Parallel & Perpendicular Lines

Inputs

Original Line

y = mx + b Ax + By = C

Slope (m)

Rise over run

Y-intercept (b)

Where line crosses y-axis

A (coefficient of x)

B (coefficient of y)

C (constant)

Point the New Line Passes Through

x₁

y₁

Examples

Result

Enter a line and a point, then click Calculate to find the parallel or perpendicular line.

Slope of New Line

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Point-Slope Form

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Slope-Intercept Form

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⚠️ The point lies on the original line — the "parallel" line through this point is the original line.

Key Concept — Parallel & Perpendicular Slopes

Parallel: m₂ = m₁ (same slope) Perpendicular: m₂ = −1 / m₁ (negative reciprocal)

Parallel lines never intersect because they rise and run at exactly the same rate. They have identical slopes but different y-intercepts (unless they are the same line).

Perpendicular lines intersect at a 90° angle. Their slopes are negative reciprocals: multiply the two slopes and you always get −1. So if the original slope is 3/2, the perpendicular slope is −2/3.

Special cases: a horizontal line (m = 0) is perpendicular to a vertical line (undefined slope), and any horizontal line is parallel to another horizontal line.

If m₁ · m₂ = −1, the lines are perpendicular. Check your answer by multiplying the slopes!

Real-World Applications

Architecture & construction — walls must be perpendicular to floors; roof ridges run parallel to eaves.
Road design — highway lanes run parallel; on-ramps and side streets meet at specific angles.
Perpendicular bisectors — used in geometry proofs, circumcenter constructions, and Voronoi diagrams.
Navigation & mapping — grid systems use parallel and perpendicular streets to define blocks.
Physics — velocity and the normal force are perpendicular on inclined planes.
Computer graphics — normal vectors are perpendicular to surfaces for lighting calculations.

Parallel & Perpendicular Lines

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