# Lesson 9

Speedy Delivery

- Let’s use perpendicular bisectors.

### 9.1: Notice and Wonder: Dots in a Square

What do you notice? What do you wonder?

### 9.2: Who Is Closest?

Here is a square city with 3 locations of the same store.

- The company wants to break the city down into regions so that whenever someone orders from an address, their order is sent to the store closest to their home. They have hired you to decide how to partition the city between the 3 stores. Explain or show your reasoning.
- If there are 100 employees, how should they be distributed among the 3 locations?
- Is there anywhere in the city that has the same distance to all 3 stores?
- Now a fourth store opens. Partition the city again.

In 1854, there was an outbreak of cholera in London. A physician named John Snow thought the water supply might be responsible. He made a map showing the location of all the water pumps in the city and the locations of all the deaths due to cholera in the city. How could he have used the ideas in this activity to help isolate the cause of the outbreak? The diagrams you made in the activity and that Snow made are called Voronoi diagrams, and are still actively studied by mathematicians.

### 9.3: Now Who is Closest?

- Use dynamic geometry software to create a Voronoi diagram from a map.
- Who might be interested in this information?
- Write a letter to the person or organization, explaining what the diagram tells them about the map you chose.

### 9.4: Another Layer

Your teacher will assign you a** tessellation**.

- Imagine that each point is a store from the “Who Is Closest?” activity. Repeat the process you used there to define the regions that are closest to each of the points.
- Use the Polygon Tool and the color selector in the Style Bar to enhance your design.

A.

B.

C.

### Summary

A **tessellation** is an arrangement of figures that covers the entire plane without gaps or overlaps. A simple example is a square grid. So that means graph paper is a tessellation. Here is another tessellation made of quadrilaterals. Can you see how repeating this pattern could cover the entire plane?

One way to draw a new tessellation is to decompose the plane into regions that are closest to each vertex. This method uses perpendicular bisectors and is called a Voronoi diagram. It is also a tessellation. What would this pattern look like when it is extended to cover the entire plane?

### 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.