Use the image to answer the question. .

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.

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 .

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 relate triangles ABC and A''B''C'', let's analyze the coordinates and their relationships.

1. Triangle ABC has vertices:

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

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

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

To determine if one triangle can be transformed into the other through congruent transformations, we must examine their positions:

• Translating triangle ABC to the right and downward towards the origin can be identified by shifting the positions.
• Checking the reflection across the x-axis flips the vertical positions of the points, which will affect the orientation.

After analyzing the transformations proposed:

• The statement regarding the translation of (2, 0) followed by a reflection across the x-axis makes sense because:
  • By first translating the triangle ABC to the right by 2 units and then reflecting it across the x-axis should yield triangle A''B''C''.

Thus, the correct response would be:

Translation of (2,0) and then reflection across the x-axis shows that triangle ABC is congruent to triangle A′′B′′C′′.