Question

How would you describe this series of transformations? (1 point) Responses 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 . 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.

Answer 1

The series of transformations described involves a combination of reflections, translations, and rotations applied to triangle ABC to produce triangle A''B''C''.

1. Transformation 1: Reflection across the y-axis and then a translation of (-3, -5):

   • This transformation reflects triangle ABC across the y-axis, which changes the orientation of the triangle. Following the reflection, the translation shifts the triangle 3 units to the left and 5 units down. The result is triangle A''B''C'', which is congruent to triangle ABC since reflections and translations preserve distance and shape.

2. Transformation 2: Translation of (2, 0) and then reflection across the x-axis:

   • In this case, triangle ABC is first translated 2 units to the right (which does not affect the orientation), and then it is reflected across the x-axis, which flips the triangle over the x-axis, changing its orientation. This results in triangle A''B''C'', which is congruent to triangle ABC due to the properties of translations and reflections.

3. Transformation 3: Rotation of 90 degrees clockwise and then reflection across the x-axis:

   • Here, triangle ABC is rotated 90 degrees clockwise, which changes its orientation. Subsequently, reflecting it across the x-axis further alters the orientation, resulting in triangle A''B''C''. This final triangle is indeed congruent to triangle ABC despite the changes in orientation as both transformations preserve the triangle's size and shape.

In summary, all mentioned transformations result in a triangle that maintains congruence with the original triangle ABC, though some steps may alter its orientation. However, it's noted that triangle ABC and triangle A''B''C'' do not have the same orientation, emphasizing that congruence relies not only on equality in shape and size but also orientation in certain contexts.