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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-axis shows that triangle ABC is congruent to triangle A′′B"C"

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

Since triangles ABC and A′′B"C" 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"

1 answer

To analyze the transformations between triangle ABC and triangle A′′B′′C′′, let's first look at the coordinates of the triangles:

  • Triangle ABC:

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

  • Triangle A′′B′′C′′:

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

Step 1: Find the transformations.

  1. Reflection across the x-axis: This changes the y-coordinates by negating them.

    • A (4, -5), B (1, -3), C (3, -1) — this does not match A′′B′′C′′.

  2. Reflection across the y-axis: This changes the x-coordinates by negating them.

    • A (4, 5), B (1, 3), C (3, 1) — this does not match A′′B′′C′′.

  3. Rotation: We can test various rotations, but we also need to see if straight translation might work.

Step 2: Check translations.

To find if a more complex transformation like a combination of transformation works, we can check for a simple translation first.

  • If we take triangle ABC's coordinates:
    • A (-4, 5) to A′′ (1, 0) — needs a transformation of (5, -5).
    • B (-1, 3) to B′′ (-2, -2) — needs a transformation of (-1, -5).
    • C (-3, 1) to C′′ (0, -4) — needs a transformation of (3, -5).

The differences show they don't align with simple translations of the original triangle directly.

Conclusion

Looking at the provided options:

  • Rotation of 90 degrees clockwise and then reflection across the x-axis shows that triangle ABC is congruent to triangle A′′B′′C′′: Not true based on coordinates.

  • Translation of (2,0) and then reflection across the x-axis shows that triangle ABC is congruent to triangle A′′B′′C′′: Not true based on coordinates.

  • Since triangles ABC and A′′B′′C′′ do not have the same orientation, they are not congruent: This is likely true as their orientations differ.

  • Reflection across the y-axis and then a translation of (−3,−5) shows that triangle ABC is congruent to triangle A′′B′′C′′: Not valid.

Thus, the best response is: Since triangles ABC and A′′B′′C′′ do not have the same orientation, they are not congruent.