Perpendicular lines are lines that intersect to form right angles: their intersection makes a 90-degree angle. The symbol ⊥ means is perpendicular to. On a coordinate plane, two perpendicular lines that are not vertical have slopes that multiply to −1, such as 2 and −12.
Crossing is not enough: perpendicular lines must meet at exactly 90°. On a graph, their slopes are negative reciprocals, so they multiply to−1.
Formal definition and notation
Two lines are perpendicular if they intersect and form a right angle, an angle of 90°. When two lines meet at one right angle, all four angles at the intersection are right angles. In the coordinate plane, non-vertical lines with slopesm1 and m2 are perpendicular exactly when m1 × m2 = −1, so each slope is the negative reciprocal of the other. A horizontal line and a vertical line are always perpendicular. Segments or rays are perpendicular when the lines containing them are.
Perpendicular lines make four right angles, not just one.
Adjacent sides of a rectangle or square are perpendicular, and the legs of a right triangle are perpendicular.
The shortest path from a point to a line is the segment perpendicular to the line.
Every point on the perpendicular bisector of a segment is the same distance from the segment’s two endpoints.
Slope test for perpendicular lines
Slope test for perpendicular lines
m1 × m2 = −1, so m2 = −1m1
m1
the slope of the first line (not 0, not undefined)
m2
the slope of the perpendicular line
To get the perpendicular slope, flip the fraction and change its sign: 23 becomes −32, and −4 becomes 14. Horizontal (slope 0) and vertical (undefined) lines are perpendicular even though the product rule does not apply.
Perpendicular Lines examples, step by step
Example 1: Find a perpendicular slope
Warm-up
What is the slope of a line perpendicular to y = 3x − 4?
The given slope is 3, or 31.
Flip it: 13. Change the sign: −13.
Check: 3 × (−13) = −1.
Answer: The perpendicular slope is −13.
Example 2: Test two equations
Core
Are the lines y = 23x + 1 and 3x + 2y = 10 perpendicular?
Solve the second equation for y: 2y = −3x + 10, so y = −32x + 5.
The slopes are 23 and −32.
Multiply: 23 × (−32) = −1.
Answer: Yes, the lines are perpendicular: their slopes multiply to −1.
Example 3: Write a perpendicular line through a point
Test-style
Write an equation of the line through (8, 3) that is perpendicular to y = −4x + 1.
The given slope is −4, so the perpendicular slope is 14.
Parallel lines never meet and have equal slopes (2 and 2). Perpendicular lines meet at 90° and have slopes that multiply to −1 (2 and −12).
Perpendicular Linesvs.Intersecting lines
Intersecting lines share one point at any angle. Perpendicular lines are the special case where the angle is 90°.
Perpendicular Linesvs.Perpendicular bisector
A perpendicular bisector is a particular perpendicular line: it crosses a segment at a right angle and also through its midpoint.
Quick check: 3 questions
Question 1 of 3
A line has slope 5. What is the slope of a line perpendicular to it?
Show answer and explanation
Answer: D. Flip and change the sign: −15, because 5 × (−15) = −1. Choice A is the parallel slope, B only changes the sign, and C only flips.
Question 2 of 3
Which pair of lines is perpendicular?
Show answer and explanation
Answer: C.12 × (−2) = −1, so C is perpendicular. A is a parallel pair, B has slopes that only differ in sign (product −9), and D’s slopes multiply to 2.
Question 3 of 3
A line passes through (0, 0) and (2, 6). What is the slope of a line perpendicular to it?
Show answer and explanation
Answer: B. The slope of the line is 6 − 02 − 0 = 3, so the perpendicular slope is −13. Choice A is the line’s own slope, C only changes the sign, and D only flips it.
Perpendicular Lines: frequently asked questions
Do perpendicular lines have to cross?
Yes. Perpendicular lines meet at a point and form right angles there. Two segments can be called perpendicular even if they do not touch, as long as the lines they lie on would meet at 90°. Lines in space that never meet are not perpendicular, even if they point in perpendicular directions.
Why do the slopes of perpendicular lines multiply to -1?
Turning a line by 90° swaps its rise and run and flips one direction. A slope of 23 (up 2, right 3) becomes up 3, left 2, which is −32. The two slopes are negative reciprocals, and a number times its negative reciprocal is always −1.
Are the x-axis and the y-axis perpendicular?
Yes. The x-axis is horizontal and the y-axis is vertical, and they cross at the origin at a right angle. Every horizontal line is perpendicular to every vertical line for the same reason. The slope rule cannot test this pair, because a vertical line’s slope is undefined, but the right angle is still there.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 4.G.A.1, CCSS HSG-CO.A.1, CCSS HSG-GPE.B.5
Cite this entry
Effortless Math Team. “Perpendicular Lines.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/perpendicular-lines/
Flashcard
TermPerpendicular Linesm ⊥ n
DefinitionLines that meet at a right angle. Their slopes multiply to −1: y = 2x is perpendicular to y = −12x + 5.
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