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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

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 .

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.

Answer 1

To understand the transformations between the triangles, we need to analyze the coordinates and the relationships between them.

1. Triangle ABC

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

2. Triangle A'B'C'

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

3. Triangle A''B''C''

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

Observations:

• Triangle A'B'C' appears to be a reflection of triangle ABC across the y-axis. This is evident because the x-coordinates of A, B, and C become the negatives when reflected (A to A', B to B', C to C').
• A" (1, 0) and the other points belong to a triangle that appears to be translated and possibly reflected downwards from triangle A'B'C'.

Transformations between triangles:

Upon reflection and transformation, we see:

• From ABC to A'B'C', there is a reflection across the y-axis and a translation.
• From A'B'C' to A''B''C'', a translation downward and another transformation.

Conclusion:

After examining the transformations, the best description matching these observations is:

"Reflection across the y-axis and then a translation of (−3,−5) shows that triangle ABC is congruent to triangle A′′B"C."

Thus, the correct response would be: "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."