Perpendicular Line Calculator: Find the Equation

Perpendicular Line

This calculator helps you find the equation of a line that is perpendicular to another line and passes through a specific point. It's a handy tool for algebra and geometry problems.

Slope of the Given Line (m)

Enter the slope of the line you want to be perpendicular to.

X-coordinate of the Point (x₁)

Enter the X-coordinate of the point the new line passes through.

Y-coordinate of the Point (y₁)

Enter the Y-coordinate of the point the new line passes through.

How it works

This calculator helps you find the equation of a line that is perpendicular to another line and passes through a specific point. It's a handy tool for algebra and geometry problems.

The Formula

Given a line with slope 'm' and a point (x₁, y₁):
1. The perpendicular slope (mₚ) is -1/m (if m ≠ 0). 2. Use the point-slope form: y - y₁ = mₚ(x - x₁). 3. Convert to slope-intercept form: y = mₚx + b.

Worked Example

Example: Find the Perpendicular Line
Let's say you have a line with a slope of 2, and you need to find a line perpendicular to it that passes through the point (3, 5). 1. The given slope (m) is 2. 2. The perpendicular slope (mₚ) is -1/2. 3. Using the point (3, 5) and mₚ = -1/2 in point-slope form: y - 5 = (-1/2)(x - 3) y - 5 = -1/2x + 3/2 y = -1/2x + 3/2 + 5 y = -1/2x + 13/2 So, the equation of the perpendicular line is y = -0.5x + 6.5.

Tips, Assumptions & Limitations

Remember that perpendicular lines have slopes that are negative reciprocals of each other.
If the given line is horizontal (slope = 0), the perpendicular line will be vertical (equation x = x₁).
If the given line is vertical (undefined slope), the perpendicular line will be horizontal (equation y = y₁).
Always double-check your calculations, especially with negative signs and fractions.

FAQ

A perpendicular line is a line that intersects another line at a 90-degree (right) angle. In coordinate geometry, their slopes are negative reciprocals of each other, unless one is horizontal and the other is vertical.

If the original line has a slope 'm', the perpendicular line's slope (mₚ) is -1/m. If the original line is horizontal (m=0), the perpendicular line is vertical (undefined slope). If the original line is vertical, the perpendicular line is horizontal (mₚ=0).

Yes, if you enter a slope of 0 for a horizontal line, the calculator will correctly identify the perpendicular line as vertical (x = x₁). If the given line is vertical, you'd typically know its equation (x=C) and infer its slope is undefined, leading to a horizontal perpendicular line (y = y₁).

The point-slope form (y - y₁ = m(x - x₁)) is incredibly useful because it allows you to write the equation of a line directly if you know its slope and any point it passes through. It's often the first step before converting to slope-intercept form (y = mx + b).