# Lesson 11

Perpendicular Lines in the Plane

### Lesson Narrative

In previous courses, students studied slopes of lines. In previous units, students studied perpendicular lines. In this lesson, students connect these ideas and prove that non-vertical and non-horizontal perpendicular lines have slopes with opposite reciprocals—that is, that they have a product of -1. Students begin by rotating a figure 90 degrees. Then they collect data about the slopes of the perpendicular lines in the original figure and the rotated image. They use their data to make a conjecture about slopes of perpendicular lines. Finally, they have an opportunity to construct a viable argument (MP3) by proving their conjecture is true for all lines.

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

### Learning Goals

Teacher Facing

• Prove the slope criterion for perpendicular lines and use it to solve geometric problems.

### Student Facing

• Let’s analyze the slopes of perpendicular lines.

### Required Preparation

Though graph paper is not specifically called for, students may find it useful, especially in the lesson synthesis.

### Student Facing

• I can prove that the slopes of perpendicular lines are opposite reciprocals.
• I can use slopes of perpendicular lines to solve problems.

Building Towards

### Glossary Entries

• opposite

Two numbers are opposites of each other if they are the same distance from 0 on the number line, but on opposite sides.

The opposite of 3 is -3 and the opposite of -5 is 5.

• point-slope form

The form of an equation for a line with slope $$m$$ through the point $$(h,k)$$. Point-slope form is usually written as $$y-k = m(x-h)$$. It can also be written as $$y = k + m(x-h)$$.

If $$p$$ is a rational number that is not zero, then the reciprocal of $$p$$ is the number $$\frac{1}{p}$$.