Similar figures are geometric shapes that have the same proportions, meaning they have the same shape but may differ in size. They can be scaled up or down without changing their overall appearance, maintaining the same angles and ratios between corresponding sides.
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The ratio of the lengths of corresponding sides in similar figures is constant, known as the scale factor.
Similar figures have the same angles, and the ratios of the lengths of their corresponding sides are equal.
The area of similar figures is proportional to the square of their scale factor, while the volume is proportional to the cube of the scale factor.
Similar figures can be used to solve problems involving proportions, scale drawings, and real-world applications such as map reading and architectural design.
Identifying and working with similar figures is a crucial skill in solving problems related to 8.7 Solve Proportion and Similar Figure Applications.
Review Questions
Explain the relationship between the scale factor and the corresponding side lengths of similar figures.
The scale factor is the ratio of the lengths of corresponding sides in similar figures. If the scale factor is $k$, then the lengths of the corresponding sides in the two figures are in the ratio $k:1$. For example, if the scale factor is 3, then the corresponding sides of the similar figures will have a ratio of 3:1, meaning one figure is three times the size of the other.
Describe how the area and volume of similar figures are related to the scale factor.
The area of similar figures is proportional to the square of the scale factor. If the scale factor is $k$, then the ratio of the areas of the similar figures is $k^2:1$. Similarly, the volume of similar figures is proportional to the cube of the scale factor, so the ratio of the volumes is $k^3:1$. This means that as the size of a figure increases, its area and volume increase at a faster rate.
Explain how the concept of similar figures can be applied to solve problems related to 8.7 Solve Proportion and Similar Figure Applications.
The properties of similar figures, such as the constant ratio of corresponding sides and the relationships between area and volume, can be used to solve a variety of problems in the context of 8.7 Solve Proportion and Similar Figure Applications. These problems may involve scale drawings, map reading, architectural design, and other real-world situations where the proportions of geometric shapes need to be understood and applied. By recognizing and working with similar figures, students can effectively solve problems that require them to set up and solve proportions, as well as apply the principles of similar figures to practical situations.