Slopes of Perpendicular Lines
Perpendicular lines meet at 90°. On the coordinate plane, their slopes are negative reciprocals — flip the fraction and change the sign.
Introduction
Parallel lines have equal slopes. Perpendicular lines have a different relationship: their slopes multiply to .
Past Knowledge
Slope formula (3.2.1). Parallel slopes (3.2.2). Perpendicular def. (3.1.1).
Today's Goal
Determine whether two lines are perpendicular by testing negative reciprocal slopes.
Future Success
Distance from a point to a line (3.2.6) requires drawing a perpendicular segment.
Key Concepts
Perpendicular Lines Slope Theorem
Two non-vertical lines are perpendicular if and only if the product of their slopes is :
Negative Reciprocal Quick Reference
| Slope of Line 1 | Perp. Slope (Line 2) |
|---|---|
| 0 (horizontal) | undefined (vertical) |
Interactive Diagram — Desmos Geometry
Two lines that meet at a 90° angle. Their slopes are negative reciprocals — verify the marked right angle in the construction.
Worked Examples
Checking Perpendicularity
Line 1 has slope . Line 2 has slope . Are they perpendicular?
Yes, perpendicular. Product = −1.
From Points
Line 1: to . Line 2: to . Parallel, perpendicular, or neither?
Product:
Perpendicular.
Right Triangle on the Plane
Show that , , form a right triangle.
Product:
AB ⊥ BC, so the right angle is at B. Triangle ABC is a right triangle.
Common Pitfalls
Forgetting to Flip AND Negate
The negative reciprocal of is , not (just flipped) or (just negated). You must do both.
Real-Life Applications
GPS & Navigation
GPS systems calculate the shortest distance from your location to a road by dropping a perpendicular from the point to the line of the road — using the negative reciprocal slope to find the direction of that perpendicular segment.
Practice Quiz
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