Lesson Plan | Traditional Methodology | Right Triangle: Metric Relationships

Objectives

Duration: 10 - 15 minutes

The purpose of this stage is to provide students with a clear understanding of the learning objectives of the lesson, detailing the specific skills they will acquire. This helps focus the lesson and ensures that students know what to expect and what is expected of them.

Main Objectives

1. Describe the metric relationships in a right triangle, stemming from similarity.

2. Calculate the legs, the hypotenuse, and the height relative to the hypotenuse in a right triangle using the metric relationships.

Introduction

Duration: 10 - 15 minutes

The purpose of this stage is to capture students' interest and contextualize the lesson topic, showing the relevance and practical application of metric relationships in right triangles. This helps create a connection between theoretical content and the real world, making learning more meaningful and motivating for students.

Context

To begin the lesson, explain that geometry is a fundamental part of mathematics that applies to various fields of knowledge and everyday life. Give examples of how this area of mathematics is essential for engineering, architecture, design, and even in sports. For instance, the construction of bridges and tall buildings relies on precise calculations of angles and measurements, many of which are based on the properties of right triangles. This demonstrates to students the practical importance of what they are learning.

Curiosities

Did you know that the famous Pythagorean theorem, which is one of the most well-known metric relationships in a right triangle, was used by ancient civilizations like the Egyptians and Babylonians? They utilized these relationships to build pyramids and temples with surprising precision, long before we had modern technology. Additionally, in many cultures, the right triangle has been associated with mystical and symbolic properties.

Development

Duration: 50 - 60 minutes

The purpose of this stage is to deepen students' knowledge about metric relationships in right triangles, using an expository and practical approach. By explaining theoretical concepts and solving detailed examples, the teacher helps students understand and apply the metric formulas in different situations. Solving questions in class allows students to practice and reinforce what they have learned, ensuring a solid understanding of the content.

Covered Topics

1. Definition and Properties of a Right Triangle: Explain that a right triangle is one that has a right angle (90 degrees). Detail the basic properties, such as the existence of two legs and one hypotenuse. 2. Pythagorean Theorem: Present the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. Formula: c² = a² + b². 3. Metric Relationships in the Right Triangle: Detail the main metric relationships that can be derived from the similarity of triangles formed by the height relative to the hypotenuse. Explain the formulas: Height relative to the hypotenuse: h² = m * n; Larger leg: b² = a * m; Smaller leg: a² = b * n; Product of the legs: a * b = c * h. 4. Practical Examples: Solve practical examples using the metric relationships. Show how to calculate the hypotenuse, the legs, and the height relative to the hypotenuse from the metric relationships. Give clear and detailed numerical examples so that students can follow along and understand the resolution process.

Classroom Questions

1. In a right triangle, the hypotenuse measures 13 cm, and one of the legs measures 5 cm. What is the measure of the other leg? 2. In triangle ABC, the height relative to the hypotenuse measures 6 cm, and the hypotenuse measures 10 cm. Determine the segments m and n on the hypotenuse. 3. If the legs of a right triangle measure 8 cm and 15 cm, calculate the height relative to the hypotenuse.

Questions Discussion

Duration: 20 - 25 minutes

The purpose of this stage is to review and reinforce the content learned, ensuring the understanding of metric relationships in right triangles. Through the discussion of the resolved questions, the teacher can identify and clarify any potential doubts, providing a collaborative environment where students can share their strategies and reflections, enriching collective learning.

Discussion

• Question 1: In a right triangle, the hypotenuse measures 13 cm, and one of the legs measures 5 cm. What is the measure of the other leg? To solve this question, apply the Pythagorean Theorem: c² = a² + b². Substitute the known values: 13² = 5² + b². Then, 169 = 25 + b². Subtract 25 from both sides: 169 - 25 = b², resulting in 144 = b². Therefore, the value of b is the square root of 144, which is 12 cm.

• Question 2: In triangle ABC, the height relative to the hypotenuse measures 6 cm, and the hypotenuse measures 10 cm. Determine the segments m and n on the hypotenuse. We know that h² = m * n, where h is the height, and m and n are the segments of the hypotenuse. Substitute the known values: 6² = m * n, resulting in 36 = m * n. We also know that m + n = 10, as the sum of the segments equals the hypotenuse. We now have a system of equations: 1) m * n = 36; 2) m + n = 10. Solving this system, we find that m = 4 and n = 6 or vice versa.

• Question 3: If the legs of a right triangle measure 8 cm and 15 cm, calculate the height relative to the hypotenuse. First, find the hypotenuse using the Pythagorean Theorem: c² = a² + b². Substitute the known values: c² = 8² + 15². Thus, c² = 64 + 225, resulting in c² = 289. Therefore, the hypotenuse c is the square root of 289, which is 17 cm. Now, use the height metric relationship: h² = (a * b) / c. Substitute the values: h² = (8 * 15) / 17, resulting in h² = 120 / 17. Therefore, h = √(120 / 17), which is approximately 2.65 cm.

Student Engagement

1. What difficulties did you encounter when solving the presented questions? 2. How do metric relationships help solve practical problems in geometry? 3. Can you think of other real-world examples where these metric relationships can be applied? 4. What is the importance of understanding metric relationships in a right triangle for other fields of knowledge, such as physics and engineering? 5. Does anyone want to share a different strategy used to solve one of the questions?

Conclusion

Duration: 10 - 15 minutes

The purpose of this stage is to review and consolidate the content learned, ensuring that students have a clear and summarized view of the key points discussed during the lesson. It also aims to reinforce the connection between theory and practice, highlighting the importance and application of metric relationships in right triangles in the real world.

Summary

• Definition and properties of a right triangle.
• Pythagorean theorem and its formula c² = a² + b².
• Metric relationships in a right triangle: h² = m * n, b² = a * m, a² = b * n and a * b = c * h.
• Practical examples of calculating the hypotenuse, legs, and height relative to the hypotenuse.
• Problem-solving using metric relationships.

The lesson connected theory with practice by presenting the metric relationships in right triangles and solving detailed practical examples. This allowed students to see how theoretical formulas can be applied to solve real problems, reinforcing their understanding of the concepts covered.

Understanding the metric relationships in right triangles is essential not only for mathematics but also for various fields of knowledge such as physics, engineering, and architecture. Additionally, these relationships are applied in everyday situations, such as in construction, object design, and even in trajectory analysis, demonstrating the practical relevance of the content.