Bot GPT 4-o mini To determine whether the line segment \( A'B' \) passes through the center of dilation, we need to analyze the relationship between the points before and after dilation.
- The original points are \( A(4, -2) \) and \( B(12, -4) \).
- The dilated points are \( A'(0.5, 0.5) \) and \( B'(3, -1) \).
To find the scale factor of the dilation, we need to calculate the distance between the original coordinates and the dilated coordinates.
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The vector representing \( AB \) is given by: \[ B - A = (12 - 4, -4 - (-2)) = (8, -2) \]
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The vector representing \( A'B' \) is: \[ B' - A' = (3 - 0.5, -1 - 0.5) = (2.5, -1.5) \]
Now to find the scale factor:
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The length of the segment \( AB \) can be calculated using the distance formula: \[ d_{AB} = \sqrt{(12 - 4)^2 + (-4 + 2)^2} = \sqrt{8^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68} = 2\sqrt{17} \]
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The length of the segment \( A'B' \): \[ d_{A'B'} = \sqrt{(3 - 0.5)^2 + (-1 - 0.5)^2} = \sqrt{(2.5)^2 + (-1.5)^2} = \sqrt{6.25 + 2.25} = \sqrt{8.5} \]
Next, to find the scale factor: \[ \text{Scale Factor} = \frac{d_{A'B'}}{d_{AB}} = \frac{\sqrt{8.5}}{2\sqrt{17}} \]
This suggests that the scale factor is less than 1, indicating a reduction.
Furthermore, for the line segments to pass through the center of dilation, the segments must be along the same line. To check for parallelism, observe that both segments \( AB \) and \( A'B' \) must have the same direction vector for \( A'B' \) to pass through the center.
Considering these points, the responses can be evaluated. Since \( A'B' \) does not scale proportionally and does not preserve the structure of \( AB \) through dilation, we can conclude:
The correct response is:
A′B′ does not pass through the center of dilation because it is taken as a line parallel to AB. Line segment A′B′ does not pass through the center of dilation because it is taken as a line parallel to line segment AB.
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