To determine whether a dilation passes through the center of dilation, we need to understand what dilation is in the context of geometry.
A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. This transformation involves a center of dilation and a scale factor.
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Center of Dilation: This is a fixed point in the plane (usually denoted as O). All points are expanded or contracted from this center based on the scale factor.
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Scale Factor: This is a positive number that determines how much larger or smaller the image will be compared to the original shape. A scale factor greater than 1 enlarges the figure, while a scale factor between 0 and 1 reduces it.
For a point A being dilated to A', by the center O and scale factor k, the following relationship holds:
- A' is located by using the formula:
\[ A' = O + k(A - O) \]
This means that the point A' is determined by a vector that originates at the center O and points towards A, scaled by the factor k.
Analysis:
To see if the dilation "passes through" the center of dilation (point O), we need to check the following:
- When A is equal to O, the dilation would yield:
\[ A' = O + k(O - O) = O + k(0) = O \]
So, if the original point A is the same as the center of dilation O, then the dilated image A' will also be at O.
- When A is not equal to O (meaning A is any point other than O), A' will not be O unless k = 1 (which keeps point A at O), and the point will move away from O depending on the scale factor.
Conclusion:
- A dilation passes through the center of dilation if and only if at least one point being dilated is the center of dilation, or the scale factor is specifically 1 (which means the dilation does not change the size of the figure). Otherwise, generally, the image produced by the dilation will not pass through the center.