Overview:  In this Summary, the teacher acknowledges that students worked on the problem with many different strategies, and then the teacher tells students to disregard different ideas and to focus on a single solution.  The teacher illustrates the solution strategy of selecting two points and making an isosceles triangle on the map (as in Explore 3).  The teacher introduces the perpendicular bisector and illustrates it with the perpendicular bisector of the base of the isosceles triangle.  Because the triangle is isosceles, the perpendicular bisector also goes through the opposite vertex.  The teacher illustrates that students could continue constructing perpendicular bisectors to find a point equidistant from the other points on the triangle.  The teacher tells students this point is the “circumcenter” of the triangle.

Prior knowledge:  The teacher establishes knowledge of the school mathematics concept of perpendicular bisector.  The teacher uses students’ ideas of making an isosceles triangle to illustrate what happens when constructing the perpendicular bisectors of the sides of that triangle.  The teacher introduces the vocabulary of perpendicular bisector, as well as circumcenter.

Other points of interest:  The teacher begins his discussion of the perpendicular bisector with a solution that students came up with to solve the problem.  However, many students who used the isosceles triangle method to locate a new grocery store placed the new store at the vertex of the triangle.  The teacher uses the triangle to locate the circumcenter of the triangle.  Students may not see the justification for locating the circumcenter in terms of solving the grocery store problem, because they have already located a point for the new store with the vertex of the triangle.  In addition, because an isosceles triangle is a special case in which one of the perpendicular bisectors goes through the opposite vertex, the teacher may promote an understanding that perpendicular bisectors always or often go through opposite vertices in triangles.