Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Discover the answers you need from a community of experts ready to help you with their knowledge and experience in various fields. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To determine which composition of similarity transformations maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex], we need to consider the given options systematically.
### Options
1. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation:
- Dilation: If we apply a dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex], the size of the polygon [tex]\(ABCD\)[/tex] will shrink to [tex]\(\frac{1}{4}\)[/tex] of its original size.
- Rotation: After shrinking, if we apply a rotation, the new polygon would be the same size but rotated. This doesn't account for translation, only a change in orientation and size.
2. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation:
- Dilation: When we apply the dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex], the polygon [tex]\(ABCD\)[/tex] will again shrink to [tex]\(\frac{1}{4}\)[/tex] of its original size.
- Translation: Following the dilation, if we then apply a translation (a shift in position without altering the shape or size), we can place the resized polygon in the correct location to align with [tex]\(A'B'C'D'\)[/tex].
3. A dilation with a scale factor of 4 and then a rotation:
- Dilation: Applying a dilation with a scale factor of 4 will enlarge the polygon [tex]\(ABCD\)[/tex] to four times its original size.
- Rotation: After enlarging, a rotation will only change the orientation and not address any positional shift needed for alignment with [tex]\(A'B'C'D'\)[/tex].
4. A dilation with a scale factor of 4 and then a translation:
- Dilation: With a scale factor of 4, the polygon [tex]\(ABCD\)[/tex] will expand to four times its initial size.
- Translation: A subsequent translation will relocate the enlarged polygon, but this composition doesn’t align with the typically observed need for reduction in size to meet a smaller target polygon, as outlined in the problem description.
### Conclusion
Given the options, the transformation that effectively adjusts the size correctly to [tex]\(\frac{1}{4}\)[/tex] and subsequently places the resized polygon in the desired position accurately without altering orientation is:
A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
Thus, the correct composition of similarity transformations to map polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex] is:
[tex]\(\boxed{2}\)[/tex]
### Options
1. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation:
- Dilation: If we apply a dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex], the size of the polygon [tex]\(ABCD\)[/tex] will shrink to [tex]\(\frac{1}{4}\)[/tex] of its original size.
- Rotation: After shrinking, if we apply a rotation, the new polygon would be the same size but rotated. This doesn't account for translation, only a change in orientation and size.
2. A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation:
- Dilation: When we apply the dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex], the polygon [tex]\(ABCD\)[/tex] will again shrink to [tex]\(\frac{1}{4}\)[/tex] of its original size.
- Translation: Following the dilation, if we then apply a translation (a shift in position without altering the shape or size), we can place the resized polygon in the correct location to align with [tex]\(A'B'C'D'\)[/tex].
3. A dilation with a scale factor of 4 and then a rotation:
- Dilation: Applying a dilation with a scale factor of 4 will enlarge the polygon [tex]\(ABCD\)[/tex] to four times its original size.
- Rotation: After enlarging, a rotation will only change the orientation and not address any positional shift needed for alignment with [tex]\(A'B'C'D'\)[/tex].
4. A dilation with a scale factor of 4 and then a translation:
- Dilation: With a scale factor of 4, the polygon [tex]\(ABCD\)[/tex] will expand to four times its initial size.
- Translation: A subsequent translation will relocate the enlarged polygon, but this composition doesn’t align with the typically observed need for reduction in size to meet a smaller target polygon, as outlined in the problem description.
### Conclusion
Given the options, the transformation that effectively adjusts the size correctly to [tex]\(\frac{1}{4}\)[/tex] and subsequently places the resized polygon in the desired position accurately without altering orientation is:
A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation.
Thus, the correct composition of similarity transformations to map polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex] is:
[tex]\(\boxed{2}\)[/tex]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.