Perpendicular Line Calculator

Find the equation of a line perpendicular to another, passing through a given point.

Define the original line, specify the point it must pass through, and calculate the perpendicular line's equation.

Examples

See how to use the calculator with common scenarios.

Using Slope-Intercept Form

slope-intercept

Find the line perpendicular to y = 2x + 3 that passes through the point (4, 5).

slope: 2

yIntercept: 3

pointX: 4

pointY: 5

Using Two-Point Form

two-point

Find the line perpendicular to the one passing through (1, 1) and (3, 5), which also passes through (2, 6).

x1: 1

y1: 1

x2: 3

y2: 5

pointX: 2

pointY: 6

Using Standard Form

standard

Find the line perpendicular to 3x + 4y = 8 that passes through (-1, 2).

coeffA: 3

coeffB: 4

coeffC: 8

pointX: -1

pointY: 2

Case: Horizontal Line

slope-intercept

Find the line perpendicular to the horizontal line y = 5 that passes through (3, -2). The result should be a vertical line.

slope: 0

yIntercept: 5

pointX: 3

pointY: -2

Other Titles
Understanding the Perpendicular Line Calculator: A Comprehensive Guide
An in-depth look at perpendicular lines, their properties, and how to calculate them.

What is a Perpendicular Line?

  • Core Definition
  • The Slope Relationship
  • Special Cases: Horizontal and Vertical Lines
In geometry, two lines are considered perpendicular if they intersect at a right angle (90 degrees). This relationship is a fundamental concept in Euclidean geometry and has significant applications in various fields like engineering, architecture, and computer graphics. The defining characteristic of perpendicular lines (that are not vertical or horizontal) is the relationship between their slopes.
The Slope Relationship: Negative Reciprocals
The slope of a line measures its steepness. If a line has a slope of 'm', any line perpendicular to it will have a slope of '-1/m'. This is known as the 'negative reciprocal'. For example, if a line has a slope of 2, a perpendicular line will have a slope of -1/2. The product of their slopes will always be -1 (m * -1/m = -1).
Special Cases
A horizontal line has a slope of 0. A line perpendicular to it is a vertical line, which has an undefined slope. Conversely, a line perpendicular to a vertical line is a horizontal line. In these cases, the negative reciprocal rule doesn't directly apply, but the 90-degree intersection principle holds true.

Basic Examples

  • Line y=2x+1 has a perpendicular line y=-0.5x+c
  • A horizontal line (y=k) is perpendicular to a vertical line (x=h)
  • Finding the altitude of a triangle from a vertex to the opposite side
  • Used in construction and architecture for right angles

Step-by-Step Guide to Using the Perpendicular Line Calculator

  • Choosing Your Method
  • Entering the Values
  • Interpreting the Results
Our calculator is designed to be intuitive. Here's how to use it effectively:
1. Choose Your Line Definition Method
Start by selecting how your original line is defined from the dropdown menu. You have three options: Slope-Intercept Form (y = mx + b), Two-Point Form, or Standard Form (Ax + By = C).
2. Enter the Line and Point Details
Based on your chosen method, the necessary input fields will appear. Fill them in accurately. Then, enter the coordinates (x₀, y₀) of the point that the new perpendicular line must pass through.
3. Calculate and Interpret the Results
Click the 'Calculate' button. The tool will display the slope of the original line, the calculated perpendicular slope, and the final equation of the perpendicular line in slope-intercept form (y = mx + b) or as x = c for vertical lines.

Real-World Applications of Perpendicular Lines

  • Architecture and Construction
  • Computer Graphics and Game Development
  • Navigation and Robotics
The concept of perpendicularity is not just an academic exercise; it's everywhere.
Architecture and Construction
Walls must be perpendicular to the floor to create stable, upright structures. The corners of rooms, windows, and doors are all practical examples of right angles formed by perpendicular lines and planes.
Computer Graphics
In 2D and 3D graphics, perpendicular vectors (lines) are used to calculate lighting, shadows, and object orientation. For instance, a 'normal vector' is a vector perpendicular to a surface that helps determine how light reflects off it.
Navigation
When plotting courses, sailors and pilots use perpendicular lines to determine bearings and make course corrections. The grid lines on maps (latitude and longitude) are perpendicular at their intersections.

Common Misconceptions and Correct Methods

  • Confusing Perpendicular and Parallel
  • Forgetting the 'Negative' in Negative Reciprocal
  • Errors with Vertical and Horizontal Lines
Perpendicular vs. Parallel
A common mistake is to confuse perpendicular with parallel. Parallel lines have the same slope and never intersect. Perpendicular lines have negative reciprocal slopes and intersect at a 90-degree angle.
Remember: Parallel Slope = m, Perpendicular Slope = -1/m.
Forgetting the Negative Sign
It's easy to calculate the reciprocal (1/m) but forget to negate it. The slope must be the negative reciprocal. If the original slope is positive, the perpendicular slope must be negative, and vice-versa.
Handling Zero and Undefined Slopes
A horizontal line (slope = 0) has a vertical perpendicular line (undefined slope). You cannot calculate -1/0. You must recognize this special case: if the original line is y = c, the perpendicular line passing through (x₀, y₀) will be x = x₀.

Mathematical Derivation and Examples

  • Deriving the Perpendicular Slope
  • Finding the Equation using Point-Slope Form
  • Worked Example
Derivation
Given a line L1 with equation y = m₁x + b₁, we want to find the equation of a line L2 (y = m₂x + b₂) that is perpendicular to L1 and passes through a point P(x₀, y₀).
1. First, we find the slope of the perpendicular line: m₂ = -1 / m₁.
2. Next, we use the point-slope form of a linear equation, which is y - y₁ = m(x - x₁).
3. Substitute the perpendicular slope (m₂) and the given point (x₀, y₀) into this formula: y - y₀ = m₂(x - x₀).
4. Finally, we rearrange this equation into the slope-intercept form (y = mx + b) to get our answer.
Worked Example
Let's find the equation of the line perpendicular to y = -3x + 2 that passes through the point (6, 4).
  • Original slope (m₁): -3
  • Perpendicular slope (m₂): -1 / (-3) = 1/3
  • Use point-slope form: y - 4 = (1/3)(x - 6)
  • Distribute the slope: y - 4 = (1/3)x - 2
  • Isolate y: y = (1/3)x - 2 + 4
  • Final Equation: y = (1/3)x + 2