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Parallel & Perpendicular Line Calculator

Enter a slope and a point to get the parallel and perpendicular line equations through that point.
Last updated: June 22, 2026 · Reviewed by TG we-Calculate Editorial Team
The lines will be parallel / perpendicular to this slope.
Through point (x, y)
Parallel line
y = 2x + 1
Perpendicular line
y = -0.5x + 3.5
Parallel slope
2
Perpendicular slope
-0.5
(x, y)
Step by step
  1. 1

    Parallel slope

    m = 2 = 2
    A parallel line keeps the same slope as the reference line.
  2. 2

    Parallel y-intercept

    3 − 2 × 1 = 1
  3. 3

    Perpendicular slope

    −1 ÷ 2 = -0.5
  4. 4

    Perpendicular y-intercept

    3 − -0.5 × 1 = 3.5000
Results are estimates for general information only and are not professional advice — always verify important results independently before relying on them. Read the full disclaimer.
Formula
Parallel slope = m; Perpendicular slope = −1 ÷ m; y = slope·(x − x₁) + y₁
How this is calculated

Two lines are parallel when they have exactly the same slope, and perpendicular when their slopes are negative reciprocals of each other — that is, the two slopes multiply to −1. So given a reference slope m and a point the new lines must pass through, the parallel line keeps the slope m, while the perpendicular line takes slope −1 ÷ m.

Each equation is then pinned to the point: substituting the point into y = slope·(x − x₁) + y₁ gives the y-intercept, and the calculator reports both lines in slope-intercept form.

The horizontal/vertical case is handled specially. If the reference slope is 0 the line is horizontal; its parallel is also horizontal (y = y₁) and its perpendicular is the vertical line x = x₁, which has no defined slope.

Examples
InputResult
slope 2 through (1, 3)Parallel: y = 2x + 1; Perpendicular: y = −0.5x + 3.5

Frequently asked questions

The negative reciprocal: if a line has slope m, a perpendicular line has slope −1/m. The two slopes multiply to −1.

A vertical line. Its slope is undefined and its equation is x = constant.

Also known as

parallel line calculator
perpendicular line calculator
parallel and perpendicular lines
negative reciprocal slope
perpendicular slope
parallel line equation
perpendicular gradient calculator

Sources & references

APA

TG we-Calculate Editorial Team. (2026). Parallel & Perpendicular Line Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/parallel-perpendicular-line-calculator

Chicago

TG we-Calculate Editorial Team. "Parallel & Perpendicular Line Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/parallel-perpendicular-line-calculator.

IEEE

TG we-Calculate Editorial Team, "Parallel & Perpendicular Line Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/parallel-perpendicular-line-calculator

BibTeX

@misc{wecalculate_parallel_perpendicular_line_calculator, title = {Parallel & Perpendicular Line Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/parallel-perpendicular-line-calculator}}, year = {2026}, note = {TG we-Calculate} }

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