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)Responses Rotation of 90 degrees clockwise and then reflection across the x-axis shows that triangle ABC is 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 .Translation of (2,0)and then reflection across the x-axis shows that triangle ABC is 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 .Reflection across the y-axis and then a translation of (−3,−5) shows that triangle ABC is 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.

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)Responses    Rotation of 90 degrees clockwise and then reflection across the x-axis shows that triangle ABC is 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 .Translation of (2,0)and then reflection across the x-axis shows that triangle ABC is 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 .Reflection across the y-axis and then a translation of (−3,−5) shows that triangle ABC is 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.

GPT-4o mini · Answer

To determine the transformations between triangles ABC, A'B'C', and A''B''C'', let's analyze the points of the triangles and the relationships between them.

Triangle ABC:
- A(-4, 5)
- B(-1, 3)
- C(-3, 1)
Triangle A'B'C':
- A'(4, 5)
- B'(1, 3)
- C'(3, 1)
Triangle A''B''C'':
- A''(1, 0)
- B''(-2, -2)
- C''(0, -4)

Transformations:

From Triangle ABC to A'B'C':
- If we compare A to A', B to B', and C to C', we see that A' is the reflection of A across the y-axis (since x-coordinates change sign), B' is also the reflection of B across the y-axis, and C' is the reflection of C across the y-axis. This means that triangle ABC is reflected across the y-axis to obtain triangle A'B'C'.
From Triangle A'B'C' to A''B''C'':
- Triangle A''B''C'' has a different arrangement and orientation. To understand how triangle A'B'C' can be transformed into triangle A''B''C'', we observe that A' and A'' appear to have moved down, while B' and B'' as well as C' and C'' have both shifted in various directions on the coordinate plane. The transformation likely involves a combination of reflection across the x-axis and some translation.
Conclusion:
- Triangle ABC to A'B'C' involves a reflection across the y-axis.
- Triangle A'B'C' to A''B''C'' likely involves additional transformations including reflection across the x-axis and translation.

Given these observations, the best description that matches our findings and also confirms the congruency of triangle ABC to triangle A''B''C'' (after reflecting and translating) is as follows:

“Rotation of 90 degrees clockwise and then reflection across the x-axis shows that triangle ABC is congruent to triangle A''B''C''.”

This phrase encapsulates the necessary transformations while confirming the congruency established after the transformations.