Perpendicular Line Through a Point Calculator

Enter a line equation (in slope-intercept or standard form) and a point (x₁, y₁) to find the equation of the perpendicular line passing through that point. You get the perpendicular slope, the full y = mx + b equation, and the standard form Ax + By = C — all calculated from your inputs.

Enter the slope of the given line

Enter the y-intercept of the given line

Coefficient A in Ax + By = C

Coefficient B in Ax + By = C

Constant C in Ax + By = C

X-coordinate of the point the perpendicular line passes through

Y-coordinate of the point the perpendicular line passes through

Results

Perpendicular Line Equation (Slope-Intercept)

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Perpendicular Slope (m')

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Y-Intercept of Perpendicular Line (b')

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Standard Form (Ax + By = C)

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Original Line Slope (m)

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Results Table

Frequently Asked Questions

How do you find the perpendicular line through a given point?

First, determine the slope of the original line (m). The perpendicular slope is the negative reciprocal: m' = -1/m. Then, substitute the perpendicular slope and the given point (x₁, y₁) into the point-slope formula y - y₁ = m'(x - x₁) and simplify to get the final equation.

What is the negative reciprocal and why does it matter?

Two lines are perpendicular if and only if the product of their slopes equals -1 (m × m' = -1). This means the perpendicular slope is found by flipping the original slope and changing its sign. For example, if the original slope is 2, the perpendicular slope is -1/2.

Can you find a perpendicular line if the original line is horizontal?

Yes. A horizontal line has a slope of 0. Its perpendicular line is a vertical line, which has an undefined slope. A vertical perpendicular line through point (x₁, y₁) is simply written as x = x₁.

Can you calculate the perpendicular line with no points?

No — you need at least one point to fully determine a unique perpendicular line. Without a specific point, you can calculate the perpendicular slope, but infinitely many parallel lines share that slope. A point pins down exactly which one you need.

How do I enter a line given in standard form Ax + By = C?

Select the 'Standard form: Ax + By = C' option and enter coefficients A, B, and C. The calculator derives the slope as m = -A/B, then computes the perpendicular slope as B/A and applies it through your given point.

What if the original line is vertical (undefined slope)?

A vertical line (x = constant) has an undefined slope. Its perpendicular line is horizontal with slope 0. The perpendicular line through point (x₁, y₁) would be the horizontal line y = y₁.

How does the standard form relate to the slope-intercept result?

Both forms represent the same line. The slope-intercept form y = m'x + b' is easier to read for the slope and y-intercept, while standard form Ax + By = C is useful in linear algebra and systems of equations. This calculator provides both for convenience.

Does the perpendicular line always pass through the given point?

Yes — by definition. The entire purpose of finding a perpendicular line 'through a point' is to locate the unique line that is both perpendicular to the original and passes exactly through your specified coordinates. You can verify this by substituting the point into the resulting equation.

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