Question
What set of reflections and rotations would carry rectangle ABCD onto itself?
Rectangle formed by ordered pairs A at negative 4, 1, B at negative 4, 2, C at negative 1, 2, D at negative 1, 1.
Reflect over the y-axis, reflect over the x-axis, rotate 180°
Rotate 180°, reflect over the x-axis, reflect over the line y = x
Reflect over the x-axis, rotate 180°, reflect over the x-axis
Rotate 180°, reflect over the y-axis, reflect over the line y = x
Answers
Answered by
GPT-4o mini
To determine which set of transformations would carry rectangle ABCD onto itself, we need to analyze the transformations that preserve the shape and orientation of the rectangle. Let's first identify the vertices of rectangle ABCD:
- A (-4, 1)
- B (-4, 2)
- C (-1, 2)
- D (-1, 1)
This rectangle is defined with vertical sides along x = -4 and x = -1, and horizontal sides along y = 1 and y = 2.
Now, let's examine the given options:
1. **Reflect over the y-axis, reflect over the x-axis, rotate 180°**
- **Reflect over the y-axis**: Points become A' (4, 1), B' (4, 2), C' (1, 2), D' (1, 1) - Not the same rectangle.
- **Reflect over the x-axis**: A'' (4, -1), B'' (4, -2), C'' (1, -2), D'' (1, -1) - Not the same rectangle.
- **Rotate 180°**: A''' (4, 1) - Not the same rectangle.
- **Result**: Not valid.
2. **Rotate 180°, reflect over the x-axis, reflect over the line y = x**
- **Rotate 180°**: A' (4, -1), B' (4, -2), C' (1, -2), D' (1, -1) - Not the same rectangle.
- **Reflect over the x-axis**: A'' (4, 1), etc. - Not the same rectangle.
- **Reflect over the line y = x**: A''' (1, 4) - Not the same rectangle.
- **Result**: Not valid.
3. **Reflect over the x-axis, rotate 180°, reflect over the x-axis**
- **Reflect over the x-axis**: A' (-4, -1), B' (-4, -2), C' (-1, -2), D' (-1, -1) - Not the same rectangle.
- **Rotate 180°**: Two points will change. Not valid.
- **Reflect over the x-axis again**: A'' will again change - Not the same rectangle.
- **Result**: Not valid.
4. **Rotate 180°, reflect over the y-axis, reflect over the line y = x**
- **Rotate 180°**: A' (4, -1), B' (4, -2), C' (1, -2), D' (1, -1) - Not the same rectangle.
- **Reflect over the y-axis**: Not a valid transformation back to original.
- **Result**: Not valid.
After checking all four transformations, none of the options successfully map rectangle ABCD onto itself. Each transformation set resulted in points that do not correspond to the original rectangle.
To map a rectangle onto itself, the transformations typically include:
- **Reflections** across the midpoint (perpendicular bisectors of the sides).
- **Rotations** around the center of the rectangle (180° being common).
For rectangle ABCD, the actual valid transformations should involve direct symmetries. In conclusion, none of provided transformation sets accurately return rectangle ABCD to its original position.
If you want valid transformations, consider using reflections across the vertical line (x = -2.5), horizontal line (y = 1.5), combined with 180-degree rotations around the center, instead.
- A (-4, 1)
- B (-4, 2)
- C (-1, 2)
- D (-1, 1)
This rectangle is defined with vertical sides along x = -4 and x = -1, and horizontal sides along y = 1 and y = 2.
Now, let's examine the given options:
1. **Reflect over the y-axis, reflect over the x-axis, rotate 180°**
- **Reflect over the y-axis**: Points become A' (4, 1), B' (4, 2), C' (1, 2), D' (1, 1) - Not the same rectangle.
- **Reflect over the x-axis**: A'' (4, -1), B'' (4, -2), C'' (1, -2), D'' (1, -1) - Not the same rectangle.
- **Rotate 180°**: A''' (4, 1) - Not the same rectangle.
- **Result**: Not valid.
2. **Rotate 180°, reflect over the x-axis, reflect over the line y = x**
- **Rotate 180°**: A' (4, -1), B' (4, -2), C' (1, -2), D' (1, -1) - Not the same rectangle.
- **Reflect over the x-axis**: A'' (4, 1), etc. - Not the same rectangle.
- **Reflect over the line y = x**: A''' (1, 4) - Not the same rectangle.
- **Result**: Not valid.
3. **Reflect over the x-axis, rotate 180°, reflect over the x-axis**
- **Reflect over the x-axis**: A' (-4, -1), B' (-4, -2), C' (-1, -2), D' (-1, -1) - Not the same rectangle.
- **Rotate 180°**: Two points will change. Not valid.
- **Reflect over the x-axis again**: A'' will again change - Not the same rectangle.
- **Result**: Not valid.
4. **Rotate 180°, reflect over the y-axis, reflect over the line y = x**
- **Rotate 180°**: A' (4, -1), B' (4, -2), C' (1, -2), D' (1, -1) - Not the same rectangle.
- **Reflect over the y-axis**: Not a valid transformation back to original.
- **Result**: Not valid.
After checking all four transformations, none of the options successfully map rectangle ABCD onto itself. Each transformation set resulted in points that do not correspond to the original rectangle.
To map a rectangle onto itself, the transformations typically include:
- **Reflections** across the midpoint (perpendicular bisectors of the sides).
- **Rotations** around the center of the rectangle (180° being common).
For rectangle ABCD, the actual valid transformations should involve direct symmetries. In conclusion, none of provided transformation sets accurately return rectangle ABCD to its original position.
If you want valid transformations, consider using reflections across the vertical line (x = -2.5), horizontal line (y = 1.5), combined with 180-degree rotations around the center, instead.