Perpendicular Line Calculator
The Negative Reciprocal Rule
Perpendicular lines meet at a right angle, and that geometric fact translates into a simple algebraic relationship between their slopes: multiply them together and the answer is always −1. Flip the original slope and change its sign, and you have the slope of every line perpendicular to it.
The Formula
b = y − m2 × x
Equation: y = m2x + b
Two special cases fall outside this formula: a horizontal original line (slope 0) has a vertical perpendicular, x = (point's x-coordinate), and a vertical original line has a horizontal perpendicular, y = (point's y-coordinate) — since neither 0 nor an undefined slope has a normal negative reciprocal.
Worked Example
A line with slope 4 must have a perpendicular through the point (2, 3):
| Step | Calculation | Result |
|---|---|---|
| Perpendicular slope | −1 / 4 | −0.25 |
| Intercept | 3 − (−0.25 × 2) | 3.5 |
| Equation | — | y = −0.25x + 3.5 |
Where This Matters
- Dropping a perpendicular — finding the shortest path from a point to a line, a routine step in distance and reflection problems.
- Structural and architectural drawings — constructing a wall, support, or cross-member at a true right angle to an existing edge.
- Circle tangents — a tangent line to a circle is always perpendicular to the radius at the point of contact.
How to Use This Calculator
- Choose how the original line is defined: Line From Two Points or Line From Slope.
- Enter the original line's coordinates (X1, Y1, X2, Y2) or its slope.
- Enter the coordinates of the new point (New Point X, New Point Y) the perpendicular must pass through.
- Select Calculate to get the perpendicular line's equation.
Related Calculations
For a line that keeps the same direction instead, see the Parallel Line Calculator, or check the Angle Between Lines Calculator to measure angles that aren't exactly 90°.