
Key terms
- Slope
- The steepness of a line, measured as vertical change divided by horizontal change.
- Negative reciprocal
- The result of flipping a fraction upside down and switching its sign, used to find perpendicular slopes.
- Point-slope form
- The equation y - b = m(x - a), used to write a line's equation from one point and its slope.
- Parallel lines
- Lines with equal slope that never intersect.
- Perpendicular lines
- Lines that meet at a right angle, whose slopes multiply to -1.
Watch it explained
- How it worksHow does turning a line a quarter turn flip its slope upside down and reverse its sign?
- Common mistakeWhy isn't a slope of -3 perpendicular to a slope of 3?
- Worked exampleFind the equation of the line through (4, -2) that runs perpendicular to y = (2/3)x + 5.
- Going furtherWhat happens to the perpendicular slope rule when one of the two lines lies flat?
Check yourself
End of topic.