Question

1. select all the sets of transformations that result in a congruent image when performed in any order.

□ translation, dilation with center (0, 1)
□ two translations
□ vertical translation, reflection over the y - axis
□ reflection over the y - axis, dilation with center (0, 0)
□ two reflections over the y - axis
□ two rotations about (3, 3)

Explanation:

Congruent images preserve size and shape, so only rigid transformations (translation, reflection, rotation) work—dilation changes size, so any set with dilation is invalid.

1. Translation + dilation: Dilation changes size → not congruent.
2. Two translations: Both are rigid; order doesn't affect congruence → valid.
3. Vertical translation + y-axis reflection: Both are rigid; order doesn't affect congruence → valid.
4. Reflection + dilation: Dilation changes size → not congruent.
5. Two y-axis reflections: Reflecting twice cancels out to original, which is congruent; order doesn't matter → valid.
6. Two rotations about (3,3): Rotations are rigid; composing two rotations about the same point is still a rigid transformation → valid.

Answer:

• two translations
• vertical translation, reflection over the y-axis
• two reflections over the y-axis
• two rotations about (3, 3)