To determine which transformation or sequence of transformations would produce an image that is not congruent to its pre-image, let's analyze each option:
1. A translation of [tex]\((x-2, y+9)\)[/tex]:
- Translation is a transformation that slides each point of a shape a certain distance in a specified direction. The shape, size, and orientation of the image remain the same, so the image is congruent to its pre-image.
2. A translation followed by a rotation:
- Translation and rotation are both rigid transformations. Translation will slide the shape without altering its size or angles, and rotation will turn the shape around a fixed point without changing its size or shape. Since neither transformation changes the size or shape of the figure, the resulting image is congruent to its pre-image.
3. A rotation of 270 degrees about the origin:
- Rotation is a rigid transformation that turns a figure around a point. In this case, rotating a figure 270 degrees around the origin will not change the size or shape of the figure. Thus, the image remains congruent to its pre-image.
4. A dilation followed by a translation:
- Dilation is a transformation that changes the size of a figure by a scale factor relative to a fixed point called the center of dilation. It can either enlarge or reduce the size of the figure. This change in size means that the figure will not remain congruent to its pre-image. Translation, which comes after dilation, will move the resized figure without altering its new size. Therefore, the resultant image will have a different size compared to its pre-image and hence will not be congruent.
Given this detailed analysis, the correct answer is:
A dilation followed by a translation.
