# Lesson 3

Construction Techniques 1: Perpendicular Bisectors

### Lesson Narrative

The purpose of this lesson is to lay the foundation for understanding the perpendicular bisector of a segment as both a line perpendicular to a segment passing through its midpoint (by definition) and the set of points equidistant to the endpoints. The second fact will be proven in the next unit. The perpendicular bisector plays a key role in the definition of reflection later in this unit and in the proof of the Side-Side-Side triangle congruence theorem in the next unit.

This lesson continues the theme of asking how much can be learned without using numbers to measure distance as well as building on students’ understanding of angle and perpendicular from previous grades. Students look for and make use of structure when they think about where their classmates should stand during Human Perpendicular Bisector in order to be the same distance away from two given points (MP7). The more students that correctly place themselves, the more apparent the structure. Once students determine the structure, they record it as a conjecture. A conjecture is defined as a reasonable guess that students are trying to either prove or disprove.

If students have ready access to digital materials in class, they can choose to perform all construction activities with the GeoGebra Construction tool accessible in the Math Tools or available at https://www.geogebra.org/m/VQ57WNyR

### Learning Goals

Teacher Facing

• Comprehend that a perpendicular bisector is the set of points equidistant from two given points.
• Construct a perpendicular bisector.

### Student Facing

• Let’s explore equal distances.

### Required Preparation

For Human Perpendicular Bisector, mark two points on the floor of the classroom two meters apart, using masking tape. Clear a large space around and between the two marked points.

### Student Facing

• I can construct a perpendicular bisector.
• I understand what is special about the set of points equidistant from two given points.

Building Towards

### Glossary Entries

• circle

A circle of radius $$r$$ with center $$O$$ is the set of all points that are a distance $$r$$ units from $$O$$

To draw a circle of radius 3 and center $$O$$, use a compass to draw all the points at a distance 3 from $$O$$.

• conjecture

A reasonable guess that you are trying to either prove or disprove.

• line segment

A set of points on a line with two endpoints.

• parallel

Two lines that don't intersect are called parallel. We can also call segments parallel if they extend into parallel lines.

• perpendicular bisector

The perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it.