To determine which transformation or sequence of transformations would produce an image that is not congruent to its pre-image, we need to understand the nature of each transformation and its effect on congruence.
1. A translation of [tex]\((x-2, y+9)\)[/tex]:
- Translation is the process of shifting every point of a shape the same distance in the same direction. This does not change the size or shape of the object, only its position.
- Therefore, a translation preserves congruence.
2. A translation followed by a rotation:
- A translation followed by a rotation involves two steps:
- Translation moves the shape without altering its size or angle.
- Rotation spins the shape around a point (in this case, likely the origin) but does not change its size.
- Both steps preserve congruence.
- Therefore, a translation followed by a rotation preserves congruence.
3. A rotation of 270 degrees about the origin:
- Rotation involves turning the shape around a point, and this transformation does not alter the size or shape of the object, only its orientation.
- Therefore, a rotation preserves congruence.
4. A dilation followed by a translation:
- Dilation changes the size of a shape by a scale factor, either enlarging or reducing it. This transformation does not preserve congruence, as the size of the shape is altered.
- Translation moves the shape but does not alter its size or angle.
- Because dilation alters the size of the shape, it does not preserve congruence.
- Therefore, a dilation followed by a translation does not preserve congruence.
Given these explanations, the correct answer is:
A dilation followed by a translation would produce an image that is not congruent to its pre-image.
