Context

Triangles are geometric shapes that we find everywhere around us, from works of art to bridge engineering. But have you ever wondered why triangles are so important? Why are they widely used in architecture, art, and engineering? The answer lies in 'Similarity of Triangles,' which plays a critical role in mathematics and its practical applications.

Similarity of triangles is a fundamental concept in the study of geometry. In this property, two triangles are considered similar if the corresponding angles are equal and the corresponding sides are proportional. This allows us to use a known quantity to find unknown values, a concept that is at the heart of trigonometry.

Similarity of triangles has practical applications in many areas. Architects and engineers use it when designing structures to ensure stability and safety. Artists use it to create perspective and depth in their works. Astronomers use it to measure the distances of planets and stars. They even have their place in our daily lives. For example, if you know the height of a tree and the shadow it casts, you can use the similarity of triangles to calculate the distance from the top of the tree to you.

To help you better understand this concept and its applications, I suggest consulting the following resources:

• The book 'Fundamentals of Elementary Mathematics: Plane Geometry' by Elon Lages Lima, to understand in a simple and complete way the principles of similarity of triangles.
• The Khan Academy platform, which provides several online lessons on similarity of triangles and exercises to practice.
• The channel Matemática Rio with Prof. Rafael Procopio on YouTube, which offers explanatory videos that teach and exemplify the concept of similarity of triangles in everyday life.

Practical Activity: 'Playing with Similar Triangles'

Project Objectives

The objective of the activity is for students to understand and apply the concept of similarity of triangles in practice, developing the following technical skills:

1. Recognize the necessary and sufficient conditions for two triangles to be similar.
2. Calculate angle and side measurements in two similar triangles. In addition, the activity will also develop socio-emotional skills such as time management, communication, problem-solving, creative thinking, and proactivity.

Detailed Description of the Activity

Students, organized in groups of 3 to 5, must create a model of an amusement park where the main attractions (ferris wheels, roller coasters, slides) have the shape of similar triangles in different scales. For this, they will have to recognize the sufficient conditions for two triangles to be similar and calculate the measurements of angles and sides accordingly.

Upon completion of the model, students must produce a final report presenting the theory of similarity of triangles, a detailed description of the project, the methodology used for the creation of the model, and the main results obtained.

Required Materials

1. Cardboard or paperboard for the base of the model
2. Wooden sticks or straws for the structure of the triangles
3. Ruler and compass for measurements
4. Glue and scissors
5. Paints and markers for decoration

Steps for Carrying Out the Activity

1. After forming the groups, each one must study the theory of similarity of triangles using the materials suggested in the introduction.
2. With the theory studied, the groups must start planning their model. A sketch of the model should be made, indicating what the attractions (triangles) are and their respective measurements.
3. With the planning in hand, students begin to build the attractions of the model, respecting the planned measurements so that the triangles are similar. At this stage, students must use their calculation and logical reasoning skills to ensure the similarity of the triangles.
4. After building the attractions, students must assemble the model, positioning the attractions on the cardboard or paperboard base.
5. Once the model is completed, the groups must produce the final report, presenting the theory of similarity of triangles, the project description, the methodology used for the creation of the model, and the results obtained.
6. The project presentation and report submission should be done within a week.

Project Deliverables

At the end of the project, each group must deliver:

• The amusement park model with similar triangles.
• The written report containing:
  1. Introduction: Contextualization of the theme, relevance and real-world application, and project objective.
  2. Development: Explanation of the theory of similarity of triangles, detailed description of the activity, methodology used, and presentation of the results obtained.
  3. Conclusion: Recap of the main points, lessons learned, and conclusions drawn from the project.
  4. Bibliography: Indication of the sources used for the project.