Last updated October 10, 2026 · Report an error
Perpendicular lines cross each other at a perfect right angle (90°), and algebraically, their slopes are always negative reciprocals of one another — if one line has slope m, any line perpendicular to it has slope −1/m. Given an existing line and a point not already on it, there is exactly one line through that point that meets the original at a right angle, and this calculator finds its equation directly. Finding a perpendicular line through a point is a routine construction in geometry (drawing an altitude of a triangle, or the perpendicular bisector of a segment), physics (finding the direction of a normal force, which always acts perpendicular to a surface), and CAD/drafting (extending a wall or beam at a right angle from an existing edge).
How it works
Enter two points that define the original line, plus the new point the perpendicular line must pass through. The calculator first finds the original line's slope: m = (y2 − y1)/(x2 − x1). It then computes the perpendicular slope by negating that value AND taking its reciprocal (flipping the fraction): perpendicular slope = −1/m. This new slope is plugged into point-slope form, y − y1 = (−1/m)(x − x1), using the new point's coordinates — giving the unique line through that point which crosses the original at exactly 90°.
- Enter original line — point 1 x1.
- Enter original line — point 1 y1.
- Enter original line — point 2 x2.
- Enter original line — point 2 y2.
- Enter new point — x.
- Enter new point — y.
- Click Calculate to see your results.
Examples
Line through (0,0), (2,4); new point (3,1)
Original slope = (4−0)/(2−0) = 2, so perpendicular slope = −1/2. Using point-slope form through (3,1): y − 1 = −0.5(x − 3), which simplifies to y = −0.5x + 2.5.
Line through (1,5), (4,-1); new point (0,0)
Original slope = (-1−5)/(4−1) = -2, so perpendicular slope = −1/(-2) = 0.5. Using point-slope form through (0,0): y − 0 = 0.5(x − 0), which simplifies to y = 0.5x.
Vertical line through (3,0), (3,7); new point (-2,5)
The original line is vertical (undefined slope). A line perpendicular to a vertical line is always horizontal, with slope 0. Through (-2,5), the perpendicular line is simply y = 5.
Who should use it
- Constructing a triangle's altitude or a perpendicular bisector.
- Finding a surface's normal direction in a physics problem.
- Drafting or CAD tasks requiring a right-angle extension from an existing edge.
Industry applications
- Geometry and physics education
- CAD, drafting, and structural design
Advantages
- Handles the horizontal and vertical special cases automatically.
- Directly usable for geometry constructions like altitudes and perpendicular bisectors.
Limitations
- Requires two clearly distinct points to define the original line's slope unambiguously.
Common mistakes to avoid
- Just negating the slope without also taking the reciprocal (or vice versa) — true perpendicularity requires both operations together.
- Forgetting the special cases for an originally horizontal or vertical line, where the "negative reciprocal" rule needs to be replaced by a simple vertical or horizontal answer.
- Using the new point's coordinates when computing the original line's slope instead of the two points that actually define that original line.
Best practices
- Always compute the original slope first, then apply both the negation and the reciprocal together in one step.
- Check for the horizontal/vertical special cases before applying the general negative-reciprocal formula.
- Verify perpendicularity by confirming the two slopes multiply to exactly −1.
Tips
- Need a line with the SAME slope instead? Use the Parallel Line Calculator.