Lesson 5
Triangles in Circles
Lesson Narrative
In the Constructions and Rigid Transformations unit, students used perpendicular bisectors to partition a region based on distances from given points. In the Coordinate Geometry unit, students proved that the medians of a triangle intersect at a single point. In that unit they also looked at a specific case of triangle altitudes coinciding, and in an optional activity, plotted the intersection of the perpendicular bisectors of a specific triangle’s sides in the coordinate plane.
In this lesson, students build on this previous work and construct the circumscribed circle of a triangle. First, students recall that points on the perpendicular bisector of a segment are equidistant from the vertices of the segment. Then, they use this property to conclude that all 3 perpendicular bisectors of the sides of a triangle intersect in a single point, the triangle’s circumcenter. They construct the circumcenter and circumscribed circle of a triangle, and conclude that this method would apply to any triangle. Finally, students investigate the locations of circumcenters in right, obtuse, and acute triangles.
One of the activities in this lesson works best when students have access to devices that can run the GeoGebra applet because students will benefit from seeing the relationship in a dynamic way.
Students make use of structure (MP7) as they use the properties of perpendicular bisectors to draw conclusions about triangle circumcenters.
Learning Goals
Teacher Facing
 Construct the circumscribed circle of a triangle.
 Prove (using words and other representations) that the perpendicular bisectors of a triangle’s sides are concurrent.
Student Facing
 Let’s see how perpendicular bisectors relate to circumscribed circles.
Required Materials
Required Preparation
Devices are required for the digital version of the activity Wandering Centers, ideally 1 for every 12 students.
Be prepared to display the applet in the synthesis for Wandering Centers.
Learning Targets
Student Facing
 I can construct the circumscribed circle of a triangle.
 I can explain why the perpendicular bisectors of a triangle’s sides meet at a single point.
CCSS Standards
Glossary Entries

circumcenter
The circumcenter of a triangle is the intersection of all three perpendicular bisectors of the triangle’s sides. It is the center of the triangle’s circumscribed circle.

circumscribed
We say a polygon is circumscribed by a circle if it fits inside the circle and every vertex of the polygon is on the circle.

cyclic quadrilateral
A quadrilateral whose vertices all lie on the same circle.