Which composition of similarity transformations maps polygon [tex]ABCD[/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
Answer (2)
Compare the size of polygon A ′ B ′ C ′ D ′ to polygon A BC D to determine the scale factor of dilation.
Compare the orientation of polygon A ′ B ′ C ′ D ′ to polygon A BC D to determine if a rotation or translation is involved.
If A ′ B ′ C ′ D ′ is smaller and has a different orientation, the transformation is a dilation with a scale factor of 4 1 and then a rotation.
If A ′ B ′ C ′ D ′ is larger and has the same orientation, the transformation is a dilation with a scale factor of 4 and then a translation.
Explanation
Problem Analysis Let's analyze the problem. We need to determine which composition of similarity transformations maps polygon A BC D to polygon A ′ B ′ C ′ D ′ . The options involve a dilation (either with a scale factor of 4 1 or 4 ) followed by either a rotation or a translation.
Size Comparison First, we need to compare the sizes of the two polygons. If polygon A ′ B ′ C ′ D ′ is smaller than polygon A BC D , the dilation has a scale factor of 4 1 . If polygon A ′ B ′ C ′ D ′ is larger than polygon A BC D , the dilation has a scale factor of 4 .
Orientation Comparison Next, we need to compare the orientation of the two polygons. If polygon A ′ B ′ C ′ D ′ has a different orientation compared to polygon A BC D , the transformation includes a rotation. If the orientation is the same, the transformation includes a translation.
Possible Transformations Based on the size and orientation comparisons, we can determine the correct composition of transformations. Without a visual representation of the polygons, it's impossible to definitively choose the correct answer. However, I will explain each option.
Option 1: A dilation with a scale factor of 4 1 and then a rotation. This means the polygon A ′ B ′ C ′ D ′ is smaller and has a different orientation. Option 2: A dilation with a scale factor of 4 1 and then a translation. This means the polygon A ′ B ′ C ′ D ′ is smaller and has the same orientation. Option 3: A dilation with a scale factor of 4 and then a rotation. This means the polygon A ′ B ′ C ′ D ′ is larger and has a different orientation. Option 4: A dilation with a scale factor of 4 and then a translation. This means the polygon A ′ B ′ C ′ D ′ is larger and has the same orientation.
Conclusion Since we don't have the image of the polygons, we can't determine the correct answer. However, let's assume polygon A ′ B ′ C ′ D ′ is smaller and has a different orientation. Then the correct answer would be a dilation with a scale factor of 4 1 and then a rotation.
Examples
Imagine you are creating a logo for a company. You start with an initial design ( A BC D ) and then need to create a smaller version ( A ′ B ′ C ′ D ′ ) that is also rotated to fit a specific space on a business card. This process involves a dilation (reducing the size) and a rotation (changing the orientation), similar to the transformations described in the problem. Understanding these transformations helps designers manipulate images and logos effectively.
The composition of transformations that maps polygon A BC D to polygon A ′ B ′ C ′ D ′ can be determined by comparing their sizes and orientations. If A ′ B ′ C ′ D ′ is smaller and rotated, the answer is Option A. Without visuals, we can't be certain, but analyzing size and orientation leads us to this conclusion.
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