Question
A rectangle in a coordinate plane has vertices (-1,9), (5,-1), (-1,-1) and (5,9). Identoify a reflection that carries the rectangle onto itself. A: ^r x=3 B: ^r y=5 C:^r x=-1 D: ^r y=4
Answers
Answered by
GPT-4o mini
To determine which reflection carries the rectangle onto itself, we first need to identify the center and orientation of the rectangle formed by the vertices (-1, 9), (5, -1), (-1, -1), and (5, 9).
The vertices suggest that the rectangle is aligned with the axes, as the x-coordinates (-1 and 5) are constant for vertical edges and the y-coordinates (9 and -1) are constant for horizontal edges.
The rectangle's center can be calculated using the midpoints of the opposing sides. The x-coordinate of the center can be found as the average of the x-coordinates of the left and right sides:
\[
\text{Center X} = \frac{-1 + 5}{2} = \frac{4}{2} = 2.
\]
The y-coordinate of the center is the average of the y-coordinates of the top and bottom sides:
\[
\text{Center Y} = \frac{9 + (-1)}{2} = \frac{8}{2} = 4.
\]
Thus, the center of the rectangle is at (2, 4).
Now let's analyze the given reflection options:
- **A: Reflection \( r_{x=3} \)**: This line is vertical, and it will reflect points to the right and left of x = 3.
- **B: Reflection \( r_{y=5} \)**: This line is horizontal, reflecting points above and below y = 5.
- **C: Reflection \( r_{x=-1} \)**: This line is the vertical line at x = -1, reflecting points to the left and right of -1.
- **D: Reflection \( r_{y=4} \)**: This line is the horizontal line at y = 4, reflecting points above and below 4.
To reflect the rectangle onto itself, we need a line that either goes through the center or maintains symmetry about the center (2, 4).
**Reflection D: \( r_{y=4} \)** makes the most sense because it is horizontal and passes directly through the y-coordinate of the rectangle's center. When the rectangle is reflected over this line, the points move to their corresponding positions while keeping the rectangle's alignment.
To conclude, the reflection that carries the rectangle onto itself is:
**D: \( ^r y=4 \)**.
The vertices suggest that the rectangle is aligned with the axes, as the x-coordinates (-1 and 5) are constant for vertical edges and the y-coordinates (9 and -1) are constant for horizontal edges.
The rectangle's center can be calculated using the midpoints of the opposing sides. The x-coordinate of the center can be found as the average of the x-coordinates of the left and right sides:
\[
\text{Center X} = \frac{-1 + 5}{2} = \frac{4}{2} = 2.
\]
The y-coordinate of the center is the average of the y-coordinates of the top and bottom sides:
\[
\text{Center Y} = \frac{9 + (-1)}{2} = \frac{8}{2} = 4.
\]
Thus, the center of the rectangle is at (2, 4).
Now let's analyze the given reflection options:
- **A: Reflection \( r_{x=3} \)**: This line is vertical, and it will reflect points to the right and left of x = 3.
- **B: Reflection \( r_{y=5} \)**: This line is horizontal, reflecting points above and below y = 5.
- **C: Reflection \( r_{x=-1} \)**: This line is the vertical line at x = -1, reflecting points to the left and right of -1.
- **D: Reflection \( r_{y=4} \)**: This line is the horizontal line at y = 4, reflecting points above and below 4.
To reflect the rectangle onto itself, we need a line that either goes through the center or maintains symmetry about the center (2, 4).
**Reflection D: \( r_{y=4} \)** makes the most sense because it is horizontal and passes directly through the y-coordinate of the rectangle's center. When the rectangle is reflected over this line, the points move to their corresponding positions while keeping the rectangle's alignment.
To conclude, the reflection that carries the rectangle onto itself is:
**D: \( ^r y=4 \)**.
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