Question

A rectangle in a coordinate plane has vertices (−1,9) , (5,−1) , (−1,−1) and (5,9) . Identify a reflection that carries the rectangle onto itself.

Answer 1

To find a reflection that carries the rectangle with the given vertices onto itself, we first observe the coordinates of the vertices:
- \( A(-1,9) \)
- \( B(5,−1) \)
- \( C(−1,−1) \)
- \( D(5,9) \)

We can check the symmetries of the rectangle. A rectangle has two axes of symmetry: one vertical and one horizontal, along with diagonal lines of symmetry.

### Vertical Reflection
We look for a vertical line of symmetry. The given vertices are symmetric along the line of \( x = 2 \):
- The midpoint between \( A(-1, 9) \) and \( D(5, 9) \) is \( (2, 9) \).
- The midpoint between \( B(5, -1) \) and \( C(-1, -1) \) is also \( (2, -1) \).

The rectangle is symmetric with respect to the line \( x = 2 \).

### Horizontal Reflection
Next, we look for a horizontal line of symmetry. The given vertices are also symmetric along the line of \( y = 4 \):
- The midpoint between \( A(-1, 9) \) and \( B(5, -1) \) is \( (2, 4) \).
- The midpoint between \( C(-1, -1) \) and \( D(5, 9) \) is also \( (2, 4) \).

The rectangle is symmetric with respect to the line \( y = 4 \).

### Conclusion
Thus, two reflections that carry the rectangle onto itself are:
1. Reflection across the vertical line \( x = 2 \).
2. Reflection across the horizontal line \( y = 4 \).

Both reflections will map the rectangle onto itself.