Question
Use the image to answer the question.A coordinate plane with 4 quadrants shows x and y axes ranging from negative 6 to 6 in increments of 1. Three triangles are formed by joining three plotted points each. The coordinates of the plotted points for the first triangle upper A upper B upper C joined by solid lines are upper A is left parenthesis negative 4 comma 5 right parenthesis, upper B is left parenthesis negative 1 comma 3 right parenthesis, and upper C is left parenthesis negative 3 comma 1 right parenthesis. The coordinates for the second triangle upper A prime upper B prime upper C prime joined by dotted lines are as follows: upper A prime at left parenthesis 4 comma 5 right parenthesis, upper B prime at left parenthesis 1 comma 3 right parenthesis, and upper C prime at left parenthesis 3 comma 1 right parenthesis. The coordinates of the plotted points for the third triangle upper A double prime upper B double prime upper C double prime joined by lines made of dashes and dots are as follows: upper A double prime at left parenthesis 1 comma 0 right parenthesis, upper B double prime at left parenthesis negative 2 comma negative 2 right parenthesis, and upper C double prime at left parenthesis 0 comma negative 4 right parenthesis.How would you describe this series of transformations?(1 point)ResponsesRotation of 90 degrees clockwise and then reflection across the x-axis shows that triangle ABCis congruent to triangle A′′B"C".Rotation of 90 degrees clockwise and then reflection across the x -axis shows that triangle upper A upper B upper C is congruent to triangle upper A double prime upper B double prime upper C double prime .Reflection across the y-axis and then a translation of (−3,−5)shows that triangle ABCis congruent to triangle A′′B"C".Reflection across the y -axis and then a translation of left parenthesis negative 3 comma negative 5 right parenthesis shows that triangle upper A upper B upper C is congruent to triangle upper A double prime upper B double prime upper C double prime .Since triangles ABCand A′′B"C"do not have the same orientation, they are not congruent.Since triangles upper A upper B upper C and upper A double prime upper B double prime upper C double prime do not have the same orientation, they are not congruent.Translation of (2,0)and then reflection across the x-axis shows that triangle ABCis congruent to triangle A′′B"C".Translation of left parenthesis 2 comma 0 right parenthesis and then reflection across the x -axis shows that triangle upper A upper B upper C is congruent to triangle upper A double prime upper B double prime upper C double prime .
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GPT-4o mini
To analyze the transformations involved with the triangles and their congruency, let's break down the descriptions given in the question.
1. **Triangles ABC and A'B'C'**:
- The original triangle ABC has vertices at (-4, 5), (-1, 3), and (-3, 1).
- The second triangle A'B'C' has vertices at (4, 5), (1, 3), and (3, 1).
- If you compare the two sets of coordinates, you can see that A'B'C' is a reflection of ABC across the y-axis (flipping the x-coordinates from negative to positive) since the y-coordinates remain unchanged.
- Thus, triangle ABC is congruent to triangle A'B'C' because reflections preserve distances.
2. **Triangles A'B'C' and A''B''C''**:
- Triangle A''B''C'' has vertices at (1, 0), (-2, -2), and (0, -4).
- The second triangle appears to have been translated and/or rotated from the previous triangle. However, the specific transformations to reach from A'B'C' to A''B''C'' was not straightforward. The orientations seem different across these triangles.
3. **Conclusion on congruency**:
- For triangles to be congruent, they must have the same shape and size, and could be achieved through rigid transformations such as rotation, reflection, and translations, without changing their orientation.
- Since the triangles ABC and A''B''C'' do not have the same orientation, they cannot be congruent even if they are transformations of each other.
Now, reviewing the options provided:
- The transformation that describes ABC to A'B'C' is a reflection across the y-axis.
- For the transformation from A'B'C' to A''B''C'', while they may be transformations, their orientations differ, leading to the conclusion that they are not congruent.
Thus, the most fitting response is:
**Since triangles ABC and A''B''C'' do not have the same orientation, they are not congruent.**
1. **Triangles ABC and A'B'C'**:
- The original triangle ABC has vertices at (-4, 5), (-1, 3), and (-3, 1).
- The second triangle A'B'C' has vertices at (4, 5), (1, 3), and (3, 1).
- If you compare the two sets of coordinates, you can see that A'B'C' is a reflection of ABC across the y-axis (flipping the x-coordinates from negative to positive) since the y-coordinates remain unchanged.
- Thus, triangle ABC is congruent to triangle A'B'C' because reflections preserve distances.
2. **Triangles A'B'C' and A''B''C''**:
- Triangle A''B''C'' has vertices at (1, 0), (-2, -2), and (0, -4).
- The second triangle appears to have been translated and/or rotated from the previous triangle. However, the specific transformations to reach from A'B'C' to A''B''C'' was not straightforward. The orientations seem different across these triangles.
3. **Conclusion on congruency**:
- For triangles to be congruent, they must have the same shape and size, and could be achieved through rigid transformations such as rotation, reflection, and translations, without changing their orientation.
- Since the triangles ABC and A''B''C'' do not have the same orientation, they cannot be congruent even if they are transformations of each other.
Now, reviewing the options provided:
- The transformation that describes ABC to A'B'C' is a reflection across the y-axis.
- For the transformation from A'B'C' to A''B''C'', while they may be transformations, their orientations differ, leading to the conclusion that they are not congruent.
Thus, the most fitting response is:
**Since triangles ABC and A''B''C'' do not have the same orientation, they are not congruent.**
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