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Perpendicular Line Calculator
Give a line as an equation, a slope and intercept, or two points, plus the point the new line must pass through. You get the perpendicular line in slope-intercept, point-slope and standard form, the parallel line through the same point, the foot of the perpendicular and the distance, all on a graph.
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Given line as
Given line
For example y = 2x + 1, 3x − 4y = 8, x = 2 or y − 1 = 1/2(x + 4)
Through the point
Enter your values to see the result.
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How to use the Perpendicular Line Calculator
- Choose how the given line is defined: equation, slope and intercept, or two points.
- Enter the line. Equations such as 3x − 4y = 8 or y = 2x + 1 are both fine.
- Enter the point the perpendicular line must pass through.
- Read the perpendicular line at the top, with its slope, point-slope form and the parallel line below.
- Check the graph and steps, then copy the results or download the table as CSV.
Perpendicular and parallel line formulas
| What | Formula | Example: y = 2x + 1, point (4, 1) |
|---|---|---|
| Perpendicular slope | m⊥ = −1 ÷ m | −1 ÷ 2 = −1/2 |
| Perpendicular line (point-slope) | y − y₀ = m⊥(x − x₀) | y − 1 = −1/2(x − 4) |
| Perpendicular line (slope-intercept) | y = m⊥x + (y₀ − m⊥x₀) | y = −1/2x + 3 |
| Parallel line | y − y₀ = m(x − x₀) | y = 2x − 7 |
| Standard form shortcut | Perpendicular to Ax + By = C: Bx − Ay = Bx₀ − Ay₀ | 2x − y = −1 gives x + 2y = 6 |
| Distance from point to line | |Ax₀ + By₀ − C| ÷ √(A² + B²) | |8 − 1 + 1| ÷ √5 ≈ 3.5777 |
Common slope pairs
| Slope m | Perpendicular slope | Product |
|---|---|---|
| 1 | −1 | −1 |
| 2 | −1/2 | −1 |
| 3/4 | −4/3 | −1 |
| −5 | 1/5 | −1 |
| 0 (horizontal) | undefined (vertical) | not defined |
| undefined (vertical) | 0 (horizontal) | not defined |
Tips
- To build a perpendicular bisector of a segment, enter the segment's endpoints as the two points and use its midpoint as the point.
- The foot of the perpendicular is the closest point on the line, which is useful for projecting a point onto a line.
- If you only need to know whether two lines are perpendicular, multiply their slopes: the answer is −1 for perpendicular lines.
Frequently asked questions
How do you find the equation of a perpendicular line?
Find the slope m of the given line, take the negative reciprocal m⊥ = −1 ÷ m, then use the point (x₀, y₀): y − y₀ = m⊥(x − x₀). For y = 2x + 1 through (4, 1), m⊥ = −1/2 and the line is y = −1/2x + 3.
What is a negative reciprocal slope?
It is the slope flipped upside down with the opposite sign. The negative reciprocal of 3/4 is −4/3 and of −5 is 1/5. Two non-vertical lines are perpendicular exactly when their slopes multiply to −1.
What is the perpendicular to a horizontal or vertical line?
A horizontal line y = k has slope 0, so its perpendicular through (x₀, y₀) is the vertical line x = x₀. A vertical line x = k has an undefined slope, so its perpendicular is the horizontal line y = y₀.
How do I find a parallel line through a point?
Parallel lines share the same slope. Keep the slope m of the given line and write y − y₀ = m(x − x₀). In standard form, keep A and B and set C = Ax₀ + By₀.
How far is a point from a line?
For a line Ax + By = C and a point (x₀, y₀), the shortest distance is |Ax₀ + By₀ − C| ÷ √(A² + B²). It is measured along the perpendicular, ending at the foot of the perpendicular.
What if the point is already on the line?
The perpendicular line still exists and crosses the given line at that point, so the foot of the perpendicular is the point itself and the distance is 0. The parallel line through the point is the given line itself.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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