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Lesson 10: Circles Outside of Triangles
• Construct the circumscribed circle of a triangle. Lesson Narrative
In the previous lessons, students found the incenter and constructed the incircle for various triangles. The question now turns to whether it’s possible to t a circle around the outside of any triangle. A circle that goes through all the vertices of a polygon is called the circumscribed circle of the polygon, and its center is called the circumcenter. This lesson builds on the extensive work students have done with perpendicular bisectors in the rst unit. To nd the circumcenter, students must nd a point that is the same distance away from the 3 vertices of the triangle. The set of points that are the same distance from endpoints of a segment is the perpendicular bisector of the segment, so to nd a point that is the same distance away from each vertex of a triangle, students must nd where the perpendicular bisectors of the sides of the triangle intersect. Students learn that any triangle has a circumscribed circle. In the next lesson, students will apply this same line of inquiry to learn that only certain quadrilaterals have circumscribed circles.
Students use appropriate tools strategically when they decide where to locate a school that is equidistant to 3 towns (MP5). In this case, students not only choose between physical tools, but they must also realize that the perpendicular bisector is the appropriate conceptual tool for nding the solution to the problem.
Required Materials Compasses
Geometry toolkits
Student Learning Goals
• Let’s construct circles outside of triangles and investigate the angles. 10.1 A Snug Circle
Warm Up: 5 minutes
Through experiment, students begin to make sense of the problem of nding the smallest circle that surrounds a shape (MP1). In later activities, students will leverage what they know about circles and distances to nd that any triangle has exactly one circle that goes through all of its vertices. In an upcoming lesson, students will nd that only certain quadrilaterals have this property.
Student Task Statement
Use a compass to estimate the smallest circle that will t snugly around each shape:
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Teacher Guide