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.
3 answers