# Perpendicular Line Calculator

URL: https://speedycalculator.com/calculators/math/perp-line/
Last updated: 2026-10-30

> The perpendicular line calculator finds the line perpendicular to y = mx + r through a given point, and the point where the two lines cross.

## Inputs

- First line: slope m
- First line: intercept r
- Point – x₀
- Point – y₀

## Results

- Perpendicular slope a
- Perpendicular intercept b
- Intersection – x
- Intersection – y
- Distance from the point to the first line

## Formula

```
Perpendicular slope a = −1/m,  b = y₀ + x₀/m
Intersection: solve mx + r = ax + b
```

## Explanation

Perpendicular lines meet at right angles; their slopes multiply to −1. They give the shortest distance from a point to a line, the altitudes of triangles and right-angled street layouts.

### How to use the Perpendicular Line Calculator

- **First line: slope m**
- **First line: intercept r**
- **Point – x₀**
- **Point – y₀**

Results update as you type, and everything is calculated in your browser.

### How it works

The slope is the negative reciprocal of the original slope. The intercept makes the line pass through the point, and solving the two equations gives the foot of the perpendicular.

### Inputs explained

FieldWhat to enterFirst line: slope mA numberFirst line: intercept rA numberPoint – x₀A numberPoint – y₀A number

### What each result means

- **Perpendicular slope a** = -1÷[First line: slope m]
- **Perpendicular intercept b** = [Point – y₀] + [Point – x₀]÷[First line: slope m]
- **Intersection – x** = ([Point – y₀] + [Point – x₀]÷[First line: slope m] - [First line: intercept r])÷([First line: slope m] + 1÷[First line: slope m])
- **Intersection – y** = [First line: slope m]×([Point – y₀] + [Point – x₀]÷[First line: slope m] - [First line: intercept r])÷([First line: slope m] + 1÷[First line: slope m]) + [First line: intercept r]
- **Distance from the point to the first line** = abs([First line: slope m]×[Point – x₀] - [Point – y₀] + [First line: intercept r])÷sqrt([First line: slope m]^2 + 1)

### Example values

How the main result changes when **First line: slope m** changes and everything else stays at the example values above:

First line: slope mPerpendicular slope a1-11.5-0.66672-0.52.5-0.43-0.33334-0.25

## Worked example

Inputs — First line: slope m: 2; First line: intercept r: 1; Point – x₀: 3; Point – y₀: -2.

Results — **Perpendicular slope a = -0.5**; Perpendicular intercept b = -0.5; Intersection – x = -0.6; Intersection – y = -0.2; Distance from the point to the first line = 4.02492236.

## Key terms

- **Perpendicular**: Meeting at a right angle.

## FAQ

**What is the slope of a line perpendicular to y = 2x + 1?**
−1/2.

**Why do perpendicular slopes multiply to −1?**
Rotating a direction (1, m) by 90° gives (−m, 1), whose slope is −1/m.

**What is perpendicular to a horizontal line?**
A vertical line, which has no slope.

**What is the intersection point called?**
The foot of the perpendicular.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)