Lesson 5 — Parallel & Perpendicular Lines

Parallel → same slope; perpendicular → negative reciprocal slopes. Write equations under constraints.

Compare slopes

Parallel if m₁ = m₂ (and different intercepts).

Perpendicular if m₁ · m₂ = −1 (i.e., m₂ = −1/m₁ with neither vertical/horizontal conflict).

Special cases: vertical lines (x = c) are ⟂ to horizontal lines (y = k).

Write the line

Parallel to y = m x + b through (x₀,y₀) → slope m, so y − y₀ = m(x − x₀).

Perpendicular → slope m' = −1/m (or vertical/horizontal pair) and same pattern.

In physics, parallel lines model same-direction motion; perpendicular lines model forces like gravity vs. normal force.

Examples

Parallel Lines: Line 1: y = 2x + 3, Line 2: y = 2x - 1. Both have slope 2, so they’re parallel (different intercepts ensure they don’t coincide).

Perpendicular Lines: Line 1: y = 3x + 2, Line 2: y = -1/3 x + 5. Slopes are 3 and -1/3, and 3 · (-1/3) = -1, so they’re perpendicular.

Vertical/Horizontal: Line 1: x = 4 (vertical), Line 2: y = -2 (horizontal). Vertical and horizontal lines are always perpendicular.

Sandbox — Spot the Relationship

Line 1: slope m₁ intercept b₁

or set vertical: x = c → check & enter c

vertical? c =

Line 2: slope m₂ intercept b₂

or set vertical: x = c → check & enter c

vertical? c =

Adjust lines and classify: parallel, perpendicular, or neither.

Sandbox — Build the Required Line

Given line

vertical? x =

Through point (x₀, y₀)

(, )

Relationship

We’ll draw the given line, your point, and the constructed line.

Try It — Test Your Understanding

Question 1 of 5

Lines with slopes 3 and −1/3 are perpendicular.

More Practice (unlocks as you answer correctly)

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