To determine whether line A'B' passes through the center of dilation based on the coordinates provided, let's analyze the situation:
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Coordinates:
- A (4, -2)
- B (12, -4)
- A' (1, -0.5)
- B' (3, -2)
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Finding the Center of Dilation: The center of dilation can be found by determining the scale factor and how the original points map to the new points.
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Calculating the Scale Factor:
- The dilation from A to A':
- From A (4, -2) to A' (1, -0.5) appears to be a reduction.
- The scale factor \(k\) can be calculated using the ratio of the distances from the center of dilation.
- The same applies to B to B'.
- The dilation from A to A':
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Examining Choices:
- Option A states that line A'B' is parallel to line AB, which doesn’t provide a conclusion about the center of dilation.
- Option B says it’s an enlargement, which is incorrect since we can see that A' is closer to the origin than A; this represents a reduction, not an enlargement.
- Option C mentions that line A'B' does not pass through the center of dilation because it is a reduction by a scale factor of 1/4. This is true as reductions do not pass through the center in this case.
- Option D states that it does not pass through the center because it is parallel. Parallel lines do not definitively indicate the presence or absence of a center of dilation.
Upon review, Option C is the most appropriate choice based on the nature of the dilation indicated by the coordinates.
Correct Answer: C. line A'B' does not pass through the center of dilation because it is a reduction of line AB by a scale factor of 1/4.