Bot GPT 4-o mini To find the coordinates of \( Z' \) after a reflection across the line \( y = 1 \), we first need to identify the original coordinates of point \( Z \) and then apply the reflection transformation.
Let’s say the original coordinates of point \( Z \) are \( (x, y) \).
The formula for reflecting a point \( (x, y) \) across the line \( y = k \) is given by:
- The x-coordinate remains the same: \( x' = x \)
- The new y-coordinate is: \( y' = 2k - y \)
In our case, \( k = 1 \). Therefore, the new y-coordinate after reflection across \( y = 1 \) would be: \[ y' = 2(1) - y = 2 - y \]
Assuming \( Z \) has coordinates \( (x, y) \).
To find the coordinates of \( Z' \), we can plug in the y-coordinate of point \( Z \) into the formula \( y' = 2 - y \).
Considering the options provided:
- \( Z'(-4, 0) \)
- \( Z'(6, 2) \)
- \( Z'(4, 0) \)
- \( Z'(-4, 2) \)
To find the reflection across line \( y=1 \) for an example \( Z(4,0) \):
- Original point: \( (4, 0) \)
- New coordinate would be:
- x-coordinate remains the same: \( x' = 4 \)
- y-coordinate: \( y' = 2 - 0 = 2 \)
Thus, the reflected point \( Z' = (4, 2) \).
Among the options provided, the correct reflection, assuming the original point was \( (4, 0) \), would be: \[ Z' = (4, 2) \]
Therefore, the answer is:
Z′(6,2).
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