# The perpendicular bisector: a closer look.

Introduction

Some familiar, yet rather abstruse, objects such as perpendicular bisectors acquire new meaning when we understand how and when they are applied to real life situations and how useful they can be. In this project, students discover that geometrically the world around us can be described in various ways, depending on our assumptions and definitions, and that we can then choose which description or model is most useful to us under given circumstances. This insight allows us to look at the circle and the perpendicular bisector from alternate perspectives, thus strengthening our conceptual understanding of those two objects and of geometry.

Learning objectives

The primary objective of this exploration is to familiarise the students with the concepts of the perpendicular bisector. To understand its defining characteristics in greater depth we study them in Euclidean geometry and in a particular non-Euclidean geometry system known as 'taxicab geometry'. To appreciate how useful the perpendicular bisector is in everyday life and activities, students are introduced to the Voronoi diagram--a tool that is interesting, helpful, sometimes counter-intuitive and always fun.

Project design

This exploration of the perpendicular bisector consists of two parts. In the first, we study the problem of boundaries using Voronoi diagrams. Given a set of points--called 'seeds' or 'sites' or 'generators'--a Voronoi diagram divides an area into a number of regions, each of which is associated with one of the seeds. Each region consists of all the points that are closer to that seeds than to any other. In our real-life application we want to determine the boundaries of three schools close to ours, or divide a fictional town into pizza delivery regions. As the students study the problems, they understand that the key to determining the boundaries in a Voronoi diagram is finding the perpendicular bisector of a line segment.

In the second part of this project we look at the same problems with taxicab geometry--a non-Euclidean geometry system that may be more useful under certain circumstances, for example when movements are constrained by having to follow roads. In taxicab geometry, the perpendicular bisector and the circle are defined in the same way as in Euclidean geometry, but they look quite different.

Class activity

After I introduce the perpendicular bisector, students construct it with a straightedge and a compass, which brings us to review the basic definition and characteristics of a circle. Then we look at the real-life applications of the perpendicular bisector: a quick Internet search of 'Voronoi diagram' will provide hundreds of thousands of hits, including applications to ecology (areas available to trees), zoology (distribution of insects), astronomy (identification of clusters of galaxies), geography (analysis of patterns of urban settlement) and so on. We also use Java applets (such as Chew, 2007)) to explore and understand these concepts more in depth. Finally we locate our school on a map, mark out two, then three high schools and draw the Voronoi diagrams of their boundaries. Alternatively, we utilise the NCTM Illuminations lesson Dividing a Town into Pizza Delivery Regions (Reeder, n.d.), which asks the students to divide a town into pizza delivery regions (see Figures 1, 2 and 3 for student work). In either case, the students, working in groups, quickly realise that the perpendicular bisector of the line segment connecting two pizzerias or two schools would be the best tool to divide the surrounding region in areas of influence. They then generalise their findings to three, four, and sometimes even five locations.

In the second step of this exploration, we study the same two objects--the circle and the perpendicular bisector--in a non-Euclidean framework, specifically taxicab geometry. While all students have generally been exposed to Euclidean geometry, most do not know much about non-Euclidean geometries and are very intrigued, if at times baffled, by this different perspective. I use a handout based on the book Taxicab Geometry: An Adventure in Non-Euclidean Geometry (Krause, 1987) or, alternatively, the excellent article Is That Square Really a Circle? by Christopher E. Smith (2013). We start by redefining distance, as in taxicab geometry you have to follow the street pattern, and we examine and debate the cases in which the Euclidean distance and the cab-distance are the same. We also discuss cab-equidistant paths from point A to point B, as the concept of equal cab-distance will be important when we investigate what the circle and the perpendicular bisector in taxicab geometry look like. Figures 4 and 5 show some students' work. The students work in small groups in class on problems similar to those they solved for the Euclidean part of this project, such as:

* In the following system of axes you must draw the school district boundaries so that every student is going to the closest school. The three schools and their coordinates are...

* Alice and Bob live at A(-6, 3) and B (4, -1). They want to open a business whose location is at an equally distant cab ride to each house. Find the possible store locations.

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Discussion

As the answers are quite unexpected, the students proceed cautiously: it is very surprising for students to find out that the locus of points equidistant from a center looks like a square and that the perpendicular bisector is not a straight line. However, through discussion, trial and error, peer review, revision, going back to definitions etc., they get the right answers and are thrilled when they realise they are correct.

At this point we are ready for a thorough discussion of both Euclidean and non-Euclidean models, including when and where one is more appropriate or more useful than the other. The research on various Voronoi diagram applications provides multiple examples in different fields and supports vigorous conversations.

Finally we talk about the simplifying assumptions that are made when constructing a mathematical model, such as a Voronoi diagram. For example, in the pizzeria problem we assume that the population who regularly buys pizzas is uniformly distributed; so we ask ourselves which other assumptions are made when we apply this model to different situations.

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Evaluation and assessment

Students are evaluated during the class portion of the project, which involves reading and discussing the project directions, researching and working with others in class, communicating and talking about mathematics, and sharing their solutions with their group. The evaluation focusses on their understanding of the problem and their search for a solution, their willingness to appreciate different approaches, their openness to, and understanding of, other students' directions and work, their ability to communicate mathematical concepts and ideas, their respect for others' thoughts and their work ethic. To assess the work done in class, I use teacher's observations, class discussion, small group review, and other oral assessments. I keep a tool (a clipboard, a notebook, or an iPad) with me at all times to take quick notes about each student's work and contributions. When time permits, we look at high schools in our area, or post offices, or fire stations: the students are asked to draw a Voronoi diagram using those locations as seeds on the city map and then to compare the Voronoi regions with the school districts, the postal delivery zones or the fire response zones. These findings can be presented by the students with (hardcopy or electronic) posters and evaluated with an appropriate rubric. The same study can be done with the distribution of public libraries or fresh produce markets: the ensuing conversations are very lively, touch on topics such as education policies, equity of access to resources, food choices, etc. and highlight the value of mathematics in understanding the world around us.

Conclusion

I find that this activity is particularly appropriate for a mixed-ability geometry class. This type of content and format encourages and emphasises collaboration, exploration, research, and real-life problem solving--all features that pique students' interest. The enthusiastic reception of this project by the students leads me to conclude that this is a strong project that gives the students a deeper understanding and a stronger appreciation of the usefulness and effectiveness of geometry.

References

Chew, P. (2007). Voronoi diagram/Delaunay triangulation. Retrieved from http://www.cs.cornell.edu/home/chew/Delaunay.html.

Krause, E. F. (1987). Taxicab geometry: An adventure in non-Euclidean geometry. New York: Dover Publications.

Reeder, J. (n.d.). Dividing a town into pizza delivery regions. Retrieved from http:// illuminations.nctm.org/Lesson.aspx?id=2688

Smith, C. E. (2013). Is that square really a circle? Mathematics Teacher, 106(8).

Alessandra King

Holton-Arms School, MD, USA

alessandra.king@holton-arms.edu
```Figure 3

A Voroni diagram of a set of points, or sites, is a collection of
regions dividing up a plane. Each region contains all points closest
to a given site. The Voroni regions are always convex polygons, and
depending on how large the area is that is being measured, the polygons
could extend forever. Besides their uses in geometry problems, Voroni
diagrams can also be used in real life. In some professions, it is
necessary to calculate the possible area of growth for a tree. In
biology and ecology, and even for yard work purposes, one may need to
see how far a trees root system can extend Here, a Voroni diagram could
be used to get an idea of where to plant the trees, or to see how much
space they have to grow. In this case, the trees would be represented
as points and the area around them would be their root systems. Voroni
diagrams are also useful for plotting and measuring the territories of
animals. If a site represented a species or group of animals, then the
region around each site would represent the territory of each group.
Voroni diagrams can help show the extent of a territory or animals or
of the root system of a plant, and overall they are very useful.
```
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