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## Hospital Locator

• Lesson
9-12
1

Students begin with a problem in a real-world context to motivate the need to construct circumcenters and then incenters of triangles and to make sense of these constructions in terms of bisecting sides and angles.

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:

 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.

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 an 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 discussion of finding a point that is equidistant from all of the vertices arises.

Use guiding questions to help students come to the conclusion that a circle needs to be circumscribed onto their triangle. This provides an opportunity to introduce the terms circumcircle (a triangle's circumscribed circle) and circumcenter (the center of a triangle's circumcircle). (Saving other locations suggested by students may provide ways to introduce other special points related to triangles.)

Have students experiment with trying to find the circumcircle and circumcenter, where the circumcenter is the location of the medical facility. Provide blank activity sheets as needed to help students keep a record of how their work is being modified. The following interactive can be used to explore the circumcircle further.

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 Incenter-Incircle Tool (mobile-friendly). 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.]

To wrap up the class, engage the class in discussing the mathematical answer of the hospital's location vs. its real-world plausibility. Students may realize that the answer found in the instruction is difficult to achieve due to the area's topography and geography (ex: the hospital would be too far away from any major highways).

• Computers or tablets with internet connection (1 for class OR 1 for each pair or small group of students)
• Hospital Problem Overhead (1 copy)
• Cities Triangle Overhead (at least 1 copy for each student)
• Hospital Map Activity Sheet (at least 1 copy for each student)
• Blank transparencies (several per pair or group of students)
• Compass, straight edge, ruler, protractor (1 set per group)

Assessment Options

1. Have students construct the circumcenter and the incenter of a triangle. Provide students with a triangle that is not an isosceles triangle and not oriented in the same way as the triangle from the hospital problem.
2. As a journal entry, students should write a response to the following questions: How do the circumcenter and incenter involve bisecting the triangle’s sides and angles? Why do these bisection processes makes sense for these constructions?

Extensions

1. Is it possible for the circumcenter and the incenter of a triangle to be the same point? Explain. [The circumcenter and the incenter would be the same point when all of the perpendicular bisectors of the sides and the angle bisectors of the triangle coincide. This happens when the triangle is an equilateral triangle. So, yes, it is possible.]
2. Perpendicular bisectors of the sides led to the circumcenter. The bisecting of the angles led to the incenter. What happens if we simply construct a line through the midpoint of each side and the opposite vertex? [The three lines intersect in a unique point. Subsequent discussion may lead to the description of the lines as the medians and the intersection point as the centroid. The sense of "half" comes into the discussion because each median divides the area of the original triangle in half, as can be seen in the Incenter-Incircle Tool. (Because M is the midpoint of the base, it follows that the area of triangles AMB and CMB will always be equal. Both have height BM, and the bases of the triangles are equal (AM = CM), so the areas are equal: ½(AM)(MB) = ½(CM)(MB).]

Questions for Students

1. 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.]

2. 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.]

Teacher Reflection

• Where did students struggle in executing the constructions?
• How did students use their ideas about "half" to explain why the constructions for circumcenters and incenters make sense?

### Angle Bisector

9-12

This interactive demonstrates how to bisect an angle.

### Circumcircle

9-12

Construct and explore of a circumcircle.

### Incenter-Incircle

9-12

This applet allows for the discovery of the incenter and incircle of a triangle.

### Learning Objectives

Students will:
• Construct the circumcenter of a triangle.
• Construct the incenter of a triangle.
• Explain how the circumcenter and incenter involve bisecting the triangle’s sides and angles, respectively, and why such a bisection process makes sense.

### NCTM Standards and Expectations

• Analyze properties and determine attributes of two- and three-dimensional objects.
• Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others.
• Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools.