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Which composition of similarity transformations maps polygon [tex]A B C D[/tex] to polygon [tex]A^{\prime} B^{\prime} C^{\prime} D^{\prime}[/tex]?

A. A dilation with a scale factor of [tex]\frac{1}{4}[/tex] and then a rotation
B. A dilation with a scale factor of [tex]\frac{1}{4}[/tex] and then a translation
C. A dilation with a scale factor of 4 and then a rotation
D. A dilation with a scale factor of 4 and then a translation

Sagot :

GinnyAnswer
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]
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