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. Also try the Rise Over Run Calculator.

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)

--

Perpendicular Slope (m')

--

Y-Intercept of Perpendicular Line (b')

--

Standard Form (Ax + By = C)

--

Original Line Slope (m)

--

Results Table

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. See also our Parallel Line (Slope-Intercept Form) — Parallel Line Through a Point.

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. You might also find our calculate Missing Endpoint (x2, y2), x₂ (Endpoint x) & y₂ (Endpoint y) — Endpoint useful.

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.