Question
Clayton needs to reflect the triangle below across the line y = x.
On a coordinate plane, triangle A B C has points (5.5, 7), (6, 2), (4, 4).
Which statements about the reflection are true? Check all that apply.
Clayton could use the relationship (x, y) right-arrow (y, x) to find the points of the image.
Clayton could negate both the x and y values in the points to find the points of the image.
C’ will remain in the same location as C because it is on the line of reflection.
C’ will move because all points move in a reflection.
The image and the pre-image will be congruent triangles.
The image and pre-image will not have the same orientation because reflections flip figures.
On a coordinate plane, triangle A B C has points (5.5, 7), (6, 2), (4, 4).
Which statements about the reflection are true? Check all that apply.
Clayton could use the relationship (x, y) right-arrow (y, x) to find the points of the image.
Clayton could negate both the x and y values in the points to find the points of the image.
C’ will remain in the same location as C because it is on the line of reflection.
C’ will move because all points move in a reflection.
The image and the pre-image will be congruent triangles.
The image and pre-image will not have the same orientation because reflections flip figures.
Answers
Answered by
GPT-4o mini
To determine which statements about the reflection of triangle ABC across the line \( y = x \) are true, let's analyze each statement:
1. **Clayton could use the relationship (x, y) right-arrow (y, x) to find the points of the image.**
**True**: The reflection over the line \( y = x \) indeed swaps the x and y coordinates of each point. So, \((x, y)\) becomes \((y, x)\).
2. **Clayton could negate both the x and y values in the points to find the points of the image.**
**False**: Negating the x and y values alters their signs, which is not how reflection over the line \( y = x \) operates.
3. **C’ will remain in the same location as C because it is on the line of reflection.**
**True**: If point C lies on the line \( y = x \), its reflection C' will indeed remain at the same location.
4. **C’ will move because all points move in a reflection.**
**False**: As noted in the previous statement, C does not move when it lies on the line of reflection.
5. **The image and the pre-image will be congruent triangles.**
**True**: Reflections preserve distances and angles, so the image and the pre-image will be congruent triangles.
6. **The image and pre-image will not have the same orientation because reflections flip figures.**
**True**: Reflections do change the orientation of a figure, so the image will not have the same orientation as the pre-image.
### Summary of True Statements:
- Clayton could use the relationship \((x, y) \rightarrow (y, x)\) to find the points of the image.
- C’ will remain in the same location as C because it is on the line of reflection.
- The image and the pre-image will be congruent triangles.
- The image and pre-image will not have the same orientation because reflections flip figures.
### Final Answers:
1, 3, 5, 6 are true.
1. **Clayton could use the relationship (x, y) right-arrow (y, x) to find the points of the image.**
**True**: The reflection over the line \( y = x \) indeed swaps the x and y coordinates of each point. So, \((x, y)\) becomes \((y, x)\).
2. **Clayton could negate both the x and y values in the points to find the points of the image.**
**False**: Negating the x and y values alters their signs, which is not how reflection over the line \( y = x \) operates.
3. **C’ will remain in the same location as C because it is on the line of reflection.**
**True**: If point C lies on the line \( y = x \), its reflection C' will indeed remain at the same location.
4. **C’ will move because all points move in a reflection.**
**False**: As noted in the previous statement, C does not move when it lies on the line of reflection.
5. **The image and the pre-image will be congruent triangles.**
**True**: Reflections preserve distances and angles, so the image and the pre-image will be congruent triangles.
6. **The image and pre-image will not have the same orientation because reflections flip figures.**
**True**: Reflections do change the orientation of a figure, so the image will not have the same orientation as the pre-image.
### Summary of True Statements:
- Clayton could use the relationship \((x, y) \rightarrow (y, x)\) to find the points of the image.
- C’ will remain in the same location as C because it is on the line of reflection.
- The image and the pre-image will be congruent triangles.
- The image and pre-image will not have the same orientation because reflections flip figures.
### Final Answers:
1, 3, 5, 6 are true.