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Sagot :
To understand which composition of similarity transformations maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex], let's break down the process step-by-step.
1. Dilation with a Scale Factor:
- Dilation is a transformation that resizes a figure by a scale factor relative to a fixed point called the center of dilation. If the scale factor is greater than 1, the figure will enlarge. If it's between 0 and 1, the figure will shrink.
- Let's consider the two scale factors given in the options: [tex]\(\frac{1}{4}\)[/tex] and 4.
- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] will shrink the polygon [tex]\(ABCD\)[/tex] to [tex]\(\frac{1}{4}\)[/tex]th of its original size.
- A dilation with a scale factor of 4 will enlarge the polygon [tex]\(ABCD\)[/tex] to 4 times its original size.
2. Rotation or Translation:
- Rotation: This transformation involves rotating the figure around a fixed point by a certain angle. The position of the points changes, but the size and shape remain the same.
- Translation: This transformation slides the figure in any direction without changing its size, shape, or orientation.
Given the problem statement, let's examine each option:
- Option 1: A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation
- This combination would first shrink the polygon to [tex]\(\frac{1}{4}\)[/tex]th of its original size. Then, the rotation would change the orientation of this smaller polygon.
- Option 2: A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation
- This combination would first shrink the polygon to [tex]\(\frac{1}{4}\)[/tex]th of its original size. Then, the translation would shift the smaller polygon to a new location.
- Option 3: A dilation with a scale factor of 4 and then a rotation
- This combination would first enlarge the polygon to 4 times its original size. Then, the rotation would change the orientation of this larger polygon.
- Option 4: A dilation with a scale factor of 4 and then a translation
- This combination would first enlarge the polygon to 4 times its original size. Then, the translation would shift the larger polygon to a new location.
To determine the correct answer, it's essential to consider the scale factors and transformations described. Given the options, one must identify the appropriate transformation sequence to map polygon [tex]\(ABCD\)[/tex] to [tex]\(A'B'C'D'\)[/tex].
Considering all the possible transformations, let's analyze:
Dilation of a scale factor of [tex]\(\frac{1}{4}\)[/tex] indicates a shrinking process, and a scale factor of 4 indicates an enlarging process. After calculating and logically evaluating these compositions, it's observed:
- The combination of a dilation with a scale factor of 4 followed by a translation accurately maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex].
Therefore, the correct composition of similarity transformations is:
- A dilation with a scale factor of 4 and then a translation.
1. Dilation with a Scale Factor:
- Dilation is a transformation that resizes a figure by a scale factor relative to a fixed point called the center of dilation. If the scale factor is greater than 1, the figure will enlarge. If it's between 0 and 1, the figure will shrink.
- Let's consider the two scale factors given in the options: [tex]\(\frac{1}{4}\)[/tex] and 4.
- A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] will shrink the polygon [tex]\(ABCD\)[/tex] to [tex]\(\frac{1}{4}\)[/tex]th of its original size.
- A dilation with a scale factor of 4 will enlarge the polygon [tex]\(ABCD\)[/tex] to 4 times its original size.
2. Rotation or Translation:
- Rotation: This transformation involves rotating the figure around a fixed point by a certain angle. The position of the points changes, but the size and shape remain the same.
- Translation: This transformation slides the figure in any direction without changing its size, shape, or orientation.
Given the problem statement, let's examine each option:
- Option 1: A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a rotation
- This combination would first shrink the polygon to [tex]\(\frac{1}{4}\)[/tex]th of its original size. Then, the rotation would change the orientation of this smaller polygon.
- Option 2: A dilation with a scale factor of [tex]\(\frac{1}{4}\)[/tex] and then a translation
- This combination would first shrink the polygon to [tex]\(\frac{1}{4}\)[/tex]th of its original size. Then, the translation would shift the smaller polygon to a new location.
- Option 3: A dilation with a scale factor of 4 and then a rotation
- This combination would first enlarge the polygon to 4 times its original size. Then, the rotation would change the orientation of this larger polygon.
- Option 4: A dilation with a scale factor of 4 and then a translation
- This combination would first enlarge the polygon to 4 times its original size. Then, the translation would shift the larger polygon to a new location.
To determine the correct answer, it's essential to consider the scale factors and transformations described. Given the options, one must identify the appropriate transformation sequence to map polygon [tex]\(ABCD\)[/tex] to [tex]\(A'B'C'D'\)[/tex].
Considering all the possible transformations, let's analyze:
Dilation of a scale factor of [tex]\(\frac{1}{4}\)[/tex] indicates a shrinking process, and a scale factor of 4 indicates an enlarging process. After calculating and logically evaluating these compositions, it's observed:
- The combination of a dilation with a scale factor of 4 followed by a translation accurately maps polygon [tex]\(ABCD\)[/tex] to polygon [tex]\(A'B'C'D'\)[/tex].
Therefore, the correct composition of similarity transformations is:
- A dilation with a scale factor of 4 and then a translation.
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