When performing a dilation, the center of dilation creates a relationship between the distances from the center to the vertices of the pre-image and the image. The rule about this relationship can be stated as follows:
Rule: For a dilation with a scale factor \( k \), the distance from the center of dilation to each vertex of the image is \( |k| \) times the distance from the center of dilation to the corresponding vertex of the pre-image.
In mathematical terms, if \( D \) is the center of dilation, \( P \) is a vertex of the pre-image, and \( P' \) is the corresponding vertex of the image, then the relationship can be expressed as:
\[ \text{Distance}(D, P') = |k| \times \text{Distance}(D, P) \]
This means that if the scale factor \( k \) is greater than 1, the image is enlarged, and if \( 0 < |k| < 1 \), the image is reduced. If \( k \) is negative, the image is also reflected across the center of dilation.
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