When a figure undergoes a dilation with a specific scale factor, the shape of the figure remains similar, but its size changes in proportion to the scale factor. For instance, if a figure is dilated by a scale factor of 2, each coordinate of the figure becomes twice as large.
Given the properties of dilation:
1. Reflection:
A reflection involves flipping the figure over a certain line, such as the x-axis or y-axis. This transformation changes the orientation of the figure but does not preserve the size change imparted by dilation. Therefore, a reflection alone cannot map the sides or angles of the dilated image to those of the original figure while maintaining the proportional change in size.
2. Series of Dilations:
Applying multiple dilations in an attempt to map the angles of the dilated image back to those of the original figure would not be appropriate. The angles in a figure remain constant during dilation, so additional dilations would be redundant and would change the size of the image further.
3. Rotation:
Rotation involves turning the figure around a point by a certain degree. It preserves the shape and angles of the figure but does not account for the change in size resulting from dilation. Hence, rotation alone cannot revert the sides or angles of the dilated image to match those of the original figure.
4. Series of Translations:
Translation entails moving the entire figure by a certain distance in a specific direction. This transformation preserves the size and angles but does not affect the scaling difference introduced by dilation. Thus, translation alone cannot map the angles or sides of the dilated image back to the original figure’s corresponding parts.
Since dilation changes the size of the figure while maintaining its shape, none of the given transformations—reflection, rotation, or translation—would be suitable to prove the relationship between the sides or angles of the original figure and those of the image after dilation.
Therefore, the correct answer is:
None of the transformations listed (reflection, series of dilations, rotation, or series of translations) is appropriate for proving that two sides or angles of the pre-image map to those of the image after dilation by a scale factor of 2.
