Lesson 6.11

Perpendicular Lines

Parallel lines never meet. Perpendicular lines meet in the most violent way possible: a perfect head-on 90° collision.

Introduction

"Perpendicular" is just a fancy math word for a right angle (). Walls are perpendicular to floors. Street corners are perpendicular. In Algebra, we find them by looking at their slopes.

Past Knowledge

Lesson 6.10 (Parallel). Parallel means "Same". Perpendicular is logically the "Opposite", but in a specific way.

Today's Goal

Identify perpendicular lines using "Negative Reciprocals".

Future Success

Essential for Geometry (Right Triangles) and Physics (Normal Force vectors).

Key Concepts

Negative Reciprocals

To make a line perpendicular, you must do TWO things to the slope:

FLIP

The Fraction

SWITCH

The Sign

Result: becomes .

Mathematical Definition:

Worked Examples

Example 1: Finding the Slope

Basic

Find the negative reciprocal for each slope.

Start

Perpendicular

Start

Perpendicular

Example 2: Visual Check

Visual

Are and perpendicular?

Slope 1:

Slope 2:

Yes.

They are negative reciprocals. The graph shows a perfect 90° angle.

Example 3: Geometric Application

Advanced

A triangle has vertices A(0,0), B(3,4), and C(-4,3). Is it a right triangle?

Step 1: Check Slopes

Slope AB: .

Slope AC: .

Step 2: Compare

and are negative reciprocals.

Yes. Angle A is .

Common Pitfalls

Only doing half the job

Students often flip the fraction () but forget to change the sign. Lines with positive slopes ( and ) both go UP. They can't possibly meet at a 90° angle!

Zero and Undefined

The reciprocal of zero () is undefined (). So, horizontal lines are perpendicular to vertical lines.

Real-Life Applications

City Planning:

Look at a map of New York City (Manhattan).
The Avenues run North-South. The Streets run East-West.
They are perpendicular. This maximizes building space (rectangles fit together perfectly) and makes navigation easy.