Introduction

Relevance of the Theme

The bisector and the perpendicular bisector are two fundamental concepts in Euclidean geometry. They are the basis for understanding symmetry and equidistance, respectively. Working with these tools not only expands the student's spatial perception but also develops crucial skills in analysis and logical reasoning. The study of these concepts lays the groundwork for more advanced concepts such as congruence, similarity of figures, and conics.

Contextualization

In the Mathematics curriculum of the 8th grade of Elementary School, the study of bisectors and perpendicular bisectors fits into the Flat Geometry Unit. This unit is based on basic geometric concepts and their applications, including understanding properties and relationships between lines, angles, triangles, and quadrilaterals. The bisector and the perpendicular bisector are natural extensions of these studies and provide a more complete view of space and geometric relationships.

Theoretical Development

Components

1. Bisector: A bisector of an angle is a line, line segment, or ray that divides the angle into two equal parts. The bisector is analogous to an equal divider in an angle. Regardless of the size of the angle, if a line, line segment, or ray divides it exactly in half, we call this line a bisector.

   • Triangle Bisector: Given a triangle, the bisector is the line that divides one of the triangle's angles into two equal parts. In the case of an equilateral triangle, where all sides and all angles are equal, the bisectors of all angles coincide at the same point inside the triangle, called the incenter.

2. Perpendicular Bisector: The perpendicular bisector is a line, line segment, or ray that is perpendicular to the line segment and passes through its midpoint. The concept of the perpendicular bisector is mainly used to investigate and describe the congruence of the sides of a triangle.

   • Segment Perpendicular Bisector: Given a segment, the perpendicular bisector is the line, line segment, or ray that is perpendicular to the segment and passes through its midpoint. In the case of a triangle, the perpendicular bisectors of all sides coincide at a single point, called the circumcenter of the triangle.

Key Terms

• Bisector: Derived from the French verb "bissester," which means to divide in two. In mathematics, it is the tool that divides an angle exactly in half.
• Perpendicular Bisector: In mathematics, "perpendicular bisector" comes from the Latin "mediatrix," which means "the one that is in the middle." In geometry, it is used to describe a line that is perpendicular to a line segment and passes through its midpoint.

Examples and Cases

1. Triangle Bisector: In triangle ABC, angle BAC is divided into two equal parts by the bisector AD. Regardless of the size of angle BAC, the bisector always divides it in half. Important: The bisectors of a triangle are concurrent at a point called the incenter.

2. Segment Perpendicular Bisector: In line segment DE, the perpendicular bisector is the line that is perpendicular to DE and passes through its midpoint F. The perpendicular bisector divides DE into two congruent segments, DF and FE. Important: The perpendicular bisectors of a triangle are concurrent at a point called the circumcenter.

Detailed Summary

Key Points

• Importance of Bisectors and Perpendicular Bisectors: Bisectors and perpendicular bisectors are key concepts in geometry, providing a deeper understanding of symmetry, equidistance, and their applications. By dividing angles and line segments equally, respectively, they form the basis for more advanced concepts such as congruence and similarity of figures.

• Definition of Bisector: A bisector of an angle is a line, line segment, or ray that divides the angle into two equal parts. The bisector is analogous to an equal divider in an angle.

• Definition of Perpendicular Bisector: The perpendicular bisector is a line, line segment, or ray that is perpendicular to the line segment and passes through its midpoint. This definition is fundamental for understanding the equidistance provided by perpendicular bisectors.

• Concurrent Bisectors and Perpendicular Bisectors: In the context of a triangle, the bisectors meet at a point called the incenter, while the perpendicular bisectors meet at a point called the circumcenter. These points of concurrency are essential for a complete understanding of the bisector and perpendicular bisector.

• Vocabulary: It is important to be familiar with the vocabulary associated with bisectors and perpendicular bisectors, including the terms "bisector" and "perpendicular bisector" themselves, and the terms "incenter" and "circumcenter," which refer to the points of concurrency of bisectors and perpendicular bisectors, respectively.

Conclusions

• Spatial Understanding and Logical Reasoning: The study of bisectors and perpendicular bisectors promotes a better understanding of space and how geometric elements relate to each other. Additionally, it develops logical reasoning by exploring the properties and applications of these special lines.

• Practical Applications: Bisectors and perpendicular bisectors have a variety of practical applications, from architectural projects seeking symmetry to the digital gaming industry that uses equidistance as a basis for building scenarios and characters.

• Continuity in Study: The study of bisectors and perpendicular bisectors paves the way for more advanced concepts, such as congruence of figures and conics, allowing for continuous progress in the study of geometry and its various fields of application.

Suggested Exercises

1. Draw any triangle ABC. Draw the three bisectors. Then, verify if all bisectors meet at a single point. This point is called the incenter.

2. Take any square LMNO. Find the midpoint of each side of the square. Draw the perpendicular bisectors from each midpoint. Verify if the perpendicular bisectors meet at a single point. This point is called the circumcenter.

3. Given angle BAC = 120°, draw the angle bisector and identify the point where the bisector intersects side AC.