Which composition of similarity transformations maps polygon ABCD to polygon \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \) ? a dilation with a scale factor of \( \frac{1}{4} \) and then a rotation a dilation with a scale factor of \( \frac{1}{4} \) and then a translation a dilation with a scale factor of 4 and then a rotation a dilation with a scale factor of 4 and then a translation

To determine which composition of similarity transformations maps polygon ABCD to polygon \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \), we need to analyze the transformations involved. 1. **Dilation**: A dilation changes the size of a figure but keeps the shape the same. A scale factor of \( \frac{1}{4} \) reduces the size of the polygon, while a scale factor of 4 enlarges it. 2. **Rotation**: A rotation turns the figure around a fixed point, changing its orientation but not its size. 3. **Translation**: A translation moves the figure from one location to another without changing its size or orientation. To map polygon ABCD to polygon \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \): - If \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \) is smaller than ABCD, a dilation with a scale factor of \( \frac{1}{4} \) would be appropriate. - If the orientation of \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \) is different from ABCD, a rotation would be necessary after the dilation. - If the position of \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \) is simply a shift from ABCD without changing size or orientation, a translation would be appropriate after the dilation. Given the options: 1. A dilation with a scale factor of \( \frac{1}{4} \) and then a rotation 2. A dilation with a scale factor of \( \frac{1}{4} \) and then a translation 3. A dilation with a scale factor of 4 and then a rotation 4. A dilation with a scale factor of 4 and then a translation If we assume that \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \) is smaller than ABCD and possibly rotated, the correct choice would be: **A dilation with a scale factor of \( \frac{1}{4} \) and then a rotation.** If you have specific information about the sizes and orientations of the polygons, that would help confirm the correct transformation.