To identify a reflection that carries the rectangle onto itself, we need to find a line of reflection that will allow the vertices to map onto their corresponding vertices.
Let's consider the vertices of the rectangle:
- A = (−1, 9)
- B = (5, −1)
- C = (−1, −1)
- D = (5, 9)
To check for potential lines of reflection, we look for the midpoints of the segments connecting opposite vertices.
For example, let's consider the midpoint between A and B, and the midpoint between C and D:
- Midpoint of A and B: \[ M_A = \left( \frac{-1 + 5}{2}, \frac{9 + (-1)}{2} \right) = \left( \frac{4}{2}, \frac{8}{2} \right) = (2, 4) \]
- Midpoint of C and D: \[ M_C = \left( \frac{-1 + 5}{2}, \frac{-1 + 9}{2} \right) = \left( \frac{4}{2}, \frac{8}{2} \right) = (2, 4) \]
Both midpoints are the same. This means that the line of reflection must pass through (2, 4).
Finding the line of reflection:
Since we are looking for a vertical reflection, we can check the potential reflections given in the responses.
- \( r_x = 3 \): This is a vertical line at x = 3. It would not carry these points onto themselves.
- \( r_x = -1 \): This is a vertical line at x = -1. It would not carry these points onto themselves.
- \( r_y = 5 \): This is a horizontal line at y = 5. It could work.
- \( r_y = 4 \): This is a horizontal line at y = 4. This also could work.
However, considering the positioning of the rectangle and the fact that (2, 4) is in between, we can best conclude that \( r_y = 4 \) is the line that will reflect the vertices onto each other.
Thus, the correct reflection is: \( r_y = 4 \).