Quick answer

Perpendicular slope a = -1/m when m ≠ 0. Horizontal ↔ vertical pairing when m = 0 or undefined.

Formula

  • a = -1/m
  • m · a = -1
  • m = 0 → vertical perpendicular

Introduction

Start every perpendicular problem by asking for the perpendicular slope. The Perpendicular Line Calculator finds it automatically when you continue to the full equation.

A slope mistake early in the problem cannot be fixed later, so this step deserves its own practice set.

The perpendicular line formula explains why the product m1 · m2 = -1 works, which makes the flip rule easier to remember.

After you have a, move to the perpendicular line equation article to place the slope through a point in proper form.

Undefined and special slopes

Vertical lines have undefined slope in slope-intercept class, but you can write x = k.

Horizontal lines use m = 0 and pair with vertical perpendiculars.

Negative base slopes still follow the same flip-and-negate process; watch signs when both values are negative.

Fraction slopes such as 3/5 become -5/3. Multiply to confirm the product is -1.

Formula and relationships

  • a = -1/m
  • m · a = -1
  • m = 0 → vertical perpendicular

Use the product test m · a = -1 as a quick checker after you think you have a.

Do not confuse perpendicular slope with the angle of inclination measured in degrees.

Parallel slopes are equal; perpendicular slopes are negative reciprocals. Label the problem before calculating.

Step-by-step guide

  1. Identify m. From the base line or two points on that line.
  2. Flip and negate. Use the negative reciprocal.
  3. Handle zeros. Switch to vertical or horizontal language when m = 0.
  4. Test the product. Multiply slopes to see -1.
  5. Only then find b. Use the slope with the given point in the next step of the problem.

Worked examples

m = 3/5 → a = -5/3. Check: (3/5)(-5/3) = -1.

m = -2 → a = 1/2. Check: (-2)(1/2) = -1.

m = 0 on y = 4 means the perpendicular through (2, 7) is the vertical line x = 2.