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

1 answer

The series of transformations described can be summarized as follows:

  1. Transformation Steps: Each transformation involves a combination of reflections, translations, and rotations applied to triangle ABC to produce new triangles (A" B" C", A"B" C, and A"B"C").

  2. Congruence Conditions: The statements highlight that congruence between triangles is not solely determined by having the same shape and size but also by their orientation in space. For example, even if two triangles might have the same dimensions, if one has a different orientation, they are not considered congruent.

  3. Types of Transformations:

    • Reflection: Flipping the triangle over a specific axis (y-axis or z-axis) can change its orientation.
    • Translation: Shifting the triangle without altering its shape or size.
    • Rotation: Rotating the triangle around a point affects its orientation.

  4. Final Result: The last transformation (translation and reflection across the r-axis) suggests that after these steps, triangle A"B"C" retains congruence with triangle ABC, meaning they have the same shape and size despite potential differences in orientation introduced by earlier steps.

Overall Conclusion: The transformations involve a careful consideration of how spatial manipulation affects the congruence of geometric figures, particularly emphasizing the importance of orientation in determining congruence.