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• “How does this angle compare, then, to the right angle?” (The associated central angle traces out less than radians. Since the angle in question measures half of its associated central angle, it must measure less than radians, making it an acute angle.)
Finally, examine the obtuse angle of the triangle whose circumcenter lies outside the triangle. Ask,
• “Does this angle trace out more than half the circle, or less than half the circle?” (The angle traces out more than half the circle.)
• “How does this angle compare, then, to the right angle?” (The associated central angle traces out more than radians. Since the angle in question measures half of its associated central angle, it must measure more than radians, making it an obtuse angle.)
Lesson Synthesis
In this lesson, students have constructed the circumcenter for various triangles. Ask students,
• “How are the circumcenter and the incenter of a triangle di erent? How are they the same? When might you use one rather than another?” (The circumcenter is the point that is the same distance away from the three vertices of a triangle, whereas the incenter is the point that is the same distance away from the three sides of the triangle. They both have to do with being an equal distance away from three objects. The circumcenter is obtained by nding the intersection of perpendicular bisectors, whereas the incenter is obtained by nding the intersection of angle bisectors. Any problems that require a point to be the same distance away from other points will use perpendicular bisectors, whereas any problem that requires a point to be the same distance away from two sides will use angle bisectors.)
10.4 Airport Location
Cool Down: 5 minutes
Student Task Statement
There are three nearby towns—Washington, Franklin, and Spring eld—with straight roads connecting each pair of town centers. The towns wish to build an airport to be shared by all of them.
Unit 7 Lesson 10: Circles Outside of Triangles 105