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 effect each type of transformation has on a figure:
1. Rotation of 180 degrees about the origin:
- A rotation is a type of rigid transformation where the shape and size of the figure remain the same. Rotating a figure 180 degrees about the origin will produce a figure that is congruent to the original figure. Therefore, this transformation does not change congruency.
2. A dilation followed by a reflection:
- A dilation is a transformation that alters the size of a figure by a scale factor. This changes the size of the figure, making it either larger or smaller than the original. Therefore, a dilation produces a figure that is not congruent to the pre-image because congruent figures must have the same size and shape.
- A reflection is a rigid transformation that flips a figure over a line, but does not change its size or shape.
- Combining a dilation (which alters the size) and a reflection (which changes the orientation but keeps the shape and size post-dilation) results in a figure that is not congruent to the original figure due to the size change caused by the dilation.
3. A translation of [tex]\((x+5, y-2)\)[/tex]:
- A translation moves every point of a figure the same distance in the same direction. This type of transformation does not change the size or shape of the figure, so the figure remains congruent to the original.
4. A translation followed by a reflection:
- A translation, as mentioned, does not change the size or shape of the figure.
- A reflection will flip the figure, changing its orientation but not its size or shape.
- Therefore, the combination of a translation followed by a reflection results in a figure that is congruent to the original.
Based on this analysis, the only transformation or sequence of transformations that produces an image that is not congruent to its pre-image is:
A dilation followed by a reflection
Hence, the answer is:
Option 2: A dilation followed by a reflection
