Enter a line equation (using coefficients A, B, and C in the form Ax + By + C = 0) and a point (x₀, y₀) through which the perpendicular must pass. The Perpendicular Line Calculator returns the perpendicular line equation, its slope, and the intersection point coordinates of the two lines.
Perpendicular Line Equation
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Original Line Slope (m₁)
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Perpendicular Slope (m₂)
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Y-Intercept of Perpendicular Line
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Intersection Point X
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Intersection Point Y
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A perpendicular line is a line that intersects another line at a 90-degree (right) angle. In two-dimensional space, two lines are perpendicular if the product of their slopes equals −1, meaning m₁ × m₂ = −1.
If the original line has slope m₁, the perpendicular slope m₂ is the negative reciprocal: m₂ = −1 / m₁. For example, if the original slope is 2, the perpendicular slope is −1/2 = −0.5.
First calculate the perpendicular slope m₂ = −1/m₁. Then use the point-slope form: y − y₀ = m₂(x − x₀), where (x₀, y₀) is the given point. Rearrange to slope-intercept form y = m₂x + b by solving for b.
Rewrite the line in slope-intercept form y = ax + b by rearranging the standard form Ax + By + C = 0 to get slope a = −A/B. The perpendicular slope is then −1/a = B/A.
Multiply the slopes of the two lines together. If the result equals exactly −1, the lines are perpendicular. For instance, slopes 3 and −1/3 satisfy 3 × (−1/3) = −1, confirming perpendicularity.
No. For two lines to be perpendicular, their slopes must multiply to −1. Here 5 × 0.2 = 1, not −1. The line perpendicular to y = 5x would need slope −1/5 = −0.2, not +0.2.
In standard two-dimensional Euclidean geometry, yes — two non-parallel lines always intersect at exactly one point. Since perpendicular lines have different slopes, they will always meet at one intersection point.
A horizontal line (slope = 0) has a perpendicular that is vertical (undefined slope), and vice versa. In those special cases, the perpendicular line equation is simply x = x₀ or y = y₀, where (x₀, y₀) is the given point.