Calculator form
Formula used
Perpendicular gradient rule: m⊥ = -1 / m
From two points: m = (y₂ - y₁) / (x₂ - x₁)
From standard form: For Ax + By + C = 0, m = -A / B
The gradient of a line perpendicular to another line is the negative reciprocal of the original gradient. If the original line has gradient 2, the perpendicular gradient is -1/2.
There are two important special cases. A horizontal line has gradient 0, so any perpendicular line is vertical and has an undefined gradient. A vertical line has an undefined gradient, so any perpendicular line is horizontal and has gradient 0.
How to use this calculator
- Choose how you want to enter the original line: gradient, two points, or standard equation.
- Enter the required values in the visible fields.
- Add a reference point if you want one exact perpendicular line equation.
- Pick the number of decimal places you want in the output.
- Press the calculate button to view the result above the form.
- Review the working steps, line equations, angles, and graph.
- Use the CSV and PDF buttons to export your current result.
Example data table
| Case | Input | Original gradient | Perpendicular gradient | Example perpendicular line |
|---|---|---|---|---|
| Known gradient | m = 2, reference point (0, 0) | 2 | -1/2 | y = -0.5x |
| Two points | (1, 2) and (5, 10) | 2 | -1/2 | y - 2 = -0.5(x - 1) |
| Horizontal line | m = 0, reference point (3, 4) | 0 | Undefined | x = 3 |
| Standard equation | 2x + 4y - 8 = 0 | -1/2 | 2 | y = 2x + 2 |
Frequently asked questions
1) What is the gradient of a perpendicular line?
It is the negative reciprocal of the original gradient. If a line has gradient m, a perpendicular line has gradient -1/m, except for horizontal and vertical special cases.
2) What happens when the original gradient is zero?
A zero gradient means the original line is horizontal. Any perpendicular line is vertical, so its gradient is undefined rather than a finite number.
3) What if the original line is vertical?
A vertical line does not have a finite gradient. Any perpendicular line is horizontal, so the perpendicular gradient becomes 0.
4) Can I use two points instead of a known gradient?
Yes. The calculator first finds the original gradient using the two-point formula, then applies the perpendicular rule to produce the required gradient.
5) Can I use a standard equation such as Ax + By + C = 0?
Yes. The calculator converts standard form to gradient form using m = -A/B when B is not zero, then finds the perpendicular gradient.
6) Why is a reference point useful?
A gradient gives only a direction, not one unique line. A reference point lets the calculator draw one exact perpendicular line and write its equation.
7) Does the graph show both lines accurately?
Yes. The graph plots the original line, the perpendicular line, and the reference point. Equal axis scaling helps the right-angle relationship look visually correct.
8) Are fractions accepted in the calculator fields?
Yes. You can enter fractions such as 3/4 or -5/2 in most fields. The calculator also shows decimal output and fraction-style interpretations where possible.