Question
A path between 3 cities is drawn on a map, creating the outlined region shown in the graph below.
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
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
Answers
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Answered by
GPT-4o mini
To determine which sequence of transformations would map the outlined region (created by the path between the 3 cities) onto itself, we need to analyze each option step-by-step.
1. **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 will change the position of the region.
- A 270-degree counterclockwise rotation is equivalent to a 90-degree clockwise rotation, which also changes the position of the region back to where it originally was when combined with the first transformation.
Result: No, it does not map onto itself.
2. **Option B**: A rotation 180-degrees counterclockwise about the origin followed by a rotation 270-degrees counterclockwise about the origin.
- A 180-degree rotation will place the region in the opposite position.
- A subsequent 270-degree counterclockwise rotation will then change the region's position further.
Result: No, it does not map onto itself.
3. **Option C**: A rotation 90-degrees counterclockwise about the origin followed by a rotation 180-degrees counterclockwise about the origin.
- A 90-degree rotation changes the position of the region.
- A 180-degree rotation will move it again to a new position that is not the original.
Result: No, it does not map onto itself.
4. **Option D**: A reflection across the y-axis followed by a reflection across the x-axis.
- A reflection across the y-axis flips the region over the y-axis.
- A reflection across the x-axis takes the already flipped position and flips it again over the x-axis.
Result: This sequence results in the region mapping onto itself, depending on the symmetry of the paths.
Ultimately, the correct answer is:
**D. a reflection across the y-axis and then a reflection across the x-axis.**
1. **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 will change the position of the region.
- A 270-degree counterclockwise rotation is equivalent to a 90-degree clockwise rotation, which also changes the position of the region back to where it originally was when combined with the first transformation.
Result: No, it does not map onto itself.
2. **Option B**: A rotation 180-degrees counterclockwise about the origin followed by a rotation 270-degrees counterclockwise about the origin.
- A 180-degree rotation will place the region in the opposite position.
- A subsequent 270-degree counterclockwise rotation will then change the region's position further.
Result: No, it does not map onto itself.
3. **Option C**: A rotation 90-degrees counterclockwise about the origin followed by a rotation 180-degrees counterclockwise about the origin.
- A 90-degree rotation changes the position of the region.
- A 180-degree rotation will move it again to a new position that is not the original.
Result: No, it does not map onto itself.
4. **Option D**: A reflection across the y-axis followed by a reflection across the x-axis.
- A reflection across the y-axis flips the region over the y-axis.
- A reflection across the x-axis takes the already flipped position and flips it again over the x-axis.
Result: This sequence results in the region mapping onto itself, depending on the symmetry of the paths.
Ultimately, the correct answer is:
**D. a reflection across the y-axis and then a reflection across the x-axis.**
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