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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

Rotation of 90 degrees clockwise and then reflection across the x
x
-axis shows that triangle ABC
A
B
C
is congruent to triangle A′′B"C"
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
A
B
C
and A′′B"C"
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
y
-axis and then a translation of (−3,−5)
(
−
3
,
−
5
)
shows that triangle ABC
A
B
C
is congruent to triangle A′′B"C"
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)
(
2
,
0
)
and then reflection across the x
x
-axis shows that triangle ABC
A
B
C
is congruent to triangle A′′B"C"
A
″
B
"
C
"
.

Answer 1

To analyze the transformations between the triangles ABC, A′B′C′, and A″B″C″, let's examine the coordinates and the transformations involved.

1. Triangle ABC has vertices:

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

2. Triangle A′B′C′ has vertices:

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

3. Triangle A″B″C″ has vertices:

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

From this, we can deduce:

• The transition from triangle ABC to triangle A′B′C′ involves moving the points left to right (suggests a reflection across the y-axis, moving from negative x-values to positive x-values).

• The transformation from ABC to A″B″C″ involves a more complex movement where the points seem to indicate some rotation and possibly translation.

However, triangle A′′B′′C′′ does not seem to maintain the same orientation as triangle ABC—they appear to point in different directions.

The best fit for these transformations would consider the movements and reflections across axes, particularly focusing on the orientation and congruence statements provided.

Therefore, the correct description of the transformation series that leads to the conclusion about the congruence of triangle ABC with triangle A″B″C″ is:

Since triangles ABC and A″B″C″ do not have the same orientation, they are not congruent.

This description aligns with the transformations outlined and confirms the resulting congruence (or lack thereof) based on orientation.