To find the coordinates of point \( R' \) after reflecting point \( R \) over a vertical line of reflection, we first need to determine the vertical line of reflection based on point \( P \) and its reflected point \( P' \).
The coordinates of point \( P \) are \( (-8, 9) \), and its reflection \( P' \) is \( (-6, 9) \).
The vertical line of reflection is placed halfway between the x-coordinates of \( P \) and \( P' \). We can calculate the x-coordinate of the line of reflection as follows:
\[ \text{Line of reflection} = \frac{x_P + x_{P'}}{2} = \frac{-8 + (-6)}{2} = \frac{-14}{2} = -7 \]
Now that we know the line of reflection is \( x = -7 \), we can reflect point \( R \) located at \( (-11, 1) \) over this line.
To reflect a point over a vertical line, we can find the distance from the point to the line and then move the same distance on the opposite side of the line. The original x-coordinate of \( R \) is \( -11 \), and the x-coordinate of the line is \( -7 \).
The distance from \( R \) to the line is:
\[ \text{Distance} = -11 - (-7) = -11 + 7 = -4 \]
Now, we reflect point \( R \) across the line by moving 4 units to the right (towards the positive x-axis):
\[ x_{R'} = -7 + 4 = -3 \]
The y-coordinate remains unchanged during reflection across a vertical line. Thus, the y-coordinate of point \( R' \) will still be \( 1 \).
So, the coordinates of point \( R' \) after the reflection are:
\[ R' = (-3, 1) \]
Therefore, the coordinates of point \( R' \) are \( \boxed{(-3, 1)} \).
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