Question

Triangle PQR is reflected over a vertical line of reflection to create triangle P′Q′R′. What are the coordinates of point R′ and Q'? (Coordinates P(-8, 9), Q(-12, 7), R(-11, 1), P'(-6, 9))

Answer 1

To find the coordinates of points \( R' \) and \( Q' \) after the reflection of triangle \( PQR \) over a vertical line of reflection, we first need to determine the line of reflection based on the provided coordinates.

We know that the coordinates of point \( P \) are \( P(-8, 9) \) and its reflected point is \( P'(-6, 9) \). Since \( P \) and \( P' \) have the same y-coordinate, we can conclude that the line of reflection is a vertical line.

The x-coordinate of the line of reflection can be found by averaging the x-coordinates of points \( P \) and \( P' \):

\[
\text{Line of reflection} = \frac{-8 + (-6)}{2} = \frac{-14}{2} = -7
\]

Now, we have the line of reflection as \( x = -7 \).

To find the coordinates of the points \( Q' \) and \( R' \):

1. **Reflect point Q (-12, 7):**
The distance from \( Q \) to the line \( x = -7 \):
\[
\text{Distance} = -7 - (-12) = -7 + 12 = 5
\]
Therefore, the x-coordinate of the reflected point \( Q' \) will be:
\[
-7 + 5 = -2
\]
The y-coordinate remains the same (since it's a vertical reflection):
\[
Q' = (-2, 7)
\]

2. **Reflect point R (-11, 1):**
The distance from \( R \) to the line \( x = -7 \):
\[
\text{Distance} = -7 - (-11) = -7 + 11 = 4
\]
Therefore, the x-coordinate of the reflected point \( R' \) will be:
\[
-7 + 4 = -3
\]
The y-coordinate remains the same:
\[
R' = (-3, 1)
\]

Thus, the coordinates of the reflected points are:
- \( Q' = (-2, 7) \)
- \( R' = (-3, 1) \)

So, the final answers are:
- \( R'(-3, 1) \)
- \( Q'(-2, 7) \)