In this activity, students recall that points on perpendicular bisectors of segments are the same distance from each endpoint of the segment. In the next activity, students will construct the remaining perpendicular bisectors of this triangle and then construct the triangle’s circumscribed circle.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

The goal of this discussion is to make sure students understand that points on the perpendicular bisector of a segment are equidistant from the segment’s endpoints. Display this image for all to see. Tell students that the dashed line is the perpendicular bisector of segment .

For each of the points , , and , ask students if the point is closer to or closer to , and ask how they know. (Point is closer to . Point is closer to . Point is the same distance from as it is to . All points to the left of the perpendicular bisector are closer to , points to the right are closer to , and points on the bisector are equidistant from and .)

Students construct the circumcenter, or the point where the perpendicular bisectors of the sides of a triangle intersect, of the triangle from the previous activity. Students determine that the circumcenter is equidistant from the vertices of the triangle using the properties of perpendicular bisectors. Finally, they construct the circumscribed circle, observing that this is possible because of the equal distances already noted, making use of the structure of a circle (MP7).

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

In this activity, students compare the circumcenters of acute, obtuse, and right triangles. They find 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.

In the digital version of the activity, students use an applet to make conclusions about the circumcenters of acute, obtuse, and right triangles. The applet allows students to move the vertices of a circumscribed triangle and observe how the location of the triangle’s circumcenter changes. 

This activity works best when each student has access to devices that can run the applet because students will benefit from seeing the relationship in a dynamic way.