Question
Given quadrilateral P (-3, 2) Q (3, -1) R (3, -4) S (-3, -7) in the diagram, find a line of reflection that will flip the figure onto itself.
Answers
GPT-4o mini
To find a line of reflection that will flip the quadrilateral PQR onto itself, we can identify the midpoints of pairs of opposite sides.
1. **Identify the midpoints** of the pairs of opposite sides:
- Midpoint of PQ:
\[
M_{PQ} = \left( \frac{-3 + 3}{2}, \frac{2 + (-1)}{2} \right) = (0, 0.5)
\]
- Midpoint of RS:
\[
M_{RS} = \left( \frac{3 + (-3)}{2}, \frac{-4 + (-7)}{2} \right) = (0, -5.5)
\]
2. **Find the midpoint of QR:**
\[
M_{QR} = \left( \frac{3 + 3}{2}, \frac{-1 + (-4)}{2} \right) = (3, -2.5)
\]
3. **Find the midpoint of PS:**
\[
M_{PS} = \left( \frac{-3 + (-3)}{2}, \frac{2 + (-7)}{2} \right) = (-3, -2.5)
\]
4. **Next, identify the line that bisects these midpoints**: The line of reflection will run perpendicular to the line connecting each pair of midpoints and will pass through the midpoints.
5. **Finding the slope of the lines connecting the midpoints (PS and QR)**:
- Slope of line connecting \( M_{PS} \) and \( M_{QR} \):
The slope is found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
\[
m = \frac{-2.5 - (-2.5)}{3 - (-3)} = \frac{0}{6} = 0
\]
Thus, the line is horizontal because the slope is 0.
6. Since the line segment connecting the midpoints is horizontal, the line of reflection will be a vertical line. The x-coordinate of the reflecting line can be described using the average of the x-coordinates of any two opposite points of the quadrilateral.
7. **Finding the average of \( x_{coordinates} \) of points**:
\[
x_{avg} = \frac{-3 + 3}{2} = 0
\]
Hence, the line of reflection that will flip quadrilateral PQR onto itself is:
\[
\boxed{x = 0}
\]
This line is the y-axis, and reflection across this line will map the points correctly to each other thus indicating that the quadrilateral is symmetric about this line.
1. **Identify the midpoints** of the pairs of opposite sides:
- Midpoint of PQ:
\[
M_{PQ} = \left( \frac{-3 + 3}{2}, \frac{2 + (-1)}{2} \right) = (0, 0.5)
\]
- Midpoint of RS:
\[
M_{RS} = \left( \frac{3 + (-3)}{2}, \frac{-4 + (-7)}{2} \right) = (0, -5.5)
\]
2. **Find the midpoint of QR:**
\[
M_{QR} = \left( \frac{3 + 3}{2}, \frac{-1 + (-4)}{2} \right) = (3, -2.5)
\]
3. **Find the midpoint of PS:**
\[
M_{PS} = \left( \frac{-3 + (-3)}{2}, \frac{2 + (-7)}{2} \right) = (-3, -2.5)
\]
4. **Next, identify the line that bisects these midpoints**: The line of reflection will run perpendicular to the line connecting each pair of midpoints and will pass through the midpoints.
5. **Finding the slope of the lines connecting the midpoints (PS and QR)**:
- Slope of line connecting \( M_{PS} \) and \( M_{QR} \):
The slope is found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
\[
m = \frac{-2.5 - (-2.5)}{3 - (-3)} = \frac{0}{6} = 0
\]
Thus, the line is horizontal because the slope is 0.
6. Since the line segment connecting the midpoints is horizontal, the line of reflection will be a vertical line. The x-coordinate of the reflecting line can be described using the average of the x-coordinates of any two opposite points of the quadrilateral.
7. **Finding the average of \( x_{coordinates} \) of points**:
\[
x_{avg} = \frac{-3 + 3}{2} = 0
\]
Hence, the line of reflection that will flip quadrilateral PQR onto itself is:
\[
\boxed{x = 0}
\]
This line is the y-axis, and reflection across this line will map the points correctly to each other thus indicating that the quadrilateral is symmetric about this line.