A rectangle in a coordinate plane has vertices (−1,9)

Question

, (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

rx=−1
r subscript x equals negative 1 baseline

ry=5
r subscript y equals 5 baseline

ry=4

Answer 1

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: $$M_A=(\frac{-1+5}{2},\frac{9+(-1)}{2})=(\frac{4}{2},\frac{8}{2})=(2,4)$$
• Midpoint of C and D: $$M_C=(\frac{-1+5}{2},\frac{-1+9}{2})=(\frac{4}{2},\frac{8}{2})=(2,4)$$

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: $r_y=4$.