To show that circle A is similar to circle B, we need to demonstrate that circle A can be transformed into circle B through a series of geometric transformations. Two circles are similar if they have the same shape; in this case, circles are defined as similar if they can be made congruent to one another through scaling, translation, and/or rotation.
Here are the steps to show similarity:
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Translate Circle A: We first translate circle A using the rule (x + 3, y - 2). This means moving the center of circle A from (4, 5) to (4 + 3, 5 - 2), which would give a new center of (7, 3). However, translation does not affect the size or the radius; circle A still remains with a radius of 3.
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Dilate Circle A: Next, we will dilate circle A by a scale factor of 3. This will increase the radius of circle A from 3 to 3 * 3 = 9. After dilation, the new radius of circle A matches the radius of circle B, which is also 9. The center remains the same at (7, 3) after dilation.
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Rotate Circle A (Optional): We can rotate the circle, but for similarity, it's not necessary because the radius and shape have been adjusted to match circle B. The rotation does not affect similarity in this scenario.
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Reflect Circle A (Optional): Reflecting is also not necessary for proving similarity, as we have already established that scaling produces the comparable size.
So, the vital steps that demonstrate that circle A is similar to circle B are the translation and the dilation. The key transformation that shows a clear similarity is dilation, since it changes the size of circle A to exactly match the size of circle B while maintaining its overall shape.
Conclusion: Circle A can be transformed into circle B through translation and dilation, establishing their similarity based on the matching radii after the dilation. The correct answer from your options primarily focuses on the dilation step as the essential transformation for similarity.
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