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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 .

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 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.

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

Answer 1

To describe the series of transformations that shows the relationship between triangles ABC and A''B''C'', we can analyze the coordinates given for each triangle.

1. Triangle ABC has vertices at:

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

2. Triangle A''B''C'' has vertices at:

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

Now, let's evaluate the transformation step by step:

• Translation from ABC to A''B''C'':

  • If we translate each point by the vector (2, 0), we note that:
    • A would move from (-4, 5) to (-2, 5) (not matching).
    • B would move from (-1, 3) to (1, 3) (matching A'').
    • C would move from (-3, 1) to (-1, 1) (not matching).
  • Thus, translation alone doesn’t produce A''B''C'' directly.

• Checking for further transformation: If we reflect ABC across the x-axis after some translation:

  • Reflecting (-4, 5) gives (-4, -5).
  • Reflecting (-1, 3) gives (-1, -3).
  • Reflecting (-3, 1) gives (-3, -1).
  • None of these points align with A'', B'', or C''.

• Alternative reflection across y-axis and translation:

  • Reflection across the y-axis would change the signs of the x-coordinates:
    • A goes from (-4, 5) to (4, 5).
    • B goes from (-1, 3) to (1, 3).
    • C goes from (-3, 1) to (3, 1).
  • This does not simplify the transformation uniquely to A''B''C''.

• Considering the congruency and orientation: If the mentioned transformations do not yield the correct coordinates, we can conclude that two triangles might not be congruent if they do not have identical orientations.

Based on the evaluations: The best possible answer is: Translation of (2,0) and then reflection across the x-axis shows that triangle ABC is congruent to triangle A′′B′′C′′.

To confirm this, you may need to revert back and understand if they precisely yield the right outputs under successive operations. But based on the options and typical transformation properties, this is the most appropriate choice from the provided selections.