Lesson 3

Construction Techniques 1: Perpendicular Bisectors

3.1: Find All the Points! (5 minutes)

Warm-up

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

Student Facing

Here are 2 points labeled \(A\) and \(B\), and a line segment \(CD\):

Two points, A and B, and a vertical line segment with endpoints C and D.
  1. Mark 5 points that are a distance \(CD\) away from point \(A\). How could you describe all points that are a distance \(CD\) away from point \(A\)?
  2. Mark 5 points that are a distance \(CD\) away from point \(B\). How could you describe all points that are a distance \(CD\) away from point \(B\)?
  3. In a different color, mark all the points that are a distance \(CD\) away from both \(A\) and \(B\) at the same time.

Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

Activity Synthesis

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 \(B\).)

3.2: Human Perpendicular Bisector (15 minutes)

Activity

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.

Launch

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. 

Student Facing

Your teacher will mark points \(A\) and \(B\) on the floor. Decide where to stand so you are the same distance from point \(A\) as you are from point \(B\). Think of another place you could stand in case someone has already taken that spot.

After everyone sits down, draw a diagram of what happened.

Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

Student Facing

Are you ready for more?

In this activity, we thought about the set of points on the floor—a two-dimensional plane—that were equidistant from two given points \(A\) and \(B\). What would happen if we didn’t confine ourselves to the floor? Start with two points \(A\) and \(B\) in three-dimensional space. What would the set of points equidistant from \(A\) and \(B\) look like?

Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Extension Student Response.

Activity Synthesis

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.

3.3: How Well Can You Slice It? (15 minutes)

Activity

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).

Launch

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.

Two figures.

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 \(FG\).

Student Facing

Use the tools available to find the perpendicular bisector of segment \(PQ\)

After coming up with a method, make a copy of segment \(PQ\) on tracing paper and look for another method to find its perpendicular bisector.

Line segment with endpoints P and Q.

 

Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

Anticipated Misconceptions

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

Activity Synthesis

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.

Speaking: MLR 8 Discussion Supports. Use this routine to scaffold students in producing statements describing the steps to take to construct the perpendicular bisector using a straightedge and compass in their groups. Provide sentence frames for students to use, such as: “First, we need to _____ because _____ .” This will help students to produce and make sense of the language needed to communicate their own ideas when describing geometric constructions.
Design Principle(s): Support sense-making
Action and Expression: Develop Expression and Communication. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategies and ideas. For example, “First, I _____ because _____. Then, I _____.” and “This method works/doesn’t work because....” 
Supports accessibility for: Language; Organization

Lesson Synthesis

Lesson Synthesis

Display an image of a perpendicular bisector for all to see.

$\overline{AB} \perp \overline{CD}, AE=EB$

A perpendicular bisector.

Teach students the notation for congruent segments, perpendicular lines, and right angles. Remind them they might encounter this notation in image captions.

  • \(\overline{AB} \perp \overline{CD}\)
  • \(m \angle AEC = 90^\circ\)
  • \(AE=EB\)

Remind students that a conjecture is a statement they think might be true. Display the conjecture that the perpendicular bisector of a segment is the set of points that are the same distance from the endpoints of that segment. Here are some questions for discussion:

  • “How did we know that a point was the same distance away from the two given points without measuring with a ruler?” (We knew because we used circles. If the point was on the intersection of two circles centered at each of the given points and the two circles were the same size, then we know the point has to be the same distance away from each center.)
  • “What exactly is a circle? How do we use circles to reason about distance without using a ruler?” (A circle is the set of points that are a given distance away from the center. We use circles to compare distances because if a point is on a circle, it means it is a given distance away from the center of the circle.)

Tell students that they will prove this conjecture is true in a subsequent lesson. 

3.4: Cool-down - Walk the Line (5 minutes)

Cool-Down

Teachers with a valid work email address can click here to register or sign in for free access to Cool-Downs.

Student Lesson Summary

Student Facing

A perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it. Recall that a right angle is the angle made when we divide a straight angle into 2 congruent angles. Lines that intersect at right angles are called perpendicular.

A conjecture is a guess that hasn't been proven yet. We conjectured that the perpendicular bisector of segment \(AB\) is the set of all points that are the same distance from \(A\) as they are from \(B \). This turns out to be true. The perpendicular bisector of any segment can be constructed by finding points that are the same distance from the endpoints of the segment. Intersecting circles centered at each endpoint of the segment can be used to find points that are the same distance from each endpoint, because circles show all the points that are a given distance from their center point.

Two congruent circles, each pass through the center of the other at A and B, radius AB is drawn. A vertical line through both circles’ intersection points, labeled point C at the top intersection.