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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 analyze the transformations between the triangles ABC and A″B″C″, we note the following:

1. Triangle ABC has vertices at A(-4, 5), B(-1, 3), and C(-3, 1).
2. Triangle A″B″C″ has vertices at A"(1, 0), B"(-2, -2), and C"(0, -4).

To go from triangle ABC to triangle A″B″C″, we can look at the transformations step by step:

• The first step could involve reflecting triangle ABC across the y-axis, which would take A(-4, 5) to A' (4, 5), B (-1, 3) to B' (1, 3), and C (-3, 1) to C' (3, 1).

• The second transformation would involve translating this reflected triangle A'B'C' downwards by moving all points down by 5 units and left by 3 units:

  • A' (4, 5) → (4 - 3, 5 - 5) = (1, 0)
  • B' (1, 3) → (1 - 3, 3 - 5) = (-2, -2)
  • C' (3, 1) → (3 - 3, 1 - 5) = (0, -4)

The key here is to determine that the first step was a reflection across the y-axis, and the second step was a translation.

For triangle A'B'C' and A″B″C″, they do share the same orientation since both triangles have been reflected and translated appropriately.

Therefore, the transformations resulting in the congruence of triangle ABC with triangle A″B″C″ can be described as:

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