Lesson Plan

# Exploring Triangle Circumcenters and Incenters

Students explore points of concurrency in triangles.
Philip K.
Classroom teacher
The Academy of Science and Entrepreneurship
Bloomington, United States
My Grades 9, 10, 11, 12
My Subjects Math
Objectives

Students will be able to understand and describe how to determine the circumcenter and incenter of a triangle.

Students will understand the significance of both the circumcenter and incenter as the centers for the circumscribed and inscribed circles.

Students will understand that the circumcenter is equidistant from the vertices of the triangle, and the incenter is equidistant from the sides of the triangle.

Subjects
Math

#### 1 Construction #1 - Perpendicular bisectors.

Guide students through the construction of the circumcenter.

1. Draw a triangle using the segment tool.
2. Construct the perpendicular bisector of each side of the triangle.
3. Have students screen shot their construction (or save as an image file).  This will be used in the next step.
4. Have students save this work, as they will return to it in step #3.

#### 2 Drawing Gallery - Student Share Out

Lino
Free

Create a blank "Bulletin Board" that allows all users to post.

Have students post their images onto the Bulletin Board.

Discussion Question:  What do the drawings have in common?

Guide students to the idea of "point of concurrency".

#### 3 Circumcenter - Define and Deepen Understanding

Have students return to their constructions and create the circumscribed circle (centered at the point of concurrency, passing through a vertex).

Discussion Question:  What do you notice about the circle?  Use student input to move toward the term circumcenter - center of the circle that circumscribes the triangle.

What does this mean about how the circumcenter relates to the vertices of the triangle? Guide students to the understanding that the circumcenter is equidistant from all 3 vertices.

#### 4 Construction #2 - Angle Bisectors

Have students work in pairs (or triples) to create a new construction (see student instructions)

Depending on students' comfort level with Geogebra, varying amounts of guidance may be necessary.

Student Instructions
1. Draw a triangle using the segment tool.
2. Construct the angle bisector of each vertex.
3. Construct a point where the bisectors intersect.  This point is called the incenter.
4. Construct a perpendicular line from the incenter through one of the sides of the triangle.
5. Plot a point where the perpendicular line intersects the side.
6. Construct a circle centered at the incenter, passing through the point where the perpendicular line intersected the side.

#### 5 Discuss/Debrief

Activity: Conversing

Guide Student Discussion

• What did you notice?  Circle touches all 3 sides of the triangle.  Inscribed Circle, hence the "incenter".
• How does the incenter relate to the sides of the triangle?  Equidistant from all 3 sides.