To determine which transformation or sequence of transformations would produce an image that is not congruent to its pre-image, we need to analyze each option given:
1. A translation of [tex]\((x-2, y+9)\)[/tex]:
- Translation involves shifting the figure without altering its shape or size. Thus, the image remains congruent to its pre-image. This option cannot be the correct answer.
2. A translation followed by a rotation:
- Translation shifts the figure, and rotation spins the figure around a point, commonly the origin. Neither of these transformations changes the size or shape of the figure, maintaining congruency between the image and the pre-image. This option cannot be the correct answer.
3. A rotation of 270 degrees about the origin:
- Rotation changes the orientation of the figure but not its size or shape. As a result, the image remains congruent to the pre-image. This option cannot be the correct answer.
4. A dilation followed by a translation:
- Dilation changes the size of the figure, either enlarging or reducing it, hence altering its shape and size proportionally. This transformation affects congruency as the image is no longer the same size as the pre-image. Additionally, a subsequent translation moves the transformed figure but does not restore the original size or shape. Therefore, this sequence of transformations results in an image that is not congruent to its pre-image.
Given the analysis, the transformation that produces an image which is not congruent to its pre-image is:
A dilation followed by a translation.
Thus, the correct answer is:
4. A dilation followed by a translation.
