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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

Since triangles ABC
and 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.

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 .

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 .
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Answer 1

To describe the series of transformations between the triangles \( ABC \) and \( A''B''C'' \), we need to analyze the triangles' coordinates and how they relate to each other.

Triangle \( ABC \) has the following vertices:

• \( A(-4, 5) \)
• \( B(-1, 3) \)
• \( C(-3, 1) \)

Triangle \( A''B''C'' \) has the following vertices:

• \( A''(1, 0) \)
• \( B''(-2, -2) \)
• \( C''(0, -4) \)

Comparing the coordinates of both triangles, we can see they do not have the same orientation. Therefore, triangle \( ABC \) and triangle \( A''B''C'' \) are not congruent.

Next, we also look for a transformation that relates triangle \( ABC \) to triangles \( A'B'C' \) and \( A''B''C'' \).

1. Triangle \( ABC \) is to triangle \( A'B'C' \):

   • Triangle \( ABC \) vertices are in the left-half of the coordinate system, while triangle \( A'B'C' \) vertices are in the right-half. This indicates a reflection across the y-axis.

2. Triangle \( A'B'C' \) can be seen to transform into triangle \( A''B''C'' \), and either requires a rotation or translation.

To describe the transformations correctly specifically as responses given:

• Triangles \( ABC \) and \( A''B''C'' \) do not maintain the same orientation, and we conclude they are not congruent.
• However, a reflection followed by a translation may or may not accurately reflect the process, needing to define rotations distinctly too.

Given this description, the appropriate choice from the options provided is: "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."