Expansion and Reduction of Figures | Traditional Summary
Contextualization
The enlargement and reduction of geometric figures are fundamental concepts in mathematics, applicable in various practical situations. When we enlarge a figure, we are proportionally increasing its dimensions, which means multiplying each of its sides by a scale factor greater than 1. Similarly, when we reduce a figure, we are decreasing its dimensions, multiplying the sides by a scale factor less than 1. These processes allow for the creation of models in different sizes without altering the original shape of the figure.
Understanding how enlargement and reduction impact areas and perimeters is essential for solving practical problems. For example, when enlarging a technical drawing of a building, engineers and architects need to correctly adjust the dimensions to ensure that the final structure is safe and functional. Similarly, in everyday activities, such as adjusting the size of a digital image for printing or adapting a recipe for a different number of servings, knowledge about proportionality and scales is extremely useful.
Concept of Proportionality
Proportionality is a mathematical concept that describes the relationship between two quantities that change in a constant manner. In the context of the enlargement and reduction of geometric figures, it is fundamental to understanding how all dimensions of a figure change in the same proportion. For example, if one side of a square is increased by 50%, all the other sides must also be increased by 50% to keep the figure proportional.
When we apply proportionality to the enlargement and reduction of figures, we use a scale factor. This scale factor is a number by which we multiply the dimensions of a figure to obtain a new figure that is proportionally larger or smaller. A scale factor greater than 1 indicates enlargement, while a scale factor less than 1 indicates reduction.
Understanding proportionality and the scale factor is crucial for solving area and perimeter problems of geometric figures. By applying these concepts, we can easily calculate how much a figure will increase or decrease in size while maintaining its original proportions.
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Proportionality involves the constant change between two quantities.
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The scale factor determines whether the figure will be enlarged or reduced.
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The application of proportionality allows for precise calculations of areas and perimeters.
Enlargement of Figures
The enlargement of geometric figures occurs when we multiply the dimensions of a figure by a scale factor greater than 1. This results in a new figure that is proportionally larger than the original. For example, if we enlarge a square with sides measuring 3 cm by a scale factor of 2, each side of the new square will measure 6 cm.
When enlarging a figure, both the area and the perimeter are affected. The perimeter of an enlarged figure is simply the product of the original perimeter by the scale factor. However, the area is proportional to the square of the scale factor. Therefore, if the scale factor is 2, the area of the enlarged figure will be four times the original area.
Understanding enlargement is essential for correctly applying the concepts of proportionality in practical problems. For example, when creating an enlarged map of a region, it is necessary to ensure that all proportions are maintained so that the map is an accurate representation of the actual area.
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Enlargement uses a scale factor greater than 1.
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The enlarged perimeter is the original perimeter multiplied by the scale factor.
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The enlarged area is proportional to the square of the scale factor.
Reduction of Figures
The reduction of geometric figures is the reverse process of enlargement. It occurs when we multiply the dimensions of a figure by a scale factor less than 1. This results in a new figure that is proportionally smaller than the original. For example, if we reduce a square with sides measuring 4 cm by a scale factor of 0.5, each side of the new square will measure 2 cm.
Just like in enlargement, both the area and the perimeter of a reduced figure are affected. The perimeter of the reduced figure is the product of the original perimeter by the scale factor. The area, in turn, is proportional to the square of the scale factor. Therefore, if the scale factor is 0.5, the area of the resulting figure will be a quarter of the original area.
These concepts are important for solving practical problems that involve the reduction of figures, such as adjusting the size of a digital image to fit a certain space without distorting its proportions.
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Reduction uses a scale factor less than 1.
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The reduced perimeter is the original perimeter multiplied by the scale factor.
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The reduced area is proportional to the square of the scale factor.
Calculation of Areas and Perimeters
Calculating areas and perimeters is an essential skill for understanding how enlargement and reduction affect geometric figures. To calculate the area of a square, we use the formula side x side, while for a rectangle, we use base x height. These calculations allow us to determine how much space a figure occupies before and after enlargement or reduction.
To calculate the perimeter, we add all the sides of the figure. In the case of a square, this is simply 4 times the length of one side. For a rectangle, we add two times the base plus two times the height. These calculations are straightforward, but it is important to remember that the perimeter changes linearly with the scale factor, while the area changes quadratically.
Understanding how to calculate areas and perimeters after enlargement or reduction is crucial for solving practical problems. For example, when increasing the size of a sports field, it is essential to know how the total area will be affected to appropriately plan the use of space.
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The area of a square is calculated as side x side.
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The area of a rectangle is calculated as base x height.
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The perimeter is the sum of all sides of the figure.
To Remember
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Proportionality: Relationship between quantities that change in a constant manner.
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Scale Factor: Number by which we multiply the dimensions of a figure to enlarge or reduce it.
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Enlargement: Process of increasing the dimensions of a figure by a scale factor greater than 1.
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Reduction: Process of decreasing the dimensions of a figure by a scale factor less than 1.
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Area: Measure of the surface of a geometric figure, calculated in square units.
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Perimeter: Sum of all sides of a geometric figure.
Conclusion
In this lesson, we explored the fundamental concepts of enlargement and reduction of geometric figures, understanding how proportionality affects the dimensions of figures. We learned to calculate areas and perimeters after applying scale factors, highlighting that the area changes quadratically while the perimeter changes linearly. This knowledge is essential for solving practical and theoretical problems in various fields, such as architecture, engineering, and even everyday activities like printing images and adapting recipes.
Understanding proportionality and the scale factor allows us to create similar figures in different sizes without distorting their proportions. Knowing how to accurately calculate the new areas and perimeters is crucial to ensuring precision in projects and activities that involve dimensional changes. Additionally, applying these mathematical concepts prepares us to face practical challenges efficiently and safely.
Finally, we emphasize the importance of continuing to explore the subject, as the enlargement and reduction of geometric figures are widely applicable skills relevant in various practical situations. We encourage students to deepen their knowledge and practice the calculations presented to become more confident and proficient in applying these concepts.
Study Tips
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Review the practical examples discussed in class and try to solve additional problems involving the enlargement and reduction of geometric figures.
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Use visual resources, such as drawings and models, to better visualize how proportionality affects the dimensions of figures.
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Practice calculations of areas and perimeters with different scale factors to reinforce understanding of the concepts and gain confidence in problem-solving.
