Contextualization

The Similarity of Triangles is a fundamental theme in geometry, whose understanding is essential for progress in mathematics and its practical applications. This theory is defined by the proportionality of corresponding sides and the congruence of corresponding angles in two triangles.

It is possible to state that two triangles are similar if they have the same shape, but not necessarily the same size. Specifically, if the three sides of one triangle are respectively proportional to three sides of another triangle, then the two triangles are similar. In other words, if one triangle is a 'scaled version' of another, they are similar.

Furthermore, the similarity of triangles is closely related to other fundamental laws and concepts in trigonometry, such as the sine law and the cosine law. These are powerful tools that allow solving a wide range of problems involving triangles.

The Similarity of Triangles has a significant impact in various areas of our daily lives. In architecture, for example, the properties of similar triangles are used to maintain correct proportions when redesigning building and structure plans. In navigation and aviation, the similarity of triangles is used to calculate distances and routes between different points.

Even in art, the similarity of triangles is paramount. Artists use it to create perspective in their works, a technique that allows representing three-dimensional objects on a two-dimensional surface. Therefore, understanding the Similarity of Triangles not only aids in the acquisition of mathematical skills but also opens up a wide range of possibilities and practical applications.

I suggest that for further exploration of the topic, you consult the following references:

1. Book 'Matemática volume único' by Gelson Iezzi.
2. Website 'Brasil Escola' in the geometry section that provides examples and exercises on the similarity of triangles.
3. Khan Academy videos available on YouTube that offer detailed explanations on the topic.

Practical Activity: 'Similarities in Practice: Art, Architecture, and Navigation'

Project Objective

The objective of this project is to explore the practical application of the similarity of triangles in various areas, such as Art, Architecture, and Navigation. The activity consists of investigating, analyzing, and building models that demonstrate how the similarity of triangles is used in these areas.

Detailed Project Description

Students will be divided into groups of 3 to 5 students, and each group will choose one of the mentioned areas to deepen their investigations and build demonstrative models.

The project will be divided into two distinct parts: the research and analysis phase and the model construction and presentation phase. Each phase should take approximately 6 hours per student to complete, totaling 12 hours of work per student.

Research and Analysis

In this phase, students must research and understand how the similarity of triangles is applied in the chosen area. Students should focus on examples of real objects, structures, or phenomena, and analyze how the similarity of triangles is used in their design or operation.

Model Construction and Presentation

After completing the research, groups must choose a specific example and build a model that demonstrates how the similarity of triangles is used in it. This model can be physical or digital, depending on the nature of the chosen example.

Required Materials

Depending on the type of model the group chooses to build, different materials will be required. These may include:

• Cardboard, paper, glue, and scissors for physical models
• Graphic design or 3D modeling programs for digital models
• Tools for precise measurements (ruler, protractor, tape measure)

Detailed Steps for Carrying Out the Activity

1. Formation of groups and selection of the area of interest.
2. Research on the application of the similarity of triangles in the chosen area.
3. Analysis of real examples and detailing how the similarity of triangles is used.
4. Selection of an example for the construction of the demonstrative model.
5. Planning the construction of the model.
6. Model construction.
7. Model presentation and explanation of how the similarity of triangles is applied.

Project Deliverables and Preparation of the Written Document

After completing all stages of the activity, each group must produce a detailed report containing the following sections:

1. Introduction: Students should introduce the topic addressed, the area chosen for study, and the relevance of the similarity of triangles in that area. They should also contextualize the object, structure, or phenomenon chosen for the model construction.

2. Development: This section should discuss the theory of the similarity of triangles and how it is applied in the chosen example. The process of building the model should be described in detail, including challenges encountered and how they were overcome. Additionally, the results obtained should be presented and discussed.

3. Conclusion: Students should reflect on what they learned during the project, both in technical and socioemotional terms. They should also discuss the impact and relevance of the similarity of triangles in the chosen area.

4. Bibliography: All resources used, including books, websites, and videos, should be cited appropriately in this section.