There are many ways to define similarity. some definitions allow congruent figures to also be similar, while others do not. Examine the following definitions for similar figures.
1. figures that are the same shape but not necessarily the same size
2. figures that are the result of a similarity transformation (including rigid transformations)
Notice that both of these definitions allow congruent figures to also be called similar.
1. Write your own definition for similarity that does not allow congruent figures to also be called similar.
2. How does this definition for similarity relate to rigid transformations, dilations, and scale factors?
a. Explain whether rigid transformations would be considered similarity transformations.
b. Explain what dilations and scale factors would be considered similarity transformations.
1. Definition: "Figures that have the same shape but are not necessarily the same size, and cannot be congruent."
2a. Rigid transformations: Rigid transformations (translations, rotations, reflections) preserve size and shape, but do not change the size of the figure. Since the definition requires that similar figures should not be congruent, rigid transformations cannot be considered similarity transformations under this definition.
2b. Dilations and scale factors: Dilations involve enlarging or shrinking a figure while maintaining its shape. Scale factors determine how much larger or smaller the figure becomes. Dilations, along with scale factors, would be considered similarity transformations under this definition, as they do change the size of the figure while preserving its shape.
1. A definition for similarity that does not allow congruent figures to also be called similar could be: "Figures that have the same shape but different sizes."
2. a. Rigid transformations, such as translations, reflections, and rotations, preserve the shape and size of figures. Therefore, they would not be considered similarity transformations under this definition, as similarity requires the figures to have different sizes.
b. Dilations and scale factors, on the other hand, would be considered similarity transformations. A dilation is a transformation that changes the size of a figure but preserves its shape. It can either enlarge or reduce the figure. The scale factor determines the ratio of the sizes of the original figure to the enlarged or reduced figure. Similarity involves the concept of proportional relationships, where the corresponding sides of the figures have the same ratio (given by the scale factor), thus maintaining the same shape. Therefore, dilations and scale factors are considered similarity transformations.
1. A definition for similarity that does not allow congruent figures to be called similar could be: "Figures that have the same shape but different sizes, regardless of whether they can be transformed into each other through any similarity transformations."
2. Regarding how this definition relates to rigid transformations, dilations, and scale factors:
a. Rigid transformations (such as translation, rotation, and reflection) would generally not be considered similarity transformations under this definition. This is because these transformations do not necessarily preserve the relative sizes of the figures. For example, if two figures are congruent, they would remain congruent after a rigid transformation but may not be considered similar if size is a distinguishing factor.
b. Dilations and scale factors, on the other hand, would still be considered similarity transformations under this definition. A dilation is a transformation that changes the size of a figure while preserving its shape and proportionality, and the scale factor indicates the ratio of the corresponding lengths in the original and dilated figures. Since similarity is defined by having the same shape, dilating a figure with a scale factor would maintain its shape and thus keep it similar to the original figure. Therefore, dilations and scale factors would still be relevant in determining similarity.