Question
Find the coordinates of Z′ after a reflection across y=1.(1 point)ResponsesZ′(−4,0)upper Z prime left parenthesis negative 4 comma 0 right parenthesisZ′(6,2)upper Z prime left parenthesis 6 comma 2 right parenthesisZ′(4,0)upper Z prime left parenthesis 4 comma 0 right parenthesisZ′(−4,2)
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GPT-4o mini
To find the coordinates of a point \( Z \) after reflection across the line \( y = 1 \), we can follow these steps:
1. Identify the original coordinates of point \( Z \). (Since the point \( Z \) is not given, we need to denote its coordinates as \( (x, y) \).)
2. Determine the distance from the original point \( Z \) to the line \( y = 1 \). This distance is given by \( d = y - 1 \) if \( y \) is greater than 1 (above the line) or \( d = 1 - y \) if \( y \) is less than 1 (below the line).
3. To find the reflection \( Z' \), we can apply the following transformation:
- If \( y > 1 \): \( y' = 1 - d = 1 - (y - 1) = 2 - y \)
- If \( y < 1 \): \( y' = 1 + d = 1 + (1 - y) = 2 - y \)
In both cases, the reflection across the line \( y = 1 \) results in \( y' = 2 - y \). The x-coordinate remains unchanged.
Now, let's analyze the potential responses given:
1. \( Z'(-4, 0) \)
2. \( Z'(6, 2) \)
3. \( Z'(4, 0) \)
4. \( Z'(-4, 2) \)
Assuming we take the second response \( Z'(6, 2) \) for analysis:
- The original point is above the line: to obtain \( Z' \) we would evaluate it:
- If we consider the original location \( (6, 0) \):
- The distance to the line: \( 0 - 1 = -1 \) (below). Hence:
\[
Z' = (6, 2 - 0) = (6, 2)
\]
Based on that assumption:
The configurations resulting would suggest that:
- The point \( Z'(-4, 0) \) is from \( (x,y):(-4, 2) \) where we presumed the original \( (-4, 0) \).
Thus, if we evaluate accordingly the suitable co-ordinates in line with respects it is:
- The reflection yielding \( Z'(-4, 2) \) reflects correlation to the start position \( (-4, 0) \) under the transform.
Evaluating afterwards with \( Z' = Y \), hence integers \( y\) yielding:
**Final Corresponding Coordinates after analysis: \(\mathbf{Z'(-4, 2) }\)**.
1. Identify the original coordinates of point \( Z \). (Since the point \( Z \) is not given, we need to denote its coordinates as \( (x, y) \).)
2. Determine the distance from the original point \( Z \) to the line \( y = 1 \). This distance is given by \( d = y - 1 \) if \( y \) is greater than 1 (above the line) or \( d = 1 - y \) if \( y \) is less than 1 (below the line).
3. To find the reflection \( Z' \), we can apply the following transformation:
- If \( y > 1 \): \( y' = 1 - d = 1 - (y - 1) = 2 - y \)
- If \( y < 1 \): \( y' = 1 + d = 1 + (1 - y) = 2 - y \)
In both cases, the reflection across the line \( y = 1 \) results in \( y' = 2 - y \). The x-coordinate remains unchanged.
Now, let's analyze the potential responses given:
1. \( Z'(-4, 0) \)
2. \( Z'(6, 2) \)
3. \( Z'(4, 0) \)
4. \( Z'(-4, 2) \)
Assuming we take the second response \( Z'(6, 2) \) for analysis:
- The original point is above the line: to obtain \( Z' \) we would evaluate it:
- If we consider the original location \( (6, 0) \):
- The distance to the line: \( 0 - 1 = -1 \) (below). Hence:
\[
Z' = (6, 2 - 0) = (6, 2)
\]
Based on that assumption:
The configurations resulting would suggest that:
- The point \( Z'(-4, 0) \) is from \( (x,y):(-4, 2) \) where we presumed the original \( (-4, 0) \).
Thus, if we evaluate accordingly the suitable co-ordinates in line with respects it is:
- The reflection yielding \( Z'(-4, 2) \) reflects correlation to the start position \( (-4, 0) \) under the transform.
Evaluating afterwards with \( Z' = Y \), hence integers \( y\) yielding:
**Final Corresponding Coordinates after analysis: \(\mathbf{Z'(-4, 2) }\)**.
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