To determine the line of reflection that produces the point X' (2, 4) from point X (−2, 4), we can find the line that is equidistant from both points X and X'. The line of reflection will bisect the segment connecting these two points perpendicularly.
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Find the midpoint of segment XX': The midpoint \( M \) of segment XX' can be calculated using the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting \( X(-2, 4) \) and \( X'(2, 4) \): \[ M = \left( \frac{-2 + 2}{2}, \frac{4 + 4}{2} \right) = (0, 4) \]
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Find the slope of the line connecting points X and X': The slope \( m \) of the line between \( X \) and \( X' \) can be calculated as follows: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 4}{2 - (-2)} = \frac{0}{4} = 0 \] This indicates that the line connecting X and X' is horizontal.
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Identify the line of reflection: Since the line connecting \( X \) and \( X' \) is horizontal (slope = 0), the line of reflection must be vertical (perpendicular to the horizontal line). The reflection line must pass through the midpoint \( (0, 4) \).
The vertical line through \( (0, 4) \) is the y-axis.
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Result: Based on the reflection definitions and the calculations:
- The line of reflection that produces \( X' \) from \( X \) is the y-axis.
Thus, the correct answer is that the line of reflection is the y-axis.