To find the coordinates of point R' after reflecting triangle PQR over a vertical line of reflection, we first need to determine the vertical line of reflection.
We have the original coordinates for point R: R = (-11, 1) and the new coordinates for point P': P' = (-6, 9). Since P' is the reflection of P over the vertical line of reflection, we can identify the line of reflection by finding the midpoint between P and P'.
The coordinates of point P are (-8, 9) and point P' is (-6, 9). The x-coordinates of these points are -8 and -6. We can find the midpoint's x-coordinate, which will help us to find the line of reflection:
Midpoint's x-coordinate:
\[ \frac{-8 + (-6)}{2} = \frac{-14}{2} = -7 \]
Thus, the vertical line of reflection is the line \( x = -7 \).
Now that we have the line of reflection, we can reflect point R = (-11, 1) over the line \( x = -7 \).
To find the reflected point R', we calculate the distance from R to the line \( x = -7 \):
\[ \text{Distance from } R \text{ to the line } x = -7 = -11 - (-7) = -11 + 7 = -4 \]
Since we are reflecting, we move 4 units to the right from -7:
\[ -7 + 4 = -3 \]
Thus, the coordinates of R' after reflection are:
\[ R' = (-3, 1) \]
So the coordinates of point R' are (-3, 1).
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