Lesson 3

The purpose of this warm-up is to apply the precise definition of a circle to explore points that are equidistant from two points.

The purpose of discussion is to emphasize the definition of a circle. Here are some questions to consider:

• “Why do all the points create a circle?” (A circle is the set of points that are the same distance away from the center.)
• “What do you notice about the points that are the same distance, \(CD\), from both \(A\) and \(B\)?” (The points that are distance \(CD\) away from both \(A\) and \(B\) are the two intersection points of the circles.)
• “Could there be 3 points that are all distance \(CD\) from \(A\) and \(B\)?” (No. The points on the circle centered at \(B\) that are inside the other circle are closer to \(A\) than \(B\), and the rest of the points are farther from \(A\) than \(B\).)

The purpose of this activity is for students to develop intuition that the set of points equidistant to two given points forms a perpendicular bisector by asking them to play the role of the points. Students will formalize this conjecture in the lesson synthesis and prove it in a subsequent lesson.

Locate an area in the classroom or nearby where several students can stand together and be seen by all students. Mark two points on the floor about 2 meters apart with masking tape and clear a space between and around the points. Label one point \(A\) and one point \(B\). Invite one student to stand at \(A\) and one student to stand at \(B\).

Tell students, “The next volunteer will stand so they are the same distance from both \(A\) and \(B\). Raise your hand when you have an idea about where you will stand.”

Select a student whose hand is raised to stand in the spot they chose. Ask the class, “Are they the same distance from \(A\) as they are from \(B\)? How can we check?” Check distances using methods the students suggest.

Tell students, “The next volunteer will also stand so that their distance from \(A\) is the same as their distance from \(B\). They can’t stand in the same place as _____. Raise your hand when you have an idea about where you will stand.”

Continue calling on volunteers to stand, checking their distances, and asking for new volunteers until students don’t hesitate to find a new spot, and are standing in a straight line perpendicular to segment \(AB\). At this point, ask students to return to their seats and draw a diagram of what just happened. 

The purpose of discussion is to establish the conjecture that the perpendicular bisector of a segment is the set of points that are the same distance to each endpoint.

Invite students to look at their sketch of points whose distance from \(A\) is the same as their distance from \(B\). Ask them what they notice about the points. Ask them what they wonder about the points.

Things students may notice:

• All the equidistant points form a line.
• The line goes through the midpoint of segment \(AB\).
• The line is perpendicular to segment \(AB\).

Things students may wonder:

• Will this always be true?
• Why is the line perpendicular to segment \(AB\)?
• Does this line have a name?

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

Tell students that in mathematics, things people wonder are often referred to as conjectures. A conjecture is a statement that we wonder whether it is true. Ask students to make a conjecture about the collection of all the points whose distance from \(A\) is the same as their distance from \(B\), and select 2 or 3 to share.

The goal of this activity is to use what students know about points equidistant to two given points to develop the construction for the perpendicular bisector using a straightedge and compass.

Monitor for students who use these methods:

• paper folding
• freehand drawing
• compass and straightedge construction

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

Define the perpendicular bisector as a line through the midpoint of a segment that is perpendicular to that segment. Informally, explain that bi means two and sect means cut, and so a perpendicular bisector is literally a line perpendicular to a segment that cuts it into two congruent pieces.

Display these two figures and ask students to explain why each dashed line is not a perpendicular bisector of the segment it intersects.

Students might point out that in one, the angles are not right angles; therefore, the lines are not perpendicular. In the other, students might state that the lines are perpendicular but segment \(FG\) is not bisected since \(H\) is not the midpoint of \(FG\).

If students struggle to get started, direct them to their diagram from the warm-up. How could they construct that diagram?

The purpose of this discussion is to compare different methods for drawing a perpendicular bisector, highlighting the pros and cons of each.

Invite a student who used paper folding to demonstrate first.

Display two copies of segment \(PQ\) for all to see. Explain that the displayed segments cannot be folded. Invite a student who did a freehand drawing to demonstrate next. Follow with a student who used a compass and straightedge to make a construction.

Discuss how to check a perpendicular bisector for accuracy. Discussion may include:

• measuring to see whether the perpendicular bisector goes through the midpoint of segment \(PQ\) and forms a \(90^\circ\) angle with segment \(PQ\)
• selecting a point on the perpendicular bisector and measuring to see whether it is the same distance from \(P\) and \(Q\), repeating for multiple points
• using the compass to see whether points along the perpendicular bisector are the same distance from \(P\) and \(Q\)

If no student uses a compass to construct or check, encourage the class to consider how to use that tool. Display the image from the warm-up again and invite students to explain how to use that construction to find a perpendicular bisector.

Emphasize that both paper folding and construction with a compass and straightedge are valid, accurate methods, but freehanding only works for a sketch. Choosing which one to use will depend on the problem and tools available.