To find a line of reflection that flips the given hexagon onto itself, we need to analyze the symmetry of the hexagon based on the given vertices.
Given the vertices of the hexagon are:
- W (-n, 0)
- X (n, 0)
- Y (?, ?)
- Z (?, ?)
- U (?, -k)
- V (-m, -r)
The hexagon is plotted in quadrants 3 and 4, which means points are located in these quadrants where the x-coordinates can be negative and the y-coordinates can be negative.
From the vertices provided, we can observe:
- The points W (-n, 0) and X (n, 0) are reflections across the y-axis (line x=0).
- The y-coordinates for points U (which is (?,-k)) and V (-m,-r) suggest symmetry about the x-axis (line y=0) as long as corresponding vertices Y and Z also satisfy symmetry under that reflection.
Now, evaluating each response option:
- y = 0: This is the x-axis. If we reflect the hexagon across the x-axis, points in the third quadrant will map to the fourth quadrant, and the points on the x-axis will remain unchanged. Points U and V would match up across the x-axis.
- y = -r: This line would not symmetrically align the points and would not serve as a reflection for the hexagon.
- x = -m: This line does not match the symmetry because it would only affect the vertical placement of certain points and not align with vertices across the entire polygon.
- x = n: This line would not serve as it would only account for reflecting over one side and not serve the purpose of reflecting the hexagon onto itself.
Thus, out of the choices provided, y = 0 (the x-axis) is the correct answer because this line of reflection ensures that any point in the third quadrant maps accordingly to the fourth quadrant, thus preserving the shape of the hexagon.
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