The areas of a figure and its transformed image are the same. Which transformation could NOT have been applied to the original figure to create the image?
dilation
all the others are rigid motions, which preserve lengths
If the areas of a figure and its transformed image are the same, it means that the transformation applied to the original figure is an isometric transformation, which preserves the area.
Out of the common isometric transformations, one transformation that could NOT have been applied to the original figure to create the image is a dilation.
A dilation is a non-isometric transformation that changes the size of the figure. It either expands or contracts the figure by a scale factor. Since dilation changes the size of the figure, it would result in a different area for the transformed image compared to the original figure.
To determine which transformation could NOT have been applied to the original figure to create the image, we should consider the properties and characteristics of different transformations.
Here are a few common transformations and their effects on the area of a figure:
1. Translation: This transformation moves the figure without changing its shape or size. A translation does not affect the area of the figure because it preserves distances.
2. Rotation: This transformation rotates the figure around a fixed point by a given angle. A rotation also does not change the area of the figure because it only changes the orientation of the shape.
3. Reflection: This transformation flips the figure across a line, known as the mirror line. A reflection preserves the area of the figure since it does not change its size or shape.
4. Dilation: This transformation scales the figure by a certain factor, either enlarging or reducing its size. However, dilation does affect the area of the figure. If the figure is enlarged, its area will increase, and if it is reduced, its area will decrease.
Based on this, the transformation that could NOT have been applied to the original figure if the areas of the figure and its transformed image are the same is dilation. Since dilation does change the area of a figure, it contradicts the premise that the areas are the same. Therefore, dilation could not have been applied to the original figure to create the image.