To find a line of reflection that will flip the quadrilateral PQRS onto itself, we can look for symmetry in the points of the quadrilateral.
The points given are:
- Point P: (-3, 2)
- Point Q: (3, -1)
- Point R: (3, -4)
- Point S: (-3, -7)
We can find a line of reflection by determining the midpoints between pairs of points that are opposite each other. The vertices P and S, and Q and R appear to be opposite pairs.
Let’s compute the midpoints of the diagonal pairs:
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Midpoint of P and S: \[ \text{Midpoint M1} = \left(\frac{-3 + (-3)}{2}, \frac{2 + (-7)}{2}\right) = \left(-3, -\frac{5}{2}\right) \]
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Midpoint of Q and R: \[ \text{Midpoint M2} = \left(\frac{3 + 3}{2}, \frac{-1 + (-4)}{2}\right) = \left(3, -\frac{5}{2}\right) \]
We see that both midpoints share the same y-coordinate, \( -\frac{5}{2} \).
Now, let’s find the line of reflection that could be the vertical line that lies directly between the x-coordinates of P and S, and Q and R.
The x-coordinates of Point P and Point S are both -3, and the x-coordinates of Point Q and Point R are both 3. The reflection line is vertical and will be the average of -3 and 3:
\[ \text{Line of reflection} = \frac{-3 + 3}{2} = 0 \]
Thus, the equation of the line of reflection is: \[ x = 0 \]
This vertical line (the y-axis) reflects Points P and S, as well as Points Q and R, onto one another, flipping quadrilateral PQRS onto itself.
Final Answer: The line of reflection that flips quadrilateral PQRS onto itself is \( x = 0 \) (the y-axis).