Question
Which sequence of transformations would result in mapping the region onto itself?
A
.
a rotation 90-degrees counterclockwise about the origin and then a rotation 270-degrees counterclockwise about the origin
B.
a rotation 180-degrees counterclockwise about the origin and then a rotation 270-degrees counterclockwise about the origin
C.
a rotation 90-degrees counterclockwise about the origin and then a rotation 180-degrees counterclockwise about the origin
D.
a reflection across the y-axis and then a reflection across the x-axis
A
.
a rotation 90-degrees counterclockwise about the origin and then a rotation 270-degrees counterclockwise about the origin
B.
a rotation 180-degrees counterclockwise about the origin and then a rotation 270-degrees counterclockwise about the origin
C.
a rotation 90-degrees counterclockwise about the origin and then a rotation 180-degrees counterclockwise about the origin
D.
a reflection across the y-axis and then a reflection across the x-axis
Answers
GPT-4o mini
To determine which sequence of transformations would result in mapping the region onto itself, we can analyze each option:
A. A rotation 90-degrees counterclockwise about the origin followed by a rotation 270-degrees counterclockwise about the origin:
- A 90-degree counterclockwise rotation followed by a 270-degree counterclockwise rotation is equivalent to a 90-degree clockwise rotation. This will not generally map the region onto itself unless the region has specific symmetry.
B. A rotation 180-degrees counterclockwise about the origin followed by a rotation 270-degrees counterclockwise about the origin:
- A 180-degree rotation followed by a 270-degree rotation is equivalent to a 90-degree rotation counterclockwise. Again, this will not necessarily map the region onto itself unless it has specific symmetry.
C. A rotation 90-degrees counterclockwise about the origin followed by a rotation 180-degrees counterclockwise about the origin:
- A 90-degree counterclockwise rotation followed by a 180-degree rotation results in a 270-degree counterclockwise rotation. This generally will not map the region onto itself.
D. A reflection across the y-axis followed by a reflection across the x-axis:
- Reflecting a figure across the y-axis and then the x-axis will result in a reflection across the origin (the diagonal line in the first quadrant will move to the third quadrant, thus mapping the region onto itself).
Based on this analysis, the transformation that would result in mapping the region onto itself is:
**D. a reflection across the y-axis and then a reflection across the x-axis.**
A. A rotation 90-degrees counterclockwise about the origin followed by a rotation 270-degrees counterclockwise about the origin:
- A 90-degree counterclockwise rotation followed by a 270-degree counterclockwise rotation is equivalent to a 90-degree clockwise rotation. This will not generally map the region onto itself unless the region has specific symmetry.
B. A rotation 180-degrees counterclockwise about the origin followed by a rotation 270-degrees counterclockwise about the origin:
- A 180-degree rotation followed by a 270-degree rotation is equivalent to a 90-degree rotation counterclockwise. Again, this will not necessarily map the region onto itself unless it has specific symmetry.
C. A rotation 90-degrees counterclockwise about the origin followed by a rotation 180-degrees counterclockwise about the origin:
- A 90-degree counterclockwise rotation followed by a 180-degree rotation results in a 270-degree counterclockwise rotation. This generally will not map the region onto itself.
D. A reflection across the y-axis followed by a reflection across the x-axis:
- Reflecting a figure across the y-axis and then the x-axis will result in a reflection across the origin (the diagonal line in the first quadrant will move to the third quadrant, thus mapping the region onto itself).
Based on this analysis, the transformation that would result in mapping the region onto itself is:
**D. a reflection across the y-axis and then a reflection across the x-axis.**