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)
Analyzes the possible transformations: dilation and either rotation or translation.
Considers both cases for dilation: scale factor of 4 1 (shrink) and 4 (enlarge).
Considers both cases for the second transformation: rotation or translation.
Concludes that without a visual representation, a definitive answer is impossible, but if polygon A ′ B ′ C ′ D ′ is larger and rotated, the answer is dilation with a scale factor of 4 and then a rotation.
Explanation
Problem Analysis Let's analyze the problem. We need to determine the composition of similarity transformations that 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. Without a visual representation of the polygons, we must consider all possibilities and make a logical deduction.
Dilation Analysis First, consider the dilation. A dilation with a scale factor less than 1 (like 4 1 ) will shrink the polygon, while a scale factor greater than 1 (like 4 ) will enlarge it. We need to determine if A ′ B ′ C ′ D ′ is larger or smaller than A BC D . Since we don't have the image, we will consider both cases.
Rotation/Translation Analysis Next, consider the second transformation. If A ′ B ′ C ′ D ′ is simply a scaled version of A BC D without any change in orientation, then a translation would be the appropriate second transformation. However, if A ′ B ′ C ′ D ′ is rotated relative to A BC D , then a rotation would be needed. Again, without the image, we must consider both cases.
Conclusion Since we don't have the visual representation of the polygons, we cannot definitively determine the correct answer. However, we can analyze the options. If polygon A ′ B ′ C ′ D ′ is smaller than polygon A BC D , the scale factor must be 4 1 . If polygon A ′ B ′ C ′ D ′ is larger than polygon A BC D , the scale factor must be 4 . If polygon A ′ B ′ C ′ D ′ is rotated compared to polygon A BC D , then a rotation is needed. If polygon A ′ B ′ C ′ D ′ is not rotated compared to polygon A BC D , then a translation is needed. Without further information, we cannot choose a single correct answer. However, if we assume that polygon A ′ B ′ C ′ D ′ is larger than polygon A BC D and also rotated, then the correct answer would be 'a dilation with a scale factor of 4 and then a rotation'.
Examples
Similarity transformations are used in computer graphics to scale, rotate, and translate objects on the screen. For example, when you zoom in on a map on your phone, the map undergoes a dilation. When you rotate an image, it undergoes a rotation. And when you move an icon on your screen, it undergoes a translation. These transformations are fundamental to creating interactive and dynamic user interfaces.
The composition of transformations that maps polygon A BC D to polygon A ′ B ′ C ′ D ′ is likely a dilation with a scale factor of 4 followed by a rotation. Without the visual representation, this conclusion assumes that polygon A ′ B ′ C ′ D ′ is larger than A BC D and may also be rotated. Therefore, the best choice is option C.
;
