Question
Triangle ABC is congruent to triangle A′′B′′C′′ .
Which sequence of transformations could have been used to transform triangle ABC to produce triangle A′′B′′C′′ ?
Responses
Triangle ABC was translated 10 units right and then reflected across the x-axis.
triangle A B C was translated 10 units right and then reflected across the x -axis.
Triangle ABC was reflected across the x-axis and then translated 9 units right.
triangle A B C was reflected across the x -axis and then translated 9 units right.
Triangle ABC was reflected across the y-axis and then translated 7 units down.
triangle A B C was reflected across the y -axis and then translated 7 units down.
Triangle ABC was translated 7 units down and then 9 units right.
Which sequence of transformations could have been used to transform triangle ABC to produce triangle A′′B′′C′′ ?
Responses
Triangle ABC was translated 10 units right and then reflected across the x-axis.
triangle A B C was translated 10 units right and then reflected across the x -axis.
Triangle ABC was reflected across the x-axis and then translated 9 units right.
triangle A B C was reflected across the x -axis and then translated 9 units right.
Triangle ABC was reflected across the y-axis and then translated 7 units down.
triangle A B C was reflected across the y -axis and then translated 7 units down.
Triangle ABC was translated 7 units down and then 9 units right.
Answers
Answered by
GPT-4o mini
To determine which sequence of transformations could have been used to transform triangle ABC into triangle A′′B′′C′′, we need to consider what transformations (translation, reflection) change the position and orientation of triangles while maintaining their congruence.
1. **Translation** moves the triangle without altering its shape and size, merely shifting it in a certain direction.
2. **Reflection** flips the triangle over a line (axis), changing its orientation but not its size or shape.
Reviewing the transformations listed:
- **First option**: "Triangle ABC was translated 10 units right and then reflected across the x-axis."
- **Second option**: "Triangle A B C was translated 10 units right and then reflected across the x-axis."
Both options 1 and 2 are the same and involve a translation followed by a reflection.
- **Third option**: "Triangle ABC was reflected across the x-axis and then translated 9 units right."
- **Fourth option**: "Triangle A B C was reflected across the x-axis and then translated 9 units right."
Again, options 3 and 4 are identical and involve a reflection followed by a translation.
- **Fifth option**: "Triangle ABC was reflected across the y-axis and then translated 7 units down."
- **Sixth option**: "Triangle A B C was reflected across the y-axis and then translated 7 units down."
Similarly, options 5 and 6 are the same and concern a reflection across the y-axis followed by a translation downwards.
- **Seventh option**: "Triangle ABC was translated 7 units down and then 9 units right." This involves only translations with no reflections.
To decide which transformation would map triangle ABC to triangle A′′B′′C′′, we need to know the specific coordinates of A′′, B′′, and C′′ compared to A, B, and C. Since we don’t have that information but we can make some logical deductions based on transformations:
1. If a triangle is reflected, its orientation changes. If the triangle changed orientation but maintained congruency, reflection would be a necessary component.
2. If there is a final position shift (like moving up, down, or sideways), translation will also feature in the transformation process.
Given the choices, if both **reflection across the x-axis** or **y-axis** and **translations** are mentioned, those combinations account for both the shape and position change.
Among the listed transformations, **either the sequences involving reflection and translation (like options 3, 4, 5, and 6)** appropriately represent potential transformations resulting in a congruent triangle with a new orientation and position.
Without knowing the actual coordinates or transformations needed, **the third option**: "Triangle ABC was reflected across the x-axis and then translated 9 units right" is a valid answer as it includes both a reflection (change of orientation) and a translation.
**Thus, I suggest option 3 or 4** based on the expected movements from the original triangle to the new triangle's position.
1. **Translation** moves the triangle without altering its shape and size, merely shifting it in a certain direction.
2. **Reflection** flips the triangle over a line (axis), changing its orientation but not its size or shape.
Reviewing the transformations listed:
- **First option**: "Triangle ABC was translated 10 units right and then reflected across the x-axis."
- **Second option**: "Triangle A B C was translated 10 units right and then reflected across the x-axis."
Both options 1 and 2 are the same and involve a translation followed by a reflection.
- **Third option**: "Triangle ABC was reflected across the x-axis and then translated 9 units right."
- **Fourth option**: "Triangle A B C was reflected across the x-axis and then translated 9 units right."
Again, options 3 and 4 are identical and involve a reflection followed by a translation.
- **Fifth option**: "Triangle ABC was reflected across the y-axis and then translated 7 units down."
- **Sixth option**: "Triangle A B C was reflected across the y-axis and then translated 7 units down."
Similarly, options 5 and 6 are the same and concern a reflection across the y-axis followed by a translation downwards.
- **Seventh option**: "Triangle ABC was translated 7 units down and then 9 units right." This involves only translations with no reflections.
To decide which transformation would map triangle ABC to triangle A′′B′′C′′, we need to know the specific coordinates of A′′, B′′, and C′′ compared to A, B, and C. Since we don’t have that information but we can make some logical deductions based on transformations:
1. If a triangle is reflected, its orientation changes. If the triangle changed orientation but maintained congruency, reflection would be a necessary component.
2. If there is a final position shift (like moving up, down, or sideways), translation will also feature in the transformation process.
Given the choices, if both **reflection across the x-axis** or **y-axis** and **translations** are mentioned, those combinations account for both the shape and position change.
Among the listed transformations, **either the sequences involving reflection and translation (like options 3, 4, 5, and 6)** appropriately represent potential transformations resulting in a congruent triangle with a new orientation and position.
Without knowing the actual coordinates or transformations needed, **the third option**: "Triangle ABC was reflected across the x-axis and then translated 9 units right" is a valid answer as it includes both a reflection (change of orientation) and a translation.
**Thus, I suggest option 3 or 4** based on the expected movements from the original triangle to the new triangle's position.