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.(1 point)
Responses

rx=3
r subscript x equals 3 baseline

ry=4
r subscript y equals 4 baseline

rx=−1
r subscript x equals negative 1 baseline

ry=5

Answer 1

To determine which reflection carries the rectangle onto itself, let's first analyze the coordinates of the vertices.

The vertices of the rectangle are:
- A (-1, 9)
- B (5, -1)
- C (-1, -1)
- D (5, 9)

From these coordinates, we can identify the midpoints and symmetry:

1. **Midpoints of opposite sides**:
- The line segment connecting A (-1, 9) and D (5, 9) is horizontal at y = 9.
- The line segment connecting B (5, -1) and C (-1, -1) is horizontal at y = -1.
- The mid-segment of the rectangle appears to be vertical with a midpoint at x = (5 + (-1)) / 2 = 2.

2. **Identifying the axes of reflection**:
- The rectangle is symmetric about vertical and horizontal lines.
- For the horizontal reflection, we notice it needs to be at the midpoint between the top edge (y = 9) and the bottom edge (y = -1). The midpoint is y = 4.
- For the vertical reflection, we notice it needs to be at the midpoint between the left edge (x = -1) and right edge (x = 5). The midpoint is x = 2.

From the options given, let's analyze them:
- \( r_{x=3} \): Vertical reflection over the line \( x = 3 \) would not keep the rectangle onto itself.
- \( r_{y=4} \): Reflection over the line \( y = 4 \) would map points on the top edge to the corresponding points on the bottom edge. So this works.
- \( r_{x=-1} \): Reflection over the line \( x = -1 \) would not keep the rectangle onto itself.
- \( r_{y=5} \): Reflection over the line \( y = 5 \) would not keep the rectangle onto itself.

Therefore, the correct reflection that carries the rectangle onto itself is:

**ry=4**