Exploring Geometry: Enlarging and Reducing Figures

Objectives

1. Understand how the enlargement and reduction of geometric figures affect their metric properties, such as area and perimeter.

2. Calculate values for areas and perimeters of enlarged and reduced figures.

3. Develop practical and applicable skills in real contexts, such as design, architecture, and engineering.

Contextualization

Imagine you are helping to plan an amusement park. The rides, food areas, and pathways between them need to be designed so that all spaces are used efficiently. To achieve this, it is essential to understand how the enlargement and reduction of geometric figures can affect the available space. This is a practical example of how mathematics and geometry are used in everyday life to solve real problems.

Relevance of the Theme

Understanding enlargement and reduction of geometric figures is fundamental in various professions, such as architecture, engineering, and graphic design. These skills are essential for creating models, building plans, calculating areas and perimeters of lands and structures, ensuring that projects are executed correctly and efficiently. In the current context, where efficiency and precision are highly valued, mastering these mathematical concepts is crucial for professional success.

Enlargement of Geometric Figures

The enlargement of geometric figures involves creating a new figure, proportional to the original, but with dimensions increased by a scale factor. This process is fundamental to understanding how metric properties, such as area and perimeter, are affected by the variation in dimensions.

• The enlarged figure retains the shape of the original figure, but its dimensions are multiplied by a scale factor.

• The perimeter of the enlarged figure is directly proportional to the scale factor applied to the sides of the original figure.

• The area of the enlarged figure is proportional to the square of the scale factor applied to the sides of the original figure.

Reduction of Geometric Figures

The reduction of geometric figures involves creating a new figure, proportional to the original, but with dimensions decreased by a scale factor. This process also illustrates how metric properties are altered as the dimensions of the figure are reduced.

• The reduced figure retains the shape of the original figure, but its dimensions are divided by a scale factor.

• The perimeter of the reduced figure is directly proportional to the scale factor applied to the sides of the original figure.

• The area of the reduced figure is proportional to the square of the scale factor applied to the sides of the original figure.

Calculation of Perimeter and Area in Enlarged and Reduced Figures

When enlarging or reducing geometric figures, calculating the perimeter and area of the new figures is essential to understand how these metric properties are affected by changes in dimensions.

• To calculate the perimeter of an enlarged or reduced figure, multiply the original perimeter by the scale factor.

• To calculate the area of an enlarged or reduced figure, multiply the original area by the square of the scale factor.

• Understanding these relationships is crucial for applying the concepts of enlargement and reduction in practical problems, such as project design and model building.

Practical Applications

• Architecture: Architects use enlargement and reduction of figures when creating plans and models of buildings at different scales, ensuring that projects are executed accurately.
• Civil Engineering: Civil engineers calculate areas and perimeters of lands and structures to plan constructions, using techniques of enlargement and reduction to work with different scales.
• Graphic Design: Graphic designers enlarge and reduce images to fit appropriately across various types of media, from business cards to banners and billboards.

Key Terms

• Enlargement: The process of increasing the dimensions of a geometric figure proportionally.

• Reduction: The process of decreasing the dimensions of a geometric figure proportionally.

• Scale Factor: The number by which the dimensions of a figure are multiplied or divided to enlarge or reduce the figure.

• Perimeter: The sum of the measures of all sides of a geometric figure.

• Area: The measure of the surface of a geometric figure, expressed in square units.

Questions

• How can the enlargement and reduction of geometric figures impact the amount of material needed in a construction project?

• In what ways can understanding scales help in professions such as architecture and engineering?

• What challenges have you encountered when calculating the new dimensions of enlarged or reduced figures during the practical activity?

Conclusion

To Reflect

Throughout this lesson, we explored how the enlargement and reduction of geometric figures impact their metric properties, such as area and perimeter. Understanding these relationships is essential not only for mathematical theory but also for practical applications in various professions in the job market. Architects, engineers, and graphic designers, for example, use these concepts to create accurate and efficient projects. We hope that the practical activities have helped to consolidate your understanding and appreciate the importance of this knowledge in real situations.

Mini Challenge - Geometric Construction Challenge

Let's put into practice what we learned about enlarging and reducing geometric figures!

• Choose a simple geometric figure (square, rectangle, triangle, or circle).
• Define a scale factor to enlarge or reduce your original figure (for example, 2:1 for enlargement or 1:2 for reduction).
• Calculate the new dimensions of the figure, including perimeter and area.
• Draw the original figure and the enlarged/reduced figure on a piece of paper.
• Compare the metric properties of the two figures and write a short paragraph explaining how the enlargement or reduction affected the perimeter and area.
• Share your findings with your classmates or with the teacher.