Calculator for Perpendicular Lines

Perpendicular Line Equation Calculator

Use this calculator to find the equation of a line that is perpendicular to a given line and passes through a specific point. Perpendicular lines intersect at a 90-degree angle, and their slopes are negative reciprocals of each other (unless one is horizontal and the other is vertical).

Slope of the Given Line (m₁):

X-coordinate of Point (x₁):

Y-coordinate of Point (y₁):

Understanding Perpendicular Lines

Two lines are considered perpendicular if they intersect at a right (90-degree) angle. This geometric relationship has a direct mathematical translation in terms of their slopes.

The Relationship Between Slopes

For any two non-vertical, non-horizontal perpendicular lines, the product of their slopes is -1. This means if the slope of one line is m₁, the slope of a line perpendicular to it, m₂, will be its negative reciprocal:

m₂ = -1 / m₁

There are special cases:

If the given line is horizontal (slope m₁ = 0), the perpendicular line will be vertical. A vertical line has an undefined slope and its equation is of the form x = constant.
If the given line is vertical (undefined slope), the perpendicular line will be horizontal. A horizontal line has a slope of 0 and its equation is of the form y = constant.

Finding the Equation of a Perpendicular Line

To find the equation of a line, you typically need two pieces of information: its slope and a point it passes through. If you have the slope of a given line and a point that the perpendicular line passes through, you can follow these steps:

Determine the slope of the perpendicular line (m₂):
- If the given line's slope (m₁) is 0, the perpendicular line is vertical, and its equation will be x = x₁ (where x₁ is the x-coordinate of the given point).
- Otherwise, calculate m₂ = -1 / m₁.
Use the point-slope form: The point-slope form of a linear equation is y - y₁ = m₂(x - x₁), where (x₁, y₁) is the point the perpendicular line passes through.
Convert to slope-intercept form (y = mx + b): Rearrange the point-slope equation to solve for y, which will give you the equation in the familiar y = m₂x + b format, where b is the y-intercept. The y-intercept b can be found using b = y₁ - m₂x₁.

Example Calculation:

Let's say the given line has a slope of 2, and the perpendicular line passes through the point (4, 3).

Given slope (m₁): 2
Point (x₁, y₁): (4, 3)
Calculate perpendicular slope (m₂): m₂ = -1 / 2 = -0.5
Calculate y-intercept (b): Using b = y₁ - m₂x₁
b = 3 - (-0.5 * 4)
b = 3 - (-2)
b = 3 + 2
b = 5
Equation of the perpendicular line: y = -0.5x + 5

If the given line had a slope of 0 and the perpendicular line passed through (5, -2):

Given slope (m₁): 0
Point (x₁, y₁): (5, -2)
Since m₁ = 0, the perpendicular line is vertical.
Equation of the perpendicular line: x = 5

How to Use the Calculator:

Simply enter the slope of your initial line in the "Slope of the Given Line (m₁)" field. Then, input the X and Y coordinates of the point that your desired perpendicular line should pass through. Click "Calculate Perpendicular Line" to see its equation.

.calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; padding: 20px; border-radius: 8px; box-shadow: 0 2px 10px rgba(0, 0, 0, 0.1); max-width: 600px; margin: 20px auto; color: #333; } .calculator-container h2 { color: #0056b3; text-align: center; margin-bottom: 20px; } .calculator-container h3 { color: #0056b3; margin-top: 25px; margin-bottom: 10px; } .calculator-container p { line-height: 1.6; margin-bottom: 10px; } .calc-input-group { margin-bottom: 15px; } .calc-input-group label { display: block; margin-bottom: 5px; font-weight: bold; color: #555; } .calc-input-group input[type="number"] { width: calc(100% – 20px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; font-size: 16px; } .calc-button { display: block; width: 100%; padding: 12px 20px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 18px; cursor: pointer; transition: background-color 0.3s ease; margin-top: 20px; } .calc-button:hover { background-color: #0056b3; } .calc-result { margin-top: 25px; padding: 15px; background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 4px; font-size: 1.1em; font-weight: bold; color: #155724; text-align: center; word-wrap: break-word; } .calculator-article { margin-top: 30px; border-top: 1px solid #eee; padding-top: 20px; } .calculator-article ol, .calculator-article ul { margin-left: 20px; margin-bottom: 10px; } .calculator-article li { margin-bottom: 5px; } .calculator-article code { background-color: #e0e0e0; padding: 2px 4px; border-radius: 3px; font-family: 'Courier New', Courier, monospace; } function calculatePerpendicularLine() { var slopeOfGivenLineInput = document.getElementById("slopeOfGivenLine").value; var pointXInput = document.getElementById("pointX").value; var pointYInput = document.getElementById("pointY").value; var resultDiv = document.getElementById("perpendicularLineResult"); var m1 = parseFloat(slopeOfGivenLineInput); var x1 = parseFloat(pointXInput); var y1 = parseFloat(pointYInput); if (isNaN(m1) || isNaN(x1) || isNaN(y1)) { resultDiv.innerHTML = "Please enter valid numbers for all fields."; resultDiv.style.backgroundColor = "#f8d7da"; resultDiv.style.borderColor = "#f5c6cb"; resultDiv.style.color = "#721c24"; return; } var m2; // Slope of the perpendicular line var b; // Y-intercept of the perpendicular line var equation = ""; if (m1 === 0) { // Given line is horizontal (y = constant), perpendicular is vertical (x = constant) equation = "Equation of Perpendicular Line: x = " + x1; } else if (m1 === -0) { // Handle negative zero if it somehow appears equation = "Equation of Perpendicular Line: x = " + x1; } else { // Calculate perpendicular slope m2 = -1 / m1; // Calculate y-intercept using point-slope form: y – y1 = m2(x – x1) => b = y1 – m2*x1 b = y1 – (m2 * x1); // Format the equation var m2_str = m2.toFixed(4); // Format slope to 4 decimal places var b_str = b.toFixed(4); // Format intercept to 4 decimal places // Clean up trailing zeros for integers if (m2 % 1 === 0) m2_str = m2.toString(); if (b % 1 === 0) b_str = b.toString(); if (b_str === "0") { equation = "Equation of Perpendicular Line: y = " + m2_str + "x"; } else if (parseFloat(b_str) < 0) { equation = "Equation of Perpendicular Line: y = " + m2_str + "x – " + Math.abs(parseFloat(b_str)); } else { equation = "Equation of Perpendicular Line: y = " + m2_str + "x + " + b_str; } } resultDiv.innerHTML = equation; resultDiv.style.backgroundColor = "#e9f7ef"; resultDiv.style.borderColor = "#d4edda"; resultDiv.style.color = "#155724"; }