In a dilation, which is a transformation that enlarges or reduces figures while keeping their shape, the center of dilation is a fixed point from which all other points transform.
To determine whether a dilation passes through the center of dilation, consider the following:
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Definition of Dilation: A dilation with center \( C \) and scale factor \( k \) transforms a point \( P \) to a new point \( P' \) such that: \[ P' = C + k(P - C) \]
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Dilation of the Center: If you apply the dilation to the center of dilation \( C \) itself, you will find that: \[ C' = C + k(C - C) = C + k(0) = C \] This means that the center does not change under the dilation, confirming that any dilation always passes through its center.
In conclusion, yes, the dilation does pass through the center of dilation by definition, since the center remains unchanged during the transformation.
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