Students may need previous experience with specific ideas to reason through the constructions in this lesson. For circumcenter, students will need to understand that the perpendicular bisector of a chord always passes through the center of the circle. For incenter, students will need to know that, if the rays of an angle are tangent to a circle, then the angle bisector passes through the center of the circle.
This lesson begins with a hypothetical problem set in the real world. The problem can be stated as follows:
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Boise, ID; Helena, MT; and Salt Lake City, UT are three large cities in the northwestern part of the United States. Although each city has a local hospital for minor needs and emergencies, an advanced medical facility is needed for transplants, research, and so forth. Imagine the potential of a high-powered, high-tech, extremely modern medical center that could be shared by the three cities and their surrounding communities!
You have been hired to determine the best location for this facility.
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To begin the lesson, project the Hospital Problem overhead, which states the problem for students:
Use a map or atlas to show the location of the three cities. (You might want to have students access a map from MapQuest, Google Maps, or another online map site, or you can project a copy of the Hospital Map activity sheet.) Point out that there are no major towns located between these cities that would be obvious locations for the medical center; therefore, the major factor in determining a location should be proximity to each city.
Distribute the Hospital Map activity sheet and a blank transparency sheet to all groups. Allow students to work on the task in groups of two or three. Each group should record their suggested solutions on a transparency sheet and be ready to explain their thinking to the class. [Students will often create a triangle using the three cities as the vertices. Students may use the Hospital Map activity sheet to trace the appropriate size triangle onto a transparency. The Cities Triangle overhead can also be used, if students need it. Students often suggest that the medical center should be located "somewhere in the center" and generally indicate a portion of the interior of the triangle. Pressing them for a more specific idea of where "in the center" the medical center should be placed may underscore the need for the group to identify some criteria for selecting a "center" of the triangle. Another common response from students is to want to locate a point that is equidistant from each of the vertices. This suggestion—or a conversation about "somewhere in the center"—leads to the following discussion of how to find the circumcenter. Saving other locations suggested by students may provide ways to introduce other special points related to triangles.]
Once the idea of having a point that is equidistant from all of the vertices arises, have students open the applet Hospital Locator. Using this activity, groups experiment with where to locate the medical facility so that it is equally distant from the three cities. While using this applet, students should notice that the perpendicular bisectors intersect. The intersection point is the center of a circle that passes through all three vertices. This provides an opportunity to introduce the terms circumcircle and circumcenter to describe what students observe.
When students arrive at an answer, have them look at other groups’ solutions. Then, conduct a discussion with the entire class based on the following questions:
- What do you notice about all of the solutions?
[The point is in about the same place in each case.]
- How would we be able to construct the point we want if we were not allowed to use measurements?
[Students may not immediately see that the perpendicular bisectors can be used, or they may mention them only because of what appears on the screen. In any event, the point of the question is to encourage students to wonder why the construction they are about to see makes sense.]
- Why does it make sense that the circumcenter would lie on the perpendicular bisectors of the sides?
[Each side can be thought of as a chord of the circle. A radius of the circle must be a perpendicular bisector of a chord.]
- If "cutting the sides in half with perpendicular bisectors" led to the circumcenter, what happens when we "cut the angle in half"?
[The angle bisectors intersect. That point is the center of a circle that is tangent to all three sides. Discussion of the observations includes introduction of the terms incircle and incenter to describe what students observe.]
After this discussion, have students explore the applet Half Angle. Explorations with this applet will lead to further discoveries by students. Use these discoveries to continue the discussion:
- Why does it make sense that the incenter would lie on the angle bisectors?
[Points on the angle bisector are equidistant from each of two sides of the angle. This situation is necessary for each ray of the bisected angle to be tangent to the inscribed circle.]
- How does "half of something" help us to make sense of how to construct circumcenters and incenters?
[Thinking "half of a side" in terms of perpendicular bisectors leads us to perpendicular bisectors of chords of the circumcircle. Thinking "half of an angle" suggests that angle bisectors could be used to identify the incircle and to locate the incenter. From half of a side or half of an angle, we get important ideas that underlie how chords and tangents are related to the constructions.]
