Introduction

Mathematics is a fundamental language with which we can describe the world around us. One of the most basic yet profound ideas in this language is the concept of similarity. The similarity of triangles is one of the basic concepts that we will study in this project.

Triangles are one of the simplest and most studied shapes in geometry, and similarity is a fundamental concept that involves these shapes. The similarity of triangles refers to two or more triangles that have the same shape but not necessarily the same size. That is, the angles of the triangles are equal and the sides are proportional.

The similarity of triangles has several important practical applications, from civil engineering, in the construction of bridges and buildings, to photography and art, in the creation of perspectives and realistic proportions. For example, engineers use the similarity of triangles to calculate inaccessible distances, such as the height of a building or the width of a river, by measuring only more easily accessible distances and applying the properties of similar triangles.

Moreover, there are aspects of our daily lives that use the principle of the similarity of triangles, even without us realizing it. For example, when taking a selfie, we intuitively adjust the distance and angle of the cell phone to obtain the best image, similar to what a portrait artist does when painting.

To aid in the development of the project and in-depth study of the theme, we suggest the following sources:

• Book: Fundamentals of Elementary Mathematics: Plane Geometry - Volume 9, by Gelson Iezzi and Carlos Murakami (ISBN: 9788526277773)
• Website: Khan Academy (https://www.khanacademy.org/math/geometry/hs-geo-similarity)
• Video: Similarity of Triangles - O Kuadro (https://www.youtube.com/watch?v=46QH_m1zvBQ)
• Website: Só Matemática (https://www.somatematica.com.br/emedio/triangulos/)

Practical Activity - "Exploring the World with Similar Triangles"

Objective of the project

In this project, students will have the opportunity to apply the concept of the similarity of triangles to solve real-world problems. The aim is for students to develop logical thinking, communication, teamwork and proactivity while reinforcing the learning of mathematical concepts.

Project description

Students will be divided into groups of 3 to 5 people. Each group will be responsible for applying the concept of the similarity of triangles to solve real-world problems. With the aid of a ruler, a compass and prior knowledge about the similarity of triangles, the groups will measure and record their observations and later prepare a report.

Required materials

1. Graph paper
2. Pencil and eraser
3. Ruler and compass
4. Calculator
5. Camera/cell phone to record the activities

Step by step

Step 1: Choose an object of interest in your home or at school that is difficult to measure directly (for example, the height of a tree, a tall bookshelf, the distance between two distant points).

Step 2: Using a ruler, measure a fixed distance on the ground from the object of interest (for example, 5 meters). From this point, use the compass to form an imaginary triangle with the object of interest.

Step 3: Choose a smaller object, which is possible to measure directly, to compare it with the object of interest. Form a triangle similar to the previous one with the new object.

Step 4: Record the measurements of the sides of the triangles formed and calculate the corresponding proportions.

Step 5: Applying the concept of the similarity of triangles, calculate the height or distance of the object of interest.

Step 6: Document the entire process with photos and detailed notes.

Step 7: After completing the practical part of the project, students should write a report.

Deliverables and Connections

Students should submit a written and organized report containing the following sections:

1. Introduction: Explain the relevance and application of the similarity of triangles in the real world, the objective of the project, and provide a brief description of the objects chosen.
2. Development: Set out the theory of the similarity of triangles and discuss how it was applied in the practical activity. Detail the step by step process, including the description of the measurements and calculations performed and the results obtained. Include the photos taken.
3. Conclusion: Return to the main points of the project, summarize the learning obtained and the conclusions about the application of the similarity of triangles in the problem proposed.
4. Bibliography: Indicate the sources used for the development of the project.

This project requires students to apply the concept of the similarity of triangles to solve practical problems and explain in their own words how the concept is applied. It also requires students to cooperate and communicate effectively within the group to organize the activities, divide tasks, summarize information and write the report.