When determining which transformation or sequence of transformations would produce an image that is not congruent to its pre-image, it's important to understand the properties of each of these transformations.
1. A translation followed by a reflection:
- Translation involves moving every point of a shape a certain distance in a specified direction without changing the shape’s size or its internal angles.
- Reflection produces a mirror image across a specified axis.
- Both translation and reflection are rigid transformations, meaning they preserve the size and shape of the figure. Hence, the resulting image will be congruent to the pre-image.
2. A rotation of 90 degrees about the origin:
- Rotation is another form of rigid transformation. When a shape is rotated, each point of the shape is turned around a fixed point (the origin, in this case) through a specified angle.
- A rotation preserves the size and shape of the figure, so the resulting image will remain congruent to the pre-image.
3. A dilation followed by a rotation:
- Dilation is a non-rigid transformation that changes the size of the figure by a certain scale factor while preserving its shape. If the scale factor is not 1, the size of the figure will change.
- Rotation, as previously explained, is a rigid transformation that maintains size and shape but changes orientation.
- Since dilation changes the size of the figure, the image produced after dilation followed by rotation will not be congruent to the pre-image.
4. A translation of [tex]\((x+3, y+4)\)[/tex]:
- This is a specific type of translation where every point is moved 3 units in the x-direction and 4 units in the y-direction.
- Translation, as noted, is a rigid transformation that does not change the size or shape of the figure.
- Therefore, the resulting image will be congruent to the pre-image.
Given the properties of these transformations, a dilation followed by a rotation will produce an image that is not congruent to its pre-image because dilation alters the size of the figure, making the final image different in size from the original. Thus, the correct sequence that does not produce a congruent image is:
Answer: A dilation followed by a rotation
