About the Parallel & Perpendicular Line Calculator
Give this calculator a line and a point and it writes the equation of the line through that point that is parallel to the original, and the one that is perpendicular to it. The original line can be entered in slope-intercept form (y = mx + b), standard form (Ax + By = C) or as two points it passes through, so you can use it straight from a textbook exercise.
Both answers are given in slope-intercept and standard form, with their slopes and y-intercepts. The calculator also finds the foot of the perpendicular — the point where the perpendicular line meets the original — and the shortest distance from your point to the original line. Vertical and horizontal lines are handled properly: a line parallel to x = 2 is another vertical line, and the perpendicular is horizontal.
Parallel lines share the same slope; perpendicular lines have slopes that are negative reciprocals, so their product is −1. That is all the calculator needs, plus the point-slope form y − y₁ = m(x − x₁).
How to use the parallel & perpendicular line calculator
- 1Choose how the original line is given: slope-intercept, standard form or two points.
- 2Enter the line’s values.
- 3Enter the point the new lines must pass through.
- 4Read the parallel and perpendicular equations in both forms.
- 5Use the foot of the perpendicular and the distance for shortest-path problems.
Formula and method
The original line is converted to the general form Ax + By = C. A parallel line keeps the same A and B (so the same slope −A/B) and must pass through (x₀, y₀), giving Ax + By = Ax₀ + By₀. A perpendicular line swaps the coefficients and changes one sign, Bx − Ay = Bx₀ − Ay₀, which makes its slope the negative reciprocal B/A. Working in this form means vertical (B = 0) and horizontal (A = 0) lines need no special handling.
Each result is then rewritten in slope-intercept form y = mx + b where possible. The foot of the perpendicular is found by projecting the point onto the line, and the point-to-line distance uses the standard formula |Ax₀ + By₀ − C| ÷ √(A² + B²).
- m
- Slope of the original line
- (x₀, y₀)
- Point the new lines pass through
- A, B, C
- Coefficients of Ax + By = C
- d
- Shortest distance from the point to the line
Worked examples
Through (3, 4), relative to y = 2x + 1
The parallel line keeps slope 2: 4 = 2(3) + b gives b = −2. The perpendicular slope is −1/2: 4 = −0.5(3) + b gives b = 5.5. The lines meet the original at (1.8, 4.6), and the point is 3/√5 ≈ 1.342 units away.
Standard form 3x + 4y = 12 through (1, 1)
The slope of 3x + 4y = 12 is −3/4. The parallel line is 3x + 4y = 7 (y = −0.75x + 1.75); the perpendicular is 4x − 3y = 1 with slope 4/3. The point is |3 + 4 − 12| ÷ 5 = 1 unit from the line.
Line through (1, 2) and (5, 10), point (0, 3)
The slope through the two points is (10 − 2) ÷ (5 − 1) = 2, giving y = 2x. Through (0, 3) the parallel is y = 2x + 3 and the perpendicular is y = −0.5x + 3.
Frequently asked questions
How do you find the equation of a parallel line?+
Use the same slope as the original line and the point-slope form y − y₀ = m(x − x₀). For y = 2x + 1 through (3, 4): y − 4 = 2(x − 3), so y = 2x − 2.
How do you find the slope of a perpendicular line?+
Take the negative reciprocal of the original slope: flip the fraction and change the sign. A slope of 2/3 becomes −3/2, and the product of the two slopes is −1.
What is perpendicular to a vertical line?+
A horizontal line. A vertical line x = a has an undefined slope, so a perpendicular through (x₀, y₀) is simply y = y₀, and a parallel one is x = x₀.
How do you tell if two lines are parallel or perpendicular?+
Compare slopes. Equal slopes (and different intercepts) mean parallel lines; slopes whose product is −1 mean perpendicular lines. Anything else intersects at another angle.
How do I find the slope from standard form Ax + By = C?+
Solve for y: the slope is −A/B and the y-intercept is C/B. For 3x + 4y = 12, the slope is −3/4 and the y-intercept is 3.