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The 3 towns are putting their resources together to build a school that they all share. The parents want the school to be the same distance away from each town.
1. Use the tools available to you to nd the location of a point that is the same distance away from each town so the towns know where to start building.
2. Use a straightedge to connect the towns to make triangle . Construct a circle centered at with radius .
Student Response
1. Student responses should include the markings for perpendicular bisectors.
2. The student response should be triangle with circumcenter and the circumscribed circle drawn.
Activity Synthesis
Ask students to share their strategies for nding the fair placement of the school. Explain that the special point at the center of a circumscribed circle is called the circumcenter. Ask students:
• “What would change about this problem if the towns were in di erent locations? What would stay the same?” (If the towns were in di erent locations, then the perpendicular bisectors would still intersect at one point, but would make di erent angles at that point. The three towns would still lie on the same circle centered at the location of the school.)
• “Do you think the three perpendicular bisectors always intersect at one point, or only sometimes?” (The perpendicular bisectors always intersect at one point because if a point is equidistant to and , and also equidistant to and , then it is equidistant to , , and .)
10.3 Wandering Centers
15 minutes
In this activity, students compare the circumcenters of acute, obtuse, and right triangles. They nd that the circumcenter of a right triangle lies on its hypotenuse, the circumcenter of an acute triangle lies inside the triangle, and the circumcenter of an obtuse triangle lies outside the triangle.
Launch
In the interest of time, you might choose to arrange students in groups of 3 and assign each member a di erent triangle to analyze.
Student Task Statement
Construct the circumscribed circle of each triangle. What do you notice about the location of the circumcenter in each triangle?
Unit 7 Lesson 10: Circles Outside of Triangles 103