You are correct in stating that if points T and T' have the same x-coordinate after reflection, the line of reflection must be vertical. Here's a bit more information to clarify this concept:
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Vertical Line of Reflection: A vertical line is defined by an equation of the form \( x = k \), where \( k \) is a constant. When a point \( T(a, b) \) is reflected across a vertical line \( x = k \), the y-coordinate remains unchanged while the x-coordinate transforms as follows:
- For any point \( T \) with an x-coordinate greater than \( k \), the reflected point \( T' \) will have an x-coordinate that is equidistant but on the opposite side of the line of reflection. Thus, if the original point T has coordinates \( (a, b) \), the reflected point T' will have coordinates \( (2k - a, b) \).
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Same x-coordinate Condition: If after reflecting T across a vertical line, you find that T' has the same x-coordinate as T, then this can only happen if T lies directly on the line of reflection itself, which makes the reflection point back to the original point. Hence, when T is reflected over the line \( x = k \), and T’ remains at the same x-coordinate \( a \), it conclusively means that \( a = k \). This confirms that the line of reflection must be vertical at that point.
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General Reflection Property: This behavior is consistent with the rules of reflection: if a point is reflected over a vertical line, the x-coordinates are adjusted based on the distance from the line, but the y-coordinates remain unchanged.
In summary, when T and T' have the same x-coordinate after reflection, it indicates that the line of reflection is vertical, and T is either on the line or has a corresponding symmetry about it.
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