Lesson Plan | Traditional Methodology | Bisector and Perpendicular Bisector

Objectives

Duration: 10 - 15 minutes

The purpose of this stage is to establish a clear and solid foundation for the students, presenting the fundamental concepts of bisector and perpendicular bisector. This will enable students to understand the importance of these geometric elements and how they apply in various contexts. By the end of this stage, students should be prepared to identify and use these concepts in geometric problems, facilitating comprehension in later stages of the lesson.

Main Objectives

1. Understand the concept of bisector and perpendicular bisector as geometric places.

2. Identify the perpendicular bisector and the bisector in different geometric figures.

3. Understand the practical application of the bisector and the perpendicular bisector in geometric problems.

Introduction

Duration: 10 - 15 minutes

 Purpose: The purpose of this stage is to establish a clear and solid foundation for the students, presenting the fundamental concepts of bisector and perpendicular bisector. This will enable students to understand the importance of these geometric elements and how they apply in various contexts. By the end of this stage, students should be prepared to identify and use these concepts in geometric problems, facilitating comprehension in later stages of the lesson.

Context

 Context: Start the lesson by presenting to the students the importance of lines and segments in geometry. Explain that when studying geometric figures, we need to understand how certain lines can divide or relate parts of these figures. Introduce the concepts of bisector and perpendicular bisector as specific types of lines that play crucial roles in geometry. Mention that today we will explore these lines in detail, understanding their properties and how to identify them in different geometric figures.

Curiosities

類 Curiosity: Did you know that the bisector is used in architecture and design to create perfect symmetries in buildings and objects? For example, when drawing the floor plan of a house, architects often use the bisector to ensure that the angles are equally divided, creating balanced and aesthetically pleasing structures. The perpendicular bisector is fundamental in civil engineering for locating equidistant points between two places, such as when building bridges or highways.

Development

Duration: 40 - 45 minutes

 Purpose: The purpose of this stage is to provide students with a detailed understanding of the concepts of bisector and perpendicular bisector, their definitions and properties. By the end of this stage, students should be able to identify and apply these concepts in different geometric figures and solve related problems, using the knowledge acquired practically.

Covered Topics

1.  Definition of Bisector: Explain that the bisector of an angle is the half-line that divides the angle into two equal parts. Highlight that the bisector is a geometric place of points equidistant from the sides of the angle. Show practical examples of bisectors, such as in triangles and other polygons.

🤓 **Properties of the Bisector:**
Detail that the bisector of an angle in a triangle divides the opposite side into segments proportional to the adjacent sides.
Explain the Internal Bisector Theorem and how it can be used to solve geometric problems.

📏 **Definition of Perpendicular Bisector:**
Explain that the perpendicular bisector of a segment is the line perpendicular to the segment and passing through the midpoint.
Highlight that the perpendicular bisector is a geometric place of points equidistant from the ends of the segment.
Show practical examples of perpendicular bisectors, such as in squares and other polygons.

🤓 **Properties of the Perpendicular Bisector:**
Detail that the perpendicular bisector of a segment in a triangle is the line that locates the point equidistant from the vertices of the triangle.
Explain the Perpendicular Bisector Theorem and how it can be used to solve geometric problems.

Classroom Questions

1. What is the location of the bisector of a 90-degree angle in an isosceles triangle? Explain your answer. 2. Given a segment AB, how can you find the perpendicular bisector of this segment? Describe the process and explain the importance of the perpendicular bisector. 3. In triangle ABC, the bisector of angle A meets side BC at D. If AB = 6 cm, AC = 8 cm, and BC = 10 cm, what are the lengths of BD and DC? Use the Bisector Theorem.

Questions Discussion

Duration: 20 - 25 minutes

 Purpose: The purpose of this stage is to consolidate the knowledge acquired by the students during the lesson, providing a moment of reflection and discussion. During this time, students will have the opportunity to clarify doubts, reinforce concepts, and apply what they have learned in new contexts. Additionally, this stage aims to encourage active participation and critical thinking among students, fostering a collaborative and enriching learning environment.

Discussion

• What is the location of the bisector of a 90-degree angle in an isosceles triangle? Explain your answer. The bisector of a 90-degree angle in an isosceles triangle divides the angle into two 45-degree angles. In an isosceles triangle, the two sides adjacent to the 90-degree angle are equal, and the bisector of this angle will be perpendicular to the base of the triangle, dividing it into two congruent right triangles.

Given a segment AB, how can you find the perpendicular bisector of this segment? Describe the process and explain the importance of the perpendicular bisector. To find the perpendicular bisector of segment AB, follow these steps: 1. Find the midpoint of AB, which is the point equidistant from A and B. 2. Draw a perpendicular line to segment AB passing through the midpoint. This line is the perpendicular bisector. The perpendicular bisector is important because it is the geometric place of the points equidistant from the ends of segment AB. This means that any point on the perpendicular bisector will be the same distance from A and B.

In triangle ABC, the bisector of angle A meets side BC at D. If AB = 6 cm, AC = 8 cm, and BC = 10 cm, what are the lengths of BD and DC? Use the Bisector Theorem. By the Internal Bisector Theorem, we know that: (\frac{AB}{AC} = \frac{BD}{DC}) Substituting the given values: (\frac{6}{8} = \frac{BD}{DC}) Simplifying, we have: (\frac{3}{4} = \frac{BD}{DC}) This means that BD is 3 parts and DC is 4 parts of a total of 7 parts of segment BC, which is 10 cm. Calculating the lengths: BD = (\frac{3}{7} \times 10) = 4.29 cm (approximately) DC = (\frac{4}{7} \times 10) = 5.71 cm (approximately)

Student Engagement

1. Can you identify the bisector and the perpendicular bisector in other geometric figures besides triangles? Give examples. 2. How would you apply the concept of perpendicular bisector to solve everyday problems, such as finding a point equidistant between two cities? 3. Discuss how the bisector can be used in construction to ensure accuracy and symmetry in architectural projects. 4. What other practical applications of the perpendicular bisector can you imagine in engineering and design? 5. If the bisector divides an angle into two equal parts, what would be the impact in terms of symmetry in reflected images?

Conclusion

Duration: 10 - 15 minutes

The purpose of this stage is to recap and consolidate the main points covered in the lesson, reinforcing the students' understanding. Moreover, it connects theory with practice, demonstrating the relevance of the concepts learned in everyday and professional situations, and encouraging the practical application of the knowledge acquired.

Summary

• The bisector of an angle is the half-line that divides the angle into two equal parts.
• The bisector is a geometric place of points equidistant from the sides of the angle.
• The perpendicular bisector of a segment is the line perpendicular to the segment and passing through the midpoint.
• The perpendicular bisector is a geometric place of points equidistant from the ends of the segment.
• The bisector of an angle in a triangle divides the opposite side into segments proportional to the adjacent sides.
• The perpendicular bisector of a segment in a triangle locates the point equidistant from the vertices of the triangle.

During the lesson, the theoretical concepts of bisector and perpendicular bisector were connected to practice through clear examples and geometric problems. Practical cases in geometric figures, such as triangles and squares, were presented, and their applications in fields such as architecture and civil engineering were discussed, reinforcing the importance of these concepts in the real world.

Understanding the bisector and the perpendicular bisector is fundamental not only to solving geometric problems but also to various practical applications. For example, the bisector is used in civil construction to ensure precision and symmetry, while the perpendicular bisector is crucial in engineering for locating equidistant points, such as in planning bridges and highways. This knowledge helps to better understand the geometry of the world around us.