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

ry=5
r subscript y equals 5 baseline

rx=3
r subscript x equals 3 baseline

ry=4
r subscript y equals 4 baseline

rx=−1

Answer 1

To determine a reflection that carries the rectangle onto itself, we can look for the line of symmetry.

The given vertices of the rectangle are:

• \((-1, 9)\)
• \((5, -1)\)
• \((-1, -1)\)
• \((5, 9)\)

First, let's identify the midpoints of the x-coordinates and the y-coordinates:

1. Finding the midpoints of the x-coordinates:

   • The x-coordinates are \(-1\) and \(5\).
   • Midpoint = \(\frac{-1 + 5}{2} = \frac{4}{2} = 2\).

2. Finding the midpoints of the y-coordinates:

   • The y-coordinates are \(9\) and \(-1\).
   • Midpoint = \(\frac{9 + (-1)}{2} = \frac{8}{2} = 4\).

The midpoint of the rectangle is therefore \((2, 4)\).

Now we can analyze the given reflection options:

• \(r_{y=5}\) reflects points over the line \(y = 5\).
• \(r_{x=3}\) reflects points over the line \(x = 3\).
• \(r_{y=4}\) reflects points over the line \(y = 4\).
• \(r_{x=-1}\) reflects points over the line \(x = -1\).

The appropriate reflection line that will carry the rectangle onto itself (which has a symmetrical center at \((2, 4)\)) will be the line \(y = 4\) (as \(y = 4\) is exactly halfway between the highest and lowest y-coordinates of the rectangle).

So, the correct response is: \(r_{y=4}\).