Contextualization

Mathematics is a discipline that permeates all aspects of our lives, even if we are not always aware of it. And one of the fundamental concepts of mathematics is the Theorem of Thales, introduced by the Greek philosopher Thales of Miletus in the 6th century BC. The Theorem of Thales establishes the basis for understanding plane geometry, a subdiscipline of mathematics that deals with shapes and figures that can be drawn on a flat surface, such as a piece of paper or a computer screen.

The Theorem of Thales is a principle that applies to triangles, a geometric shape that we find everywhere, in nature, in architecture, in design, and even in the prism of a rainbow. But what exactly does this theorem tell us? Simply put, it states that, in a triangle, if a line is drawn parallel to one of the sides of the triangle and intersects the other two sides, then this line divides these two sides into segments of proportional length.

Theorem of Thales and Its Impacts

When we think of geometry, it is easy to see it as a dry and theoretical topic that has no practical applications. However, the Theorem of Thales has a large number of practical applications in the real world. For example, engineers and architects use the Theorem of Thales to ensure that the structures they design and build are stable and safe. In addition, the Theorem of Thales is useful in a variety of other fields, such as geology, astronomy, art, photography, and even biology.

In the world of art and design, for example, the theorem is used to create perspectives and depth, key elements to make a two-dimensional image more realistic and appealing to the human eye. In geology, the theorem can be used to calculate the depth of a well or the height of a mountain. And in the world of film and photography, it is used to calculate distances and angles to achieve the perfect composition for a scene or photograph.

Below are some resources to deepen your knowledge about the theorem of Thales and its applications:

• Theorem of Thales - Mundo Educação
• Theorem of Thales - Brasil Escola
• Theorem of Thales - Só Matemática

Practical Activity

Activity Title: Building and Applying the Theorem of Thales

Objective

The objective of the project is to apply the Theorem of Thales in a real-world context, developing a deep understanding of this important mathematical concept.

Project Description

This project will have two main parts. In the first part, you will build a physical model to demonstrate the Theorem of Thales. In the second part, you will use this model to answer a practical question. This is done using the principles of the Theorem of Thales.

Groups of 3 to 5 students will participate in the project, which should have a total duration of approximately 15 to 20 hours per student.

Required Materials

• Ruler
• Graph paper
• Pencils
• Colored pens
• Calculator
• Computer with internet access
• Graphic editing software (optional)

Step by Step

1. Building the Model: Each group should build a physical model that demonstrates the Theorem of Thales. This can be done by drawing a triangle and a line parallel to one of the sides of the triangle that intersects the other two sides of the triangle.

2. Measurement and Calculations: Measure the segments of each side of the triangle that were created by the parallel line. Use the proportion to verify if the Theorem of Thales applies.

3. Model Presentation: Each group should present their model to the others, explaining its construction and the calculations performed.

4. Practical Application: The second part of the project involves the application of the Theorem of Thales. Each group should research an application of the Theorem of Thales in the real world and present a solution to a practical problem involving this mathematical concept.

5. Presentation of the Practical Application: Each group should present their solution to the practical problem, explaining how they used the Theorem of Thales to reach their solution.

Project Deliverables

Groups must deliver the following:

1. Physical Model: The physical model built to demonstrate the Theorem of Thales.

2. Written Report: A report detailing the process of building the model, the calculations performed, and the practical application of the Theorem of Thales. This report should include:

   • Introduction: Explanation of the Theorem of Thales and its relevance in the real world.
   • Development: Details about the construction of the model, the measurements and calculations performed, the methodology used, and the results obtained. Additionally, a description of the practical application of the Theorem of Thales, including the problem and the proposed solution.
   • Conclusions: Brief summary of the main points of the project, the lessons learned, and the conclusions drawn from the project.
   • Bibliography: Indication of the sources used for the project.

This report should complement the work done in the project, explaining in detail the process, calculations, and decisions made by the group members. Additionally, the report should highlight the technical skills acquired during the project, such as the application of metric relations and the use of the Theorem of Thales, as well as the socio-emotional skills, such as time management, communication, problem-solving, creative thinking, among others.