Context

To start our explorations on the concept of barycenter, it is important to understand that we live in a world full of geometric shapes. They are present everywhere, in constructions, bridge structures, architectural designs, sculptures, and even in nature. And as mathematicians, we seek mathematical descriptions and formulas of these shapes.

The barycenter is one of these descriptions, a special point within a triangle, defined as the point of intersection of the medians. Understanding the concepts of medians and what defines a barycenter, as well as its properties, is fundamental to our study. The ability to calculate barycenters on a Cartesian plane is a central skill in mathematics, and that is precisely what we will seek in this project.

But what makes the barycenter such an interesting concept? The barycenter, also known as the center of gravity, has applications in physics, engineering, computer science, architecture, and much more. It is important for balancing objects, modeling complex shapes, and can be used to solve practical problems, such as determining the best location to build a hospital in a city, so that it is as equidistant as possible from the three most populous regions of the city.

To delve deeper into the subject, I suggest consulting the book 'Euclidean Plane Geometry' by Paulo Cezar Pinto Carvalho, which contains an entire chapter dedicated to notable points of the triangle, including the barycenter. You can also watch the lecture on the topic on the Matemática Rio channel with Prof. Rafael Procopio available on YouTube.

Practical Activity

Activity Title: Barycenter in Action: Where to Build?

Project Objective: The students will apply the concept of barycenter to solve a real-world problem: the decision of where to build a public utility structure (a hospital, for example) on a map of a fictional city, so that the structure is most equidistantly accessible to the three most populous areas of the city.

Detailed Project Description:

This project involves both mathematics and geography disciplines, as it requires students to use the mathematical concept of the barycenter within a geographical context. The fictional city will be represented on a Cartesian plane, where each populous area will be represented by a point.

The terrible Covid-19 pandemic highlights how essential it is to have access to adequate health facilities. Therefore, the question of where to build a hospital is of vital importance. Let's use our mathematical skills to solve this problem!

Materials Needed:

1. Map of a fictional city on a Cartesian plane.
2. Calculators.
3. Graphic design software (such as GeoGebra).
4. Online research tools.
5. Paper and pencil.

Detailed Step-by-Step:

The groups should follow the following steps:

1. Research and study the concept of barycenter and its applications in different fields.

2. Each group will receive a map of a fictional city with three densely populated areas represented by points on the Cartesian plane.

3. The group must perform mathematical operations to calculate the barycenter of these points.

4. Use graphic design software (e.g., GeoGebra) to plot the points and the calculated barycenter and thus visualize the ideal location to build the public utility structure.

5. Each group member should contribute to writing the project report, which should include:

   • Introduction: Contextualize the situation, importance of the question asked, and project objectives.

   • Development: Explain the theory of the barycenter, describe the problem presented and the methodology used, present the calculations performed, and discuss the results obtained.

   • Conclusion: Summarize the main points of the project, present the learnings obtained, and draw conclusions.

   • Bibliography: Indicate the sources that were used to develop the activity.

Project Deliverables:

The groups must deliver:

1. The written report.
2. The city map with the points and barycenters plotted.

The project deliverables should reflect the learned and applied concepts about the barycenter. The report writing should be clear and concise, presenting logically and cohesively the introduction, development, conclusion, and bibliographic references. The work in the report should indicate a deep understanding of the topics and themes addressed and an ability to apply these concepts in a real-world context. Project progress, collaboration among group members, and individual contribution will be considered in the evaluation criteria.