Contextualization

The similarity of triangles is a central concept in geometry, allowing us to better understand the world around us. When we say that two triangles are similar, we are saying that they have the same shape, even though they may have different sizes. This happens when the corresponding angles of the two triangles are equal and the corresponding sides are proportional.

This concept is fundamental for us to understand various phenomena in our world. For example, when we look at a mountain in the distance, we are actually seeing a reduced version of it. The proportions of the mountain are the same, but the distance makes us see it as if it were smaller. This is a practical application of the concept of triangle similarity.

Triangle similarity is also an essential topic in mathematics, as it allows you to solve a variety of problems using geometric and algebraic principles. Often, we can simplify a complex problem by transforming it into a triangle similarity problem.

Introduction

This project aims to deepen your understanding of triangle similarity. We will explore the similarity criteria (angle-angle (AA), side-angle-side (SAS), and side-side-side (SSS)), and how to apply them to solve problems.

We will also learn how to calculate angle and side measurements in similar triangles. This may seem complicated, but you will see that once you understand the basic principles, you will be able to solve these problems with ease.

Finally, the project will challenge you to apply these concepts in a practical and collaborative activity. Through it, we hope that you will not only solidify your understanding of triangle similarity, but also develop teamwork, time management, and problem-solving skills.

Practical Activity: "The Art of Similarity"

Project Objective

The purpose of this project is to use triangle similarity to "reconstruct" a famous work of art. Students will choose a work of art by a famous painter and, through the application of similarity criteria, will "deconstruct" the work into a series of similar triangles and then "reconstruct" it. In addition to applying the concepts learned from triangle similarity, this project also aims to integrate Mathematics and Art, seemingly distinct areas, but which connect in various ways.

Detailed Project Description

The groups should first choose a famous work of art. This work will be the basis for the entire project. Each group will create a "triangular version" of the chosen work of art. To do this, students must divide the work into triangles and, for each identified triangle, apply the concepts of similarity, calculating measurements and proportions.

The final work should be an image of the original work of art and the group's "triangular version," side by side, demonstrating how the similarity criteria were used to "reconstruct" the work.

Students will also be stimulating other skills such as teamwork, problem-solving, creative thinking, as well as exercising their artistic sensitivity and appreciation.

The project is intended for groups of 3 to 5 students and it is estimated that the execution will take around twelve hours per student, consisting of research, study of concepts, deconstruction and reconstruction of the work of art, and report writing.

Required Materials

• Computer with internet access for research.
• Image editing software (such as Paint, GIMP, Photoshop, or similar).
• Paper and pencil for notes and calculations.
• Ruler, compass, and protractor for measuring angles and drawing lines.

Step by Step

1. The groups must choose a work of art to be the object of study. It is important to choose an image that can be easily "deconstructed" into triangles.

2. With the chosen work of art, students should research about the artist and the historical context in which the work was created. This way, students will be stimulating research skills and acquiring cultural and historical knowledge.

3. Using an image editing program, students should select the work and draw triangles on the image, trying to maintain as much as possible of the original format of the work.

4. With the defined triangles, students must, for each pair of similar triangles, calculate measurements and proportions. Students should apply the AA (angle-angle), SAS (side-angle-side), and SSS (side-side-side) criteria to verify the similarity.

5. After the calculations, students must "reconstruct" the work of art from the similar triangles, building a second image next to the first one.

6. Finally, the group should prepare a descriptive report on the process. In the report, the group should contextualize the chosen work of art, explain how the deconstruction and reconstruction of the work was carried out applying the concepts of triangle similarity, demonstrating through calculations and proportions. The report should be divided into four main sections: Introduction, Development, Conclusions, and Bibliography used.

Project Deliverables

• Image of the original work of art and the group's created "triangular version," side by side.
• Detailed report containing: Introduction (information about the chosen work of art and the reason for the choice), Development (detailed description of the process of "deconstruction" and "reconstruction" of the work using triangle similarity, with calculations and results demonstrating the application of similarity criteria), Conclusions (discussion on what was learned from the project, highlighting connections between geometry and art and the importance of teamwork), and Bibliography used.