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# The Shortest Segment

Lesson 6 of 17

## Objective: SWBAT construct the perpendicular to a line through a point not on the line. Students will understand how the properties of a perpendicular are related to the properties of bisectors.

#### Lesson Open

*9 min*

I begin the lesson by displaying the warm-up prompt using the slide show for the lesson.

The prompt asks students to describe the symmetry in a set of non-concentric circles. The purpose of the warm-up is to help ensure that students see the symmetry in the concentric circles they construct as part of the tire swing problem. It is also good practice in a skill that they learned several lessons ago.

The warm-up follows our Team Warm-up routine. I choose students at random to write the team's answer on the board.

Reviewing the teams' answers on the board, I ask where the centers of the circles all have to be. On the line of reflection.

I display the Agenda and Learning Targets for the lesson. I tell the class that today we will be using the symmetry of a figure to solve an interesting problem. At least, I hope it is interesting. Has anyone ever built a tire swing?

Without further ado, I introduce the next activity: a pair of problems.

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The purpose of this activity is to dramatize the construction of a perpendicular bisector with a real-world problem (**MP4**). As much as possible, I want students to reason for themselves that the shortest segment between a point and a line--which is what a perpendicular is, of course--can be found by constructing a circle that intersects the line at two points first, and then constructing the bisector of those points (**MP1**).

To lead students to this idea the first problem (The Tire Swing) is designed to entice students to construct a pair of circles that have too great a radius, and one that is too small. From the symmetry of the figure, I hope that they will think of constructing a bisector to find the point midway between the points of intersection (**MP7**)

As students go to work, display the instructions and circulate to answer questions or give out hints where necessary. If students find a different method of solution, I give praise while asking them to see the reasoning behind the solution I have in mind.

Students each complete their own solution, but I encourage them to work together and share their thinking.

The second problem (A Campfire Story) can be used as a check for understanding. It also gives students a second opportunity to solve the problem and see the symmetry in the construction of the perpendicular to a line.

Note that the handout for the activity is meant to be reproduced on 11" x 17" legal-size paper.

If students finish early, I give them a problem from the Bisector Challenge activity.

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#### Summarizing Perpendiculars

*15 min*

We use the Guided Notes to summarize the steps of the construction of a perpendicular. Which, of course, the name of the bisector that students have just learned to construct for themselves. I demonstrate the steps of the construction, and students follow along to create the diagrams in their notes.

For more on how I use Guided Notes in my lessons, see the article in my Strategies folder.

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**Individual Size-Up**

The lesson close follows our Individual Size-Up routine. The prompt asks students to name the shortest path between a point and a line.

**Homework**

For homework, I assign problems #18-20 of Homework Set 1 for this unit. Since there is no time in this lesson to practice the construction of a perpendicular or fully explore its properties, the homework problems review and extend concepts and skills taught in earlier lessons.

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- UNIT 1: Models and Constructions
- UNIT 2: Dimension and Structure
- UNIT 3: Congruence and Rigid Motions
- UNIT 4: Triangles and Congruence
- UNIT 5: Area Relationships
- UNIT 6: Scaling Up- Dilations, Similarity and Proportional Relationships
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- UNIT 8: Volume of Cones, Pyramids, and Spheres

- LESSON 1: Previewing Congruence and Rigid Motions
- LESSON 2: Congruence and Coincidence
- LESSON 3: Re-Discovering Symmetry
- LESSON 4: Perfect Polygons
- LESSON 5: Bisector Bonanza
- LESSON 6: The Shortest Segment
- LESSON 7: From Perpendiculars to Parallels
- LESSON 8: Reviewing Congruence
- LESSON 9: Re-Examining Reflections
- LESSON 10: Reconsidering Rotations
- LESSON 11: Taking Apart Translations
- LESSON 12: Visualizing Transformations
- LESSON 13: Reasoning About Rigid Motions
- LESSON 14: Analyzing the Symmetry of a Polygon
- LESSON 15: Reviewing Rigid Motions
- LESSON 16: Rigid Motion and Congruence Unit Quiz
- LESSON 17: Describing Precise Transformations