Unit 2, Lesson 8
The Perpendicular Bisector Theorem
Integrated Math 1
8.1
Warm-up
Which Three Go Together: Intersecting Lines
Instructional Routines
Which Three Go Together?
Materials
To Gather
Math Community Chart
Activity Narrative
This Warm-up prompts students to compare four images. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. In particular, students will be focused on the characteristics of perpendicular lines.
Launch
Arrange students in groups of 2–4. Display the figures for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three images that go together and can explain why. Next, tell each student to share their response with their group and then together find as many sets of three as they can.
Activity
NoneWhich three go together? Why do they go together?
A
B
C
D
Student Response
Activity Synthesis
Ask each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology they use, such as “perpendicular.” Also, press students on unsubstantiated claims.
Math Community
After the Warm-up, display the Math Community Chart and a list of 2–5 revisions suggested by the class in the previous exercise for all to see. Remind students that norms are agreements that everyone in the class shares responsibility for, so everyone needs to understand and agree to work on upholding the norms. Briefly discuss any revisions and make changes to the “Norms” sections of the chart as the class agrees. Depending on the level of agreement or disagreement, it may not be possible to discuss all suggested revisions at this time. If that happens, plan to discuss the remaining suggestions over the next few lessons.
Tell students that the class now has an initial list of norms or “hopes” for how the classroom math community will work together throughout the school year. This list is just a start, and over the year it will be revised and improved as students in the class learn more about each other and about themselves as math learners.
Lesson Synthesis
Display these statements for all to see:
- If a point
is the same distance from as it is from , prove that must be on the perpendicular bisector of . - If
is a point on the perpendicular bisector of , prove that the distance from to is the same as the distance from to .
Ask students what is the same about the two statements and what is different. (They contain the same words—only the “if” and “then” statements are switched.)
Remind students that when two statements are switched like this, we call them converses. Ask students if these statements are converses: “If you practice, then you’ll get better,” and “If you don’t practice, then you won’t get better.” (No, one is the negation of the other, but they don’t switch the order of what is given and what is concluded.)
Ask students for the converse of this statement: “If I won the lottery, then I am rich.” (If I am rich, then I won the lottery.)
Ask students if the converse of a true statement has to be true. (No, someone could get rich by receiving an inheritance or having an insanely popular YouTube channel.)
Say, “Today, we proved these converse statements about perpendicular bisectors are both true, which means that a point being the same distance from two endpoints of a segment and a point being on the perpendicular bisector of that segment is the exact same thing.” A statement and its converse will be true if they are from a definition, such as: “If a quadrilateral has four equal sides and four equal angles, then it is a square” and “If a quadrilateral is a square, then it has four equal sides and four equal angles.”
Display the correct, complete image from the first part of the Cool-down of the “Side-Angle-Side Triangle Congruence” lesson. Ask students what they can conclude about this image. (Point
Add the following theorem to the class reference chart, and ask students to add it to their reference charts:
If
Student Lesson Summary
The perpendicular bisector of a line segment is exactly those points that are the same distance from both endpoints of the line segment. This idea can be broken down into two statements:
- If a point is on the perpendicular bisector of a segment, then it must be the same distance from both endpoints of the line segment.
- If a point is the same distance from both endpoints of a line segment, then it must be on the perpendicular bisector of the segment.
These statements are converses of each other. Two statements are converses if the “if” parts and the “then” parts are swapped. The converse of a true statement isn’t always true, but in this case, both statements of the Perpendicular Bisector Theorem are true.
A line of reflection is the perpendicular bisector of segments connecting points in the original figure with corresponding points in the image. Therefore, the following three lines are all the same:
- The perpendicular bisector of a segment.
- The set of points equidistant from the 2 endpoints of a segment.
- The line of reflection that takes the 2 endpoints of the segment onto each other and the segment onto itself.
A figure: A large quadrilateral with a small triangle inside. The two upward facing sides of the quadrilateral each have two tick marks, the downward facing sides each have three tick marks. Two sides of the downward facing small triangle each have one tick mark. A perpendicular bisector is drawn through the figure, intersecting at three points. A horizontal line segment is drawn in the center of the figure.
It is useful to know that the perpendicular bisector of a line segment is also all the points which are the same distance from both endpoints of the line segment, because then:
- If 2 points
and are both equidistant from the endpoints of a segment, then the line through and must be the perpendicular bisector of the segment (because 2 points define a unique line). - If 2 points
and are both equidistant from the endpoints of a segment, then the line through and must be the line of reflection that takes the segment to itself and swaps the endpoints. - If a point
is on the line of reflection, then the distance from to a point on the original figure is the same as the distance from to its corresponding point on the image. - If a point
is on the perpendicular bisector of a segment, then it is the same distance from to both endpoints of the segment.
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