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:

• \( R'(-3, 1) \)
• \( Q'(-2, 7) \)