Quick answer

Perpendicular slopes satisfy m1 · m2 = -1. Through (x0, y0): y = (-1/m)x + b with b = y0 - (-1/m)x0.

Formula

  • m1 · m2 = -1
  • a = -1/m
  • b = y0 - a·x0
  • mx + r = ax + b

Introduction

Formulas turn the idea of a square corner into algebra you can repeat on tests. The Perpendicular Line Calculator applies the same rules after you enter m, r, and a point.

If the vocabulary still feels new, start with the definition of a perpendicular line article, then return here for the full symbolic story.

We build from the slope product to complete equations and intersection points, including the horizontal and vertical exceptions that textbooks mention briefly.

When you need help choosing point-slope versus slope-intercept output, the perpendicular line equation guide shows how to rewrite the same line in different forms.

Why does m<sub>1</sub> · m<sub>2</sub> = -1?

Slopes measure direction. Rotating a direction by 90 degrees flips one slope to the negative reciprocal of the other in the Cartesian plane.

That is why multiplying slopes yields -1 instead of 1, which is the parallel case.

The formula does not replace understanding; it compresses understanding into a few reliable lines of algebra.

Each symbol in the worksheet should map to a feature on the graph: base slope, passing point, perpendicular slope, and crossing location.

Formula and relationships

  • m1 · m2 = -1
  • a = -1/m
  • b = y0 - a·x0
  • mx + r = ax + b

Keep horizontal and vertical cases separate: m = 0 means the perpendicular is vertical and is written x = x0.

After you find a and b, set the base expression mx + r equal to ax + b to locate the intersection.

Fractions are common when m is not an integer. Exact fractions on homework often beat early rounding.

Step-by-step guide

  1. Write m for the base line. From y = mx + r or an equivalent form.
  2. Compute a = -1/m. Skip division when m = 0 and switch to vertical language.
  3. Solve for b with your point. Substitute (x0, y0) into y = ax + b.
  4. State the perpendicular equation. Use slope-intercept or point-slope depending on the rubric.
  5. Equate lines for intersection. Solve mx + r = ax + b for x, then back-substitute for y.
  6. Verify the slope product. Multiply slopes to check -1 when both are defined.

Worked examples

Base m = 4 gives perpendicular a = -1/4. Through (2, 5): b = 5 - (-1/4)(2) = 5.5, so the line is y = -0.25x + 5.5.

Set 4x + r equal to -0.25x + 5.5 only after you know r from the original problem statement.

Check the arithmetic with the calculator on the home page if the intersection fractions look messy.