Answer :
To determine which composition of similarity transformations maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex], we need to consider each transformation method given in the choices and identify which one correctly accomplishes the task.
### Understanding Similarity Transformations
Similarity transformations include translations, rotations, reflections, and dilations (scaling). For two polygons to be similar, their corresponding angles must be equal, and their corresponding side lengths must be proportional by a constant scale factor.
### Analyzing the Options
1. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation:
- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] will scale down all lengths of polygon [tex]\(ABCD\)[/tex] to one-fourth of their original lengths.
- After this scaling, a rotation will re-orient the smaller polygon [tex]\(ABCD\)[/tex] to match the orientation of [tex]\(A'B'C'D'\)[/tex]. This step doesn't alter the side lengths, only the orientation.
- Therefore, this combination might work, but further verification with other options is required.
2. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation:
- Similar to the previous option, the dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] will shrink the polygon [tex]\(ABCD\)[/tex] to one-fourth of its original size.
- A translation will move the polygon to a new position without rotating or changing its shape or size further.
- This ensures that the polygons [tex]\(ABCD\)[/tex] and [tex]\(A'B'C'D'\)[/tex] are not only the same shape and size but also in the correct position. Hence, this transformation is a suitable candidate.
3. A dilation with a scale factor of 4 and then a rotation:
- A dilation with a scale factor of 4 will enlarge polygon [tex]\(ABCD\)[/tex] by a factor of 4.
- Next, a rotation will reorient the larger polygon to match [tex]\(A'B'C'D'\)[/tex]'s orientation. However, the size of the polygon [tex]\(ABCD\)[/tex] has increased rather than decreased to match [tex]\(A'B'C'D'\)[/tex], hence this option would not map [tex]\(ABCD\)[/tex] to [tex]\(A'B'C'D'\)[/tex] correctly.
4. A dilation with a scale factor of 4 and then a translation:
- Similar to the previous option, this dilation will enlarge the polygon [tex]\(ABCD\)[/tex] by a factor of 4.
- The translation step will then move the enlarged polygon to a different location without adjusting the orientation.
- Again, the misalignment in the side lengths (being 4 times larger instead of a fraction) indicates that this option would not work.
### Conclusion
The suitable composition of similarity transformations that maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex] must involve shrinking the polygon by a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then translating it to the correct position.
Thus, the correct answer is:
- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
### Understanding Similarity Transformations
Similarity transformations include translations, rotations, reflections, and dilations (scaling). For two polygons to be similar, their corresponding angles must be equal, and their corresponding side lengths must be proportional by a constant scale factor.
### Analyzing the Options
1. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation:
- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] will scale down all lengths of polygon [tex]\(ABCD\)[/tex] to one-fourth of their original lengths.
- After this scaling, a rotation will re-orient the smaller polygon [tex]\(ABCD\)[/tex] to match the orientation of [tex]\(A'B'C'D'\)[/tex]. This step doesn't alter the side lengths, only the orientation.
- Therefore, this combination might work, but further verification with other options is required.
2. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation:
- Similar to the previous option, the dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] will shrink the polygon [tex]\(ABCD\)[/tex] to one-fourth of its original size.
- A translation will move the polygon to a new position without rotating or changing its shape or size further.
- This ensures that the polygons [tex]\(ABCD\)[/tex] and [tex]\(A'B'C'D'\)[/tex] are not only the same shape and size but also in the correct position. Hence, this transformation is a suitable candidate.
3. A dilation with a scale factor of 4 and then a rotation:
- A dilation with a scale factor of 4 will enlarge polygon [tex]\(ABCD\)[/tex] by a factor of 4.
- Next, a rotation will reorient the larger polygon to match [tex]\(A'B'C'D'\)[/tex]'s orientation. However, the size of the polygon [tex]\(ABCD\)[/tex] has increased rather than decreased to match [tex]\(A'B'C'D'\)[/tex], hence this option would not map [tex]\(ABCD\)[/tex] to [tex]\(A'B'C'D'\)[/tex] correctly.
4. A dilation with a scale factor of 4 and then a translation:
- Similar to the previous option, this dilation will enlarge the polygon [tex]\(ABCD\)[/tex] by a factor of 4.
- The translation step will then move the enlarged polygon to a different location without adjusting the orientation.
- Again, the misalignment in the side lengths (being 4 times larger instead of a fraction) indicates that this option would not work.
### Conclusion
The suitable composition of similarity transformations that maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex] must involve shrinking the polygon by a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then translating it to the correct position.
Thus, the correct answer is:
- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
