Contextualization

Welcome to this adventurous project involving two very exciting concepts of our beloved Mathematics: the bisector and the perpendicular bisector. But before we dive into our practical journey, let's take a theoretical trip to better understand these two concepts.

The bisector is a line that divides an angle into two equal angles. When we talk about a bisector in a triangle, we are commonly referring to the bisector of an internal angle. It has the interesting characteristic of being the locus of points in the plane that are equidistant from the sides of the angle. On the other hand, the perpendicular bisector is a line that divides a line segment in half, that is, into two equal parts. In a triangle, the perpendicular bisector of a side is the line that passes through the midpoint of that side and is perpendicular to it. It has the interesting property of being the locus of points in the plane that are equidistant from the endpoints of the segment.

An application of these concepts very present in our daily lives is the location of public facilities in a city, or the location of radio and TV signal transmission antennas. For example, the ideal place to build a hospital in a city, in order to minimize the average distance that the population will have to travel to reach it, is precisely at the intersection point of the perpendicular bisectors of the line segments that connect the farthest neighborhoods. Similarly, a TV antenna should be positioned at the intersection point of the perpendicular bisectors of the segments that connect the farthest points of the city, in order to ensure good signal quality in all regions.

The suggested sources for consultation and basis for the realization of this project are:

• Brasil Escola for information about the bisector.
• Mundo Educação for details about the perpendicular bisector.
• YouTube – Omatematico.com for a visual and didactic explanation of these two concepts.

Practical Activity

Activity Title: Building a City Geometrically

Project Objective

This activity aims to build a city using the concepts of bisector and perpendicular bisector, highlighting the application of these concepts in the location of public facilities and in the distribution of communication signals.

Detailed Project Description

Groups will create a map representing a fictional city with at least five neighborhoods. Within this city, they must decide where to position three types of structures: a hospital, a school, and a radio tower, so that each structure serves the city in the best possible way. They will use the concepts of bisector and perpendicular bisector to make these decisions.

Required Materials

1. Cardboard or cardstock (for the construction of the city map)
2. Pencils, colored pens, and rulers (for drawing and measurements)
3. Materials for building the structures (wooden sticks, clay, paper, etc.)
4. Books, internet, etc. (for research)

Detailed Step-by-Step for Carrying Out the Activity

1. Research and discuss in groups the concepts of bisector and perpendicular bisector and their practical applications, using the suggested sources and any other legitimate sources the group desires.

2. Draw on cardboard or cardstock the map of the fictional city, clearly marking the location of each neighborhood.

3. Decide where on the map each structure (hospital, school, and radio tower) should be positioned, taking into account the concepts of bisector and perpendicular bisector.

4. Draw the bisector and perpendicular bisector that led to the decision of the positioning of each structure, explaining how each influenced this decision.

5. Build the structures using the chosen materials and position them on the city map.

6. Write a report describing the experience, which should include the four main topics: Introduction, Development, Conclusions, and Bibliography used.

   1. Introduction: The group must contextualize the theme, talk about its relevance and application in the real world, and discuss the project's objective.

   2. Development: The group must explain what bisector and perpendicular bisector are, how these concepts were applied in the activity, the method used to build the city map and position the structures, and discuss the results, explaining why they chose the locations where they positioned each structure.

   3. Conclusions: The group must conclude the work, recalling its main points, what they learned from the experience, and the conclusions drawn about the project.

   4. Bibliography: The group must indicate the sources they relied on to work on the project such as books, web pages, videos, etc.

Project Delivery

The final project must include the city map with the positioned structures and the graphic explanation of the use of the concepts of bisector and perpendicular bisector, as well as the report describing the experience. In addition, students must be prepared to make an oral presentation about the project, explaining their decisions and the learning obtained.