IM Geo.2.8 Lesson: The Perpendicular Bisector Theorem

Prove that if a point  is the same distance from  as it is from , then C must be on the perpendicular bisector of . At first they were really stuck. Noah asked, “How do you prove a point is on a line?” Their teacher gave them the hint, “Another way to think about it is to draw a line that you know  is on, and prove that line has to be the perpendicular bisector.” They each drew a line and thought about their pictures. Here are their rough drafts.

Diego’s approach: “I drew a line through  that was perpendicular to  and through the midpoint of . That line is the perpendicular bisector of  and  is on it, so that proves  is on the perpendicular bisector.”



Jada’s approach: “I thought the line through  would probably go through the midpoint of  so I drew that and labeled the midpoint . Triangle  is isosceles, so angles  and  are congruent, and  and  are congruent. And  and  are congruent because  is a midpoint. That made two congruent triangles by the Side-Angle-Side Triangle Congruence Theorem. So I know angle  and angle  are congruent, but I still don’t know if  is the perpendicular bisector of ."



Noah’s approach: “In the Isosceles Triangle Theorem proof, Mai and Kiran drew an angle bisector in their isosceles triangle, so I’ll try that. I’ll draw the angle bisector of angle . The point where the angle bisector hits  will be . So triangles  and  are congruent, which means  and  are congruent, so  is a midpoint and  is the perpendicular bisector."



• What do you notice that this student understands about the problem?

• What question would you ask them to help them move forward?

Using the ideas you heard and the ways you think each student could make their explanation better, write your own explanation for why  must be on the perpendicular bisector of  and .

Elena has another approach: “I drew the line of reflection. If you reflect across C, then A and B will switch places, meaning A' coincides with B, and B' coincides with C.  will stay in its place, so the triangles will be congruent.” What feedback would you give Elena?

Write your own explanation based on Elena‘s idea.

If  is a point on the perpendicular bisector of , prove that the distance from  to  is the same as the distance from  to .

• What do you notice that your partner understands about the problem?

• What question would you ask them to help them move forward?