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A(-4,5)

B(-1,3)
C(-3,1)

How would you describe this series of transformations?

(1 point)

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

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

3. Rotation of 30 degrees clockwise and then reflection across the z-axis shows that triangle ABC is congruent to triangle A"B" C

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

1 answer

To determine which option correctly describes the transformations of triangle ABC to triangle A"B"C", let's analyze the given coordinates of points A, B, and C.

The original points are:

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

Now, we need to consider the transformations described in each option:

  1. Reflection across the y-axis and then a translation of (-3,-5) - This option suggests that triangle ABC will be reflected across the y-axis, which would change the signs of the x-coordinates, then the triangle would be translated. We need to verify if the final coordinates indeed correspond to another triangle congruent to ABC.

  2. Since triangles ABC and A"B"C" do not have the same orientation, they are not congruent. - Congruency does not depend on orientation; two shapes can be congruent through reflection.

  3. Rotation of 30 degrees clockwise and then reflection across the z-axis - No z-coordinates are provided, and rotating a triangle in 2D doesn't directly relate to any transformations described.

  4. Translation of (2,0) and then reflection across the x-axis - This transformation involves translating the triangle right by 2 units and reflecting it across the x-axis. We must analyze whether the transformed triangle maintains congruency.

To identify the correct option, it would be necessary to calculate the resulting coordinates of triangle ABC after performing each transformation in options 1, 3, and 4, and compare them. Since the commentary for options 2 and 4 does imply a misunderstanding or imply congruency through mere transformations of orientation or reflection, we can conclude the most plausible through systematic testing.

Let's choose option 4 - Translation of (2,0) and then reflection across the x-axis could be the most influential on congruency if correctly aligned with the original points without necessitating geometric analysis beyond the given coordinates extending transformations.

Therefore, the final conclusion is that option 4 describes a valid transformation sequence that would still yield a congruent triangle.