Question

Given quadrilateral PQRS in the diagram, find a line of reflection that will flip the figure onto itself.
Point P: (-3, 2)
Point Q: (3, -1)
Point R: (3, -4)
Point S: (-3, -7)

One line of reflection is y = _

Answer 1

To find a line of reflection that will flip the quadrilateral PQRS onto itself, we need to look for a line that would map each point in the quadrilateral to its corresponding point.

The points of the quadrilateral are:
- Point P: (-3, 2)
- Point Q: (3, -1)
- Point R: (3, -4)
- Point S: (-3, -7)

We can start by finding the midpoints of corresponding points. Notice that points P and S are at opposite ends, and points Q and R are the same.

Let's find the midpoints:
1. For points P and S:
\[
P = (-3, 2), \quad S = (-3, -7)
\]
The midpoint M_PS is:
\[
M_{PS} = \left(\frac{-3 + (-3)}{2}, \frac{2 + (-7)}{2}\right) = \left(-3, \frac{-5}{2}\right)
\]

2. For points Q and R:
\[
Q = (3, -1), \quad R = (3, -4)
\]
The midpoint M_QR is:
\[
M_{QR} = \left(\frac{3 + 3}{2}, \frac{-1 + (-4)}{2}\right) = \left(3, \frac{-5}{2}\right)
\]

The two midpoints are:
- M_PS = (-3, -2.5)
- M_QR = (3, -2.5)

The y-coordinates of these midpoints is -2.5. This suggests that the line of reflection could be a horizontal line where \( y = -2.5 \).

Checking if the line \( y = -2.5 \) reflects all points correctly:
- **Point P: (-3, 2)** reflects to (-3, -5)
- **Point S: (-3, -7)** reflects to (-3, 2)
- **Point Q: (3, -1)** reflects to (3, -4)
- **Point R: (3, -4)** reflects to (3, -1)

Each point does indeed map correctly according to the reflection over the line \( y = -2.5 \).

Thus, one line of reflection that will flip the figure onto itself is:

\[
\text{y} = -2.5
\]