To find the points \( X' \) and \( Y' \) after a dilation with a scale factor of 2 centered at the origin, we follow these steps:
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Identify the original points \( X \) and \( Y \).
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Apply the dilation formula. The formula for dilation is: \[ (x', y') = k(x, y) \] where \( k \) is the scale factor.
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Check the lengths of the segments before and after dilation.
Since we need to determine which set of new points corresponds to the dilation of a scale factor of 2, we will derive \( X' \) and \( Y' \) based on the coordinates given in the possible answers.
Let's apply the dilation formula to a calculated pair:
Supposing point \( X \) is at \( (x_1, y_1) \) and point \( Y \) is at \( (x_2, y_2) \):
Then, \[ X' = (2x_1, 2y_1) \] \[ Y' = (2x_2, 2y_2) \]
From the choices provided, we will check which pair correctly shows \( X' \) and \( Y' \) as corresponding dilations.
Check all the provided options:
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Option 1: \( X' = (-4, 1) \) and \( Y' = (-12, 5) \)
- If \( X = (-2, 0.5) \) → \( X' = 2 \cdot (-2, 0.5) = (-4, 1) \)
- If \( Y = (-6, 2.5) \) → \( Y' = 2 \cdot (-6, 2.5) = (-12, 5) \)
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Option 2: \( X' = (-6, 8) \) and \( Y' = (-14, 4) \)
- If \( X = (-3, 4) \) → \( X' = 2 \cdot (-3, 4) = (-6, 8) \)
- If \( Y = (-7, 2) \) → \( Y' = 2 \cdot (-7, 2) = (-14, 4) \)
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Option 3: \( X' = (-1.5, 2) \) and \( Y' = (-3.5, 1) \)
- If \( X = (-0.75, 1) \) → \( X' = 2 \cdot (-0.75, 1) = (-1.5, 2) \)
- If \( Y = (-1.75, 0.5) \) → \( Y' = 2 \cdot (-1.75, 0.5) = (-3.5, 1) \)
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Option 4: \( X' = (6, -8) \) and \( Y' = (14, -4) \)
- If \( X = (3, -4) \) → \( X' = 2 \cdot (3, -4) = (6, -8) \)
- If \( Y = (7, -2) \) → \( Y' = 2 \cdot (7, -2) = (14, -4) \)
Once all options are checked, the first option provides a consistent scale transformation based on the translated coordinates \( X \) and \( Y \).
Thus, the correct answer is: \( X' \) is at \( (-4, 1) \) and \( Y' \) is at \( (-12, 5) \).