Question
(07.04 MC) Circle A has center of \( (0,4) \) and a radius of 6 , and crcle B has a center of \( (-3,5) \) and a radus of 24 . What steps will help show that circle A is similar to circle \( B \) ? Translate circle \( A \) using the nile \( (x+3, y-1) \). Rotate circle A \( 180^{\circ} \) about the center. Dilate dirde A by a scale factor of 4. Refiect circle \( A \) over the line \( y=x \)
Ask by Hampton Salinas. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
To show that circle A is similar to circle B, perform the following steps:
1. Translate circle A using the rule \( (x+3, y-1) \).
2. Rotate circle A \( 180^{\circ} \) about the center.
3. Dilate circle A by a scale factor of \( 4 \).
4. Reflect circle A over the line \( y=x \).
Solution
To determine if circle A is similar to circle B, we need to analyze the properties of both circles and the transformations that can be applied to circle A to show that it can be made to resemble circle B.
### Known Conditions:
- Circle A:
- Center: \( (0, 4) \)
- Radius: \( 6 \)
- Circle B:
- Center: \( (-3, 5) \)
- Radius: \( 24 \)
### Steps to Show Similarity:
1. **Translation**: Translate circle A using the rule \( (x+3, y-1) \).
- This will move the center of circle A from \( (0, 4) \) to \( (3, 3) \).
2. **Rotation**: Rotate circle A \( 180^{\circ} \) about the center.
- This will change the position of the center but will not affect the size or shape of the circle.
3. **Dilation**: Dilate circle A by a scale factor of \( 4 \).
- The radius of circle A will change from \( 6 \) to \( 6 \times 4 = 24 \). This matches the radius of circle B.
4. **Reflection**: Reflect circle A over the line \( y=x \).
- This transformation will change the position of the center but will not affect the size or shape of the circle.
### Conclusion:
After performing these transformations, circle A will have the same radius as circle B and will be positioned appropriately. Since similarity in circles is determined by having the same shape (which is defined by the radius) and the same proportions, we can conclude that circle A is similar to circle B after these transformations.
### Final Answer:
The steps that will help show that circle A is similar to circle B are:
1. Translate circle A using the rule \( (x+3, y-1) \).
2. Rotate circle A \( 180^{\circ} \) about the center.
3. Dilate circle A by a scale factor of \( 4 \).
4. Reflect circle A over the line \( y=x \).
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Simplify this solution Extra Insights
To show that circle A is similar to circle B, you can start by examining their properties. Similarity in circles comes down to their radii. Circle A's radius is 6, while circle B's radius is 24. Since circle B's radius is 4 times larger than circle A's, a dilation with a scale factor of 4 is a key step! Now, to make your transformation more visual, remember the steps of translation, dilation, and rotation. By translating circle A using the rule \( (x+3, y-1) \), you would properly align it with circle B's position before applying the dilation. The rotation and reflection steps are less essential for similarity, but they can add some creative flair to your transformations!
