# Lesson 9

Speedy Delivery

### Lesson Narrative

In this lesson, students build on their experiences with perpendicular bisectors to answer questions about allocating resources in a real-world situation (MP4). To complete more steps in the mathematical modeling cycle, use the optional activity Now Who Is Closest? Then students study a tessellation (an arrangement of figures that cover the entire plane), create a Voronoi diagram by applying perpendicular bisectors, and conjecture that the Voronoi diagram of a tessellation is also a tessellation.

Some of the activities in this lesson work best when each student has access to GeoGebra Geometry from Math Tools, because students are using perpendicular bisectors to determine which regions of a map are closest to certain points. In Who Is Closest?, they do this with 3 and 4 points, but doing it with more in Now Who is Closest? will require help from technology.

### Learning Goals

Teacher Facing

• Choose geometric methods to solve design problems.
• Construct perpendicular bisectors and explain (in writing) how they are used to solve problems.

### Student Facing

• Let’s use perpendicular bisectors.

### Required Preparation

Acquire computers or tablets that can run GeoGebra Geometry from Math Tools, with one for every 2–3 students. The digital version is recommended for all classes over the paper and pencil version.

Ensure that students have at least 4 colors in their toolkits if they will be doing the paper and pencil version of Who is Closest?.

### Student Facing

• I can construct perpendicular bisectors to help solve problems.
• I can use my geometry knowledge to solve problems.

Building On

Building Towards

### Glossary Entries

• angle bisector

A line through the vertex of an angle that divides it into two equal angles.

• 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.

• inscribed

We say a polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle. We say a circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.

• 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.

• regular polygon

A polygon where all of the sides are congruent and all the angles are congruent.

• tessellation

An arrangement of figures that covers the entire plane without gaps or overlaps.