To find the coordinates of vertex \( X' \) after the reflection over a horizontal line, we need to determine the horizontal distance from the point \( X(-3, 9) \) to the line of reflection, then apply the same distance below the line to find \( X' \).
Since the triangle below the line has one vertex given as \( (3, -2) \), we know the line of reflection lies at \( y = 4 \) (the average y-coordinates of points from both triangles).
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Find the y-coordinate of \( X' \): The line of reflection is \( y = 4 \), and \( X \) is located at \( y = 9 \). The distance from \( X \) to the line is: \[ 9 - 4 = 5 \] Therefore, to find \( X' \), we subtract this distance from the line of reflection: \[ 4 - 5 = -1 \]
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Find the x-coordinate of \( X' \): The x-coordinate remains the same since this is a reflection over a horizontal line. Thus, the x-coordinate of \( X' \) is: \[ -3 \]
Putting it all together, the coordinates of vertex \( X' \) after the reflection are: \[ (-3, -1) \]
However, it seems I made a mistake in considering the distance reflected; \( 9 \) is the original position beyond the line \( 4 \), so we need to reflect it correctly considering the point \( (3, -2) \) suggests we have misidentified where the line of reflection occurs.
- Hypercorrecting as:
- Distance from \( y = 9 \) down to the line found below must yield \( y = -1 \).
- Reset the count \( 4 \) as the line split; \( 4 + x = y \) for finality.
Therefore, the corrected response for \( X' \) should be:
- Correct choices retain its form.
Adjust \( X' = (-3, -9) \), so we select: Option: \( (-3, -9) \) as the valid vertex posture.
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