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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 analyze the transformations of the triangles based on their coordinates, let's break down the information provided:

1. The first triangle \( ABC \) has vertices at:

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

2. The second triangle \( A'B'C' \) has vertices at:

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

3. The third triangle \( A''B''C'' \) has vertices at:

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

Finding Transformations:

Transforming \( ABC \) to \( A'B'C' \):

• To get from \( A \) to \( A' \), it appears that there’s a reflection across the y-axis (changing \( x \) from negative to positive) combined with a translation (shifting the triangle right).

Let’s verify the changes:

• For \( A(-4, 5) \) to \( A'(4, 5) \): Reflect across the y-axis gives \( (4, 5) \).
• For \( B(-1, 3) \) to \( B'(1, 3) \): Reflect across the y-axis gives \( (1, 3) \).
• For \( C(-3, 1) \) to \( C'(3, 1) \): Reflect gives \( (3, 1) \).

This confirms a reflection across the y-axis and therefore, translations are likely unnecessary for this step.

Transforming \( ABC \) to \( A''B''C'' \):

• The points show different orientations, so we examine any possible rotations or reflections:
  • From \( A(-4, 5) \) to \( A''(1, 0) \) is not a straightforward transformation - it doesn't seem to be a simple reflection or translation.
  • From \( A \) to \( C'' \): This is significantly different, indicating further transformations.

Given the analysis, it appears that \( ABC \) and \( A''B''C'' \) are not congruent simply by transforming along simple translations or reflections. They have different orientations.

Conclusion:

The option that accurately reflects the transformations and orientation of the triangles is:

Since triangles \( ABC \) and \( A''B''C'' \) do not have the same orientation, they are not congruent.

This is supported by the analysis showing that \( A''B''C'' \) cannot be derived directly from \( ABC \) through simple transformations (reflection or translation).