Context

Introduction

The field of geometry is one of the oldest in mathematics, with its origins and applications dating back thousands of years. However, even with all this time and numerous advancements, geometry still holds concepts and theories that are vital to understanding modern mathematics. One such concept is Cevians in Triangles.

Cevians are line segments that connect a vertex of a triangle to any point on the opposite side. As simple as they may seem, Cevians have fascinating properties and are the basis for deep theorems in Geometry.

There are three main types of Cevians: Medians, Bisectors, and Altitudes. Each of these types of Cevians has unique and distinct properties, and together they form a rich and complex landscape that reveals the beauty and depth of geometry.

Real-World Context

Cevians have practical applications in the real world, especially in Engineering and Architecture where understanding the properties of triangles is crucial. For instance, the angle bisector (a type of Cevian) is often used in architectural planning to create equal divisions of space.

Further, Cevians also find use in various areas of Mathematics, such as Optimization, a field of Mathematics that seeks the best way to do a task. For example, the Fermat Point, one of the notable points in a triangle, is found using Cevians and their properties and plays an important role in solving the ''Postman's Problem'', a famous optimization problem.

Suffice it to say, mastering Cevians and their properties opens doors to a better understanding of the world and the structures that form it, from the buildings we live in to the most complex problems in Mathematics.

Hands-on Activity

Activity Title: Discovering Cevians

Project Goal

The goal of this project is to deeply and practically understand the concept of Cevians, their types, and properties by constructing physical models of triangles and analyzing their Cevians, culminating in a well-structured project report.

Detailed Project Description

This project will be carried out in groups of 3 to 5 students and has an estimated time of completion of 2 to 4 hours per student, over the course of a week.

The students will use simple materials like paper, pencils, rulers, compasses, and colored pens to construct triangles and draw their Cevians. Additionally, they will need books, articles, and videos to study the topic further and write a comprehensive report.

Required Materials

• A3 size paper
• Pencils
• Ruler
• Compass
• Colored pens
• Internet access for research

Detailed Step-by-Step Procedure for Activity

Step 1: Students will study about Cevians, their types, and properties using the suggested resources and any others of their choice.

Step 2: They will draw, on an A3 sheet of paper, a triangle and all the Cevians they can identify. Each type of Cevian should be drawn in a different color for easier analysis later.

Step 3: The students will carry out an analysis of the Cevians drawn, identifying and explaining their properties and related notable points.

Step 4: The group will prepare a detailed report with the following structure:

• Introduction: Contextualization of the topic, its relevance and real-world applications, as well as the objective of this project.
• Development: Explanation of the theory studied on Cevians, the hands-on activity in detail, the methodology employed, and presentation and discussion of the results obtained. Images from the hands-on work can be attached in this section.
• Conclusion: The group will make their final considerations, highlighting the learnings, conclusions drawn from the project, and any difficulties they faced during its execution.
• Bibliography: List of research sources used to substantiate the project.

This report will be the final deliverable of the project, and teachers will evaluate the project based on it along with the students' critical analysis of their learnings and the results obtained.