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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

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 .

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 .

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 .
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Answer 1

To describe the series of transformations between the triangles \(ABC\) and \(A''B''C''\), we look at their coordinates:

1. Triangle \(ABC\):

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

2. Triangle \(A''B''C''\):

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

Now let's analyze their transformations to determine a correct description:

Step 1: Find a translation
To move from triangle \(ABC\) to triangle \(A''B''C''\), we can calculate the differences in the x and y coordinates:

• Moving point \(A(-4, 5)\) to \(A''(1, 0)\):

  • Change in \(x\): \(1 - (-4) = 5\)
  • Change in \(y\): \(0 - 5 = -5\)

• This suggests that if we move \(ABC\) by adding (5, -5), we would get points in the vicinity of where triangle \(A''B''C''\) exists.

Step 2: Determine if reflection is needed
After applying the translation, we determine if a reflection across the x-axis is required to align the triangles.

Considering that:

• Reflecting across the x-axis changes point \((x, y)\) to \((x, -y)\).

Finally, we notice that the transformations affect orientation between triangles.

Given that the triangles exhibit different orientation from \(ABC\) to \(A''B''C''\), it is clear that triangle \(ABC\) must go through a reflection.

Conclusion: The most accurate description of the transformations to achieve congruency between triangle \(A''B''C''\) and triangle \(ABC\) would be a translation followed by a reflection across the x-axis.

Thus, among the options, the description that fits would be:

"Translation of \((5,-5)\) and then reflection across the x-axis shows that triangle ABC is congruent to triangle A′′B′′C′′."

However, it is notable that the choices provided may not directly indicate a specific translation vector that leads perfectly. Thus, among the possible choices reflecting the alignment of vertices, I cannot find a direct match with the reflections mentioned in your options. You may need to verify the exact transformations.