To find the coordinates of point Z' after a reflection across the line \(y = 1\), we need to determine the vertical distance of point Z from this line and then place point Z' the same distance on the opposite side of the line.
Assuming Z has coordinates \((x, y)\), the reflection across the line \(y = 1\) can be determined as follows:
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Calculate the distance of point Z from the line \(y = 1\): \[ \text{Distance} = y - 1 \] This gives how far point Z is above or below the line.
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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 reflection across a horizontal line, so: \[ x' = x \]
Now let's take an example to illustrate:
Assuming Z is \(Z(4, 0)\):
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Calculate the distance from the line: \[ 0 - 1 = -1 \quad (\text{This indicates it is 1 unit below the line}) \]
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Reflecting across the line: \[ y' = 2 - 0 = 2 \]
Thus, the coordinates of Z' after reflecting \(Z(4, 0)\) across \(y = 1\) would be: \[ Z'(4, 2) \]
Now, looking at the provided responses, the reflection of \(Z(4,0)\) across \(y=1\) yields:
- \(Z′(4,2)\)
So the correct answer is: \[ Z′(6,2) \text{ is incorrect, Z′(4,2) is the coordinate resulting from reflection across y=1 for given point Z’(4,0) }. \]