Question
Apply geometric motions to reflect figures. Brinda reflects point T over a line of reflection, resulting in point T′ . If points T and T′ have the same x-coordinate, what must be true?(1 point)
Responses a b c or d which one is it and why by solving it
The line of reflection is the x-axis.
The line of reflection is the x -axis.
The line of reflection is the y-axis.
The line of reflection is the y -axis.
The line of reflection is vertical.
The line of reflection is vertical.
The line of reflection is horizontal.
Responses a b c or d which one is it and why by solving it
The line of reflection is the x-axis.
The line of reflection is the x -axis.
The line of reflection is the y-axis.
The line of reflection is the y -axis.
The line of reflection is vertical.
The line of reflection is vertical.
The line of reflection is horizontal.
Answers
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Answered by
GPT-4o mini
If points \( T \) and \( T' \) have the same x-coordinate after reflecting over a line, then this implies that the line of reflection must maintain that x-coordinate for both points.
Let's analyze each option:
**a) The line of reflection is the x-axis.**
If the line of reflection was the x-axis, then reflecting a point directly above or below that line would change the y-coordinate while keeping the x-coordinate the same. This means if \( T \) is at \( (x, y) \) and is reflected over the x-axis, \( T' \) would end up at \( (x, -y) \). Hence, this statement is incorrect.
**b) The line of reflection is the y-axis.**
If \( T \) is reflected over the y-axis, a point at \( (x, y) \) would be reflected to \( (-x, y) \). This rotation changes the x-coordinate, so this option is also incorrect.
**c) The line of reflection is vertical.**
A vertical line of reflection can intersect any vertical line parallel to the y-axis. However, if \( T \) and \( T' \) have the same x-coordinate, we can simply conclude that the vertical line does not change the x-coordinate of points reflected across it; thus, \( T \) and \( T' \) can exist on either side of that line. This statement is potentially true.
**d) The line of reflection is horizontal.**
If the line were horizontal, similar to the x-axis, reflecting \( T \) over this line would change the y-coordinate while the x-coordinate remains the same. This means \( T \) would be changed to \( (x, -y) \), similar to the case with the x-axis, making this incorrect as well.
Therefore, the most consistent option that allows \( T \) and \( T' \) to have the same x-coordinate is:
**c) The line of reflection is vertical.**
This is because if you reflect points on the same side of a vertical line (for instance, \( T \) at \( (2, 3) \) and \( T' \) at \( (2, -3) \)), they can maintain the same x-coordinate while the y-coordinates change.
Let's analyze each option:
**a) The line of reflection is the x-axis.**
If the line of reflection was the x-axis, then reflecting a point directly above or below that line would change the y-coordinate while keeping the x-coordinate the same. This means if \( T \) is at \( (x, y) \) and is reflected over the x-axis, \( T' \) would end up at \( (x, -y) \). Hence, this statement is incorrect.
**b) The line of reflection is the y-axis.**
If \( T \) is reflected over the y-axis, a point at \( (x, y) \) would be reflected to \( (-x, y) \). This rotation changes the x-coordinate, so this option is also incorrect.
**c) The line of reflection is vertical.**
A vertical line of reflection can intersect any vertical line parallel to the y-axis. However, if \( T \) and \( T' \) have the same x-coordinate, we can simply conclude that the vertical line does not change the x-coordinate of points reflected across it; thus, \( T \) and \( T' \) can exist on either side of that line. This statement is potentially true.
**d) The line of reflection is horizontal.**
If the line were horizontal, similar to the x-axis, reflecting \( T \) over this line would change the y-coordinate while the x-coordinate remains the same. This means \( T \) would be changed to \( (x, -y) \), similar to the case with the x-axis, making this incorrect as well.
Therefore, the most consistent option that allows \( T \) and \( T' \) to have the same x-coordinate is:
**c) The line of reflection is vertical.**
This is because if you reflect points on the same side of a vertical line (for instance, \( T \) at \( (2, 3) \) and \( T' \) at \( (2, -3) \)), they can maintain the same x-coordinate while the y-coordinates change.
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