Unit 1, Lesson 5
Construction Techniques 3: Perpendicular Lines and Angle Bisectors
Integrated Math 1
5.1
Warm-up
Instructional Routines
NoneMaterials
Activity Narrative
The purpose of this activity is to prepare students for constructing a perpendicular line through a point on the line. This figure of two circles of the same radius intersecting also plays a role in the angle bisector construction later in the lesson.
Launch
Arrange students in groups of 2. After quiet work time, ask students to compare their conjectures to their partner’s and decide whether they are both correct, even if they are different. Follow with a whole-class discussion.
Activity
NonePoints and are each at the centers of circles of radius .
- Compare the distance to the distance . Be prepared to explain your reasoning.
- Compare the distance to the distance . Be prepared to explain your reasoning.
- Draw line , and write a conjecture about its relationship with segment .
Activity Synthesis
The goal of this discussion is to connect this diagram to perpendicular lines. Ask for students to share their conjectures. Connect these ideas back to the previous activity “Human Perpendicular Bisector,” in which students conjectured that the set of points the same distance from points and is the perpendicular bisector of segment . In this case, that would mean that line would be the perpendicular bisector of segment .
Math Community
After the Warm-up, display the class Math Community Chart for all to see and explain that the listed “Doing Math” actions come from the sticky notes students wrote in the first exercise. Give students 1 minute to review the chart. Then invite students to identify something on the chart they agree with and hope for the class or something they feel is missing from the chart and would like to add. Record any additions on the chart. Tell students that the chart will continue to grow and that they can suggest other additions that they think of throughout today’s lesson during the Cool-down.
Lesson Synthesis
The purpose of this discussion is to connect the basic construction of two circles going through each other’s center with the constructions for a line perpendicular to a given line and an angle bisector. Display images of the construction from each activity for all to see. Ask students:
- “What was the first step in each of these constructions?” (In each construction, the first step was to create a circle that intersects the given line or angle in two places.)
- “The first circle creates two points that are the same distance away from the center. Now, where do you see the two circles that go through each other’s center in each of the constructions?” (In each construction, the 2 points created from the first step become centers of circles that go through each other’s center.)
Highlight the 2 points created from the first step of each construction, then highlight the 2 circles that go through each other’s centers in the second step.
Demonstrate the construction of an angle bisector using geometry software and increase the measure of the angle until it reaches 180 degrees. Explain that each construction is really a perpendicular bisector. For the angle bisector, connecting the two equidistant points forms an isosceles triangle. This surfaces a surprising and important connection: The angle bisector of an isosceles triangle seems to also be the perpendicular bisector of the base. Tell students that this is only a conjecture at this point, but this connection between the angle bisector of an isosceles triangle and the perpendicular bisector of its base will play an important role in the next unit.
Student Lesson Summary
We can construct a line that is perpendicular to a given line. We can also bisect a given angle using only a straightedge and compass. The line that goes through the vertex of an angle to divide it into two equal angles is called the angle bisector. Both constructions use 2 circles that go through each other’s centers:
To construct a line perpendicular to line that goes through a given point , start by finding 2 points, labeled here as and , on the given line that are the same distance from . Then create 2 circles of the same size centered at and that go through each other’s centers. Connect the intersection points of those circles to draw a perpendicular line, .
To construct an angle bisector, start by finding 2 points, labeled here as and , that are on the rays and the same distance from the vertex. Then create the 2 circles of the same size centered at and that go through each other’s centers. Connect the intersection points of those circles to draw the angle bisector.
One large circle with a center point and a central angle. Two small circles centered on the intersection of central angle and large circle. A line is drawn from the intersection point of the two small circles through the center of the large circle.
In fact, we can think of creating a perpendicular line as bisecting a 180 degree angle!