Lesson 6.13

Writing Perpendicular Equations

This is the final challenge of the unit. It combines finding slope, negative reciprocals, and point-slope form into one comprehensive problem.

Introduction

Just like with parallel lines, you are given a reference line and a point. The difference is that you don't copy the slope—you transform it.

Past Knowledge

Lesson 6.11 (Perpendicular slopes). You must remember to Flip and Switch.

Today's Goal

Find the new slope, then write the equation.

Future Success

This is frequently tested on college entrance exams (SAT/ACT) as it tests multiple skills at once.

Key Concepts

The Transformation Process

  1. Extract Old Slope.

    Example: means .

  2. Find Perpendicular Slope.

    Flip to . Switch sign to negative.

  3. Write Equation.

    Use and the new point.

Worked Examples

Example 1: The Basics

Basic

Write equation perpendicular to going through .

Step 1: Slopes

Old: .

New (Flip & Switch): .

Step 2: Write Equation

Example 2: Standard Form

Intermediate

Write equation perpendicular to going through .

Step 1: Find Old Slope

Solve for y: .

Old .

Step 2: New Slope

Flip to . Change negative to positive.

New .

Step 3: Equation

Example 3: Horizontal Lines

Advanced

Write equation perpendicular to going through .

Step 1: Identify

is horizontal.

A perpendicular line must be Vertical.

Step 2: Write Vertical Equation

Vertical means

New x-coordinate is 5.

Common Pitfalls

Forgetting to Flip

Sometimes students just change the sign () but forget to flip the fraction. That's not perpendicular!

Double Negative

If the old slope is negative (), the new slope is positive (). Sometimes students write which is confusing.

Real-Life Applications

Physics (Forces):

  • The "Normal Force" is the force a surface pushes back with.
  • It is DEFINED as being perpendicular to the surface.
  • If you are designing a ramp (slope), you need to calculate the perpendicular vector to know how much friction there will be.

Practice Quiz

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