Quick answer

Each example: find a = -1/m, build the line, solve intersection.

Formula

  • a = -1/m
  • y = ax + b
  • mx + r = ax + b

Introduction

Copy these setups into the calculator on the home page to confirm your work before you turn in an assignment.

Examples turn the abstract rule m1 · m2 = -1 into numbers you can repeat on new problems.

If any step feels unclear, review the how to calculate a perpendicular line guide, then return here with paper and a pencil.

When you are ready to plot the same relationships, the graphing perpendicular lines article connects algebra to the grid.

How to study with examples

Cover the solution, then redo it without looking. Change the point coordinates to invent new drills.

Group examples by feature: positive slope, negative slope, horizontal base, and applied context.

Say the slope product out loud after each problem. Hearing m · a = -1 builds habit.

Keep one worked page in your notebook as a template you can copy during exams.

Formula and relationships

  • a = -1/m
  • y = ax + b
  • mx + r = ax + b

Always test m · a = -1 when both slopes exist.

Intersection coordinates should satisfy both line equations if you substitute correctly.

Applied problems may hide m and r inside a word story. Translate words to y = mx + r before calculating.

Step-by-step guide

  1. Read the base line. Identify m and r from the prompt or diagram.
  2. Apply the point. Substitute into the perpendicular form.
  3. Solve intersection. Equate the two equations for y.
  4. Graph if time allows. Plot both lines to see the right angle.
  5. Log your mistakes. Note whether errors were sign, substitution, or arithmetic.

Worked examples

Example A: y = 2x + 1 through (3, 4) gives perpendicular y = -0.5x + 5.5 and intersection (1.8, 4.6).

Example B: horizontal y = 4 through (2, 7) gives perpendicular x = 2 and intersection (2, 4).

Example C: y = -3x + 2 through (1, 5) gives perpendicular y = (1/3)x + 14/3 and intersection (11/12, 13/12).

Example D: a path y = 0.25x - 1 with stake (40, 8) produces a steep perpendicular suitable for a shortest-distance sketch.