# Inscribed and Circumscribed Circles

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## Objective

SWBAT construct the inscribed and circumscribed circles of a triangle.

#### Big Idea

Stay calm and find your center. In this lesson, students learn to construct the incenter and circumcenter.

## Concept Development

10 minutes

Using the Inscribed and Circumscribed Concept Development slideshow, I give students examples and non-examples of inscribed and circumscribed figures. During the presentation I also develop the meanings of the prefixes in- and circum- and the root scribe.

## Freestyle Construction

20 minutes

For this section, I hand students a copy of Inscribed and Circumscribed Circles_Freestyle. They will be trying to inscribe and circumscribe circles using only a compass, a pencil, and their good vision. My hope is that they will begin to realize that they are being prevented from successfully completing the construction because they don't know where to locate the centers of the circles.

## Reviewing the Angle Bisector and Perpendicular Bisector Theorems

15 minutes

At this point, I'll remind students that they have just attempted to draw the inscribed and circumscribed circles of triangles. I will also remind them that the key to succeeding in that task would have been to precisely locate the centers of the circles.

I then tell them that there is some knowledge we gained earlier in the course that might be helpful in that regard. Check out the screencast for a summary of what I share with students.

After seeing the demonstration, students know enough to complete the proof. I won't provide any more hints. It's time for students to own the constructions.

## Formal Constructions with Explanations

30 minutes

Students have just been trying to locate the centers of the inscribed and circumscribed circles of a triangle using only their compass and good vision. They have hopefully realized that the key things they're missing are the centers of the inscribed and circumscribed circles.

At this time, I give them language to assign to the centers they were seeking. I explain that the incenter is the center of the inscribed circle and that the circumcenter is the center of the circumscribed circle.

Next I give each student a copy of Inscribed and Circumscribed Circles_Constructed. Students will be working independently for 15 - 20 minutes on these. When the time has elapsed, I'll randomly select non-volunteers to come to the front of the class to present.

## Extension

20 minutes

Using Exploring the Effect of Triangle Classification on Inscribed and Circumscribed Circles, students will perform classic constructions to explore the effect of triangle classification on the location of the circumcenter and incenter and make conjectures.

Students will also use Geometer's Sketchpad to verify or disprove their conjectures.