To find the coordinates of the point \( Z' \) after reflecting across the line \( y = 1 \), we can follow these steps:
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Identify the original coordinates of \( Z \), which we need to reflect. Since it's not given in your question, let's assume the original point is \( (x, y) \).
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Calculate the vertical distance from the point \( Z \) to the line \( y = 1 \). This is calculated as: \[ \text{Distance} = y - 1 \]
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Reflecting across the line \( y = 1 \) means moving that same distance below the line. Therefore, the y-coordinate of the reflected point \( Z' \) will be: \[ y' = 1 - (y - 1) = 2 - y \]
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The x-coordinate remains the same during this vertical reflection: \[ x' = x \]
So the coordinates after reflection will be \( (x, 2 - y) \).
Now, we can analyze the given options. Without knowing the original coordinates, let's consider the possible outcomes for some specific y-coordinates:
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If \( Z = (x, 0) \): \[ y' = 2 - 0 = 2 \quad \Rightarrow \quad Z' = (x, 2) \]
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If \( Z = (x, 2) \): \[ y' = 2 - 2 = 0 \quad \Rightarrow \quad Z' = (x, 0) \]
By looking at the options:
- \( Z'(-4, 2) \)
- \( Z'(6, 2) \)
- \( Z'(-4, 0) \)
- \( Z'(4, 0) \)
To reflect correctly, if the original point \( Z \) was \( (-4, 0) \) (thus resulting in \( (-4,2) \)), or \( Z \) was \( (4, 2) \) which gives \( (4,0) \) upon reflection.
If we assume the most common scenarios for the original point, the coordinates of \( Z' \) after reflecting across \( y=1 \) could reasonably be:
- \( Z'(-4, 2) \) if reflecting from \( (-4, 0) \)
- \( Z'(4, 0) \) if reflecting from \( (4, 2) \)
Since both pairs are plausible depending on the original point chosen:
- But considering typical reflection outcomes, the correct coordinates would be \( Z'(4, 0) \) if \( (4, 2) \) was originally given.
Thus, without a specific original point provided, you might often find:
\[ Z' = (x, 2) \text{ or } Z' = (x, 0) \] as valid reflection outcomes based on chosen x-coordinates.
Final answer is \( Z' (4, 0) \).
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