### Proving Circle Similarity#### Problem Statement:Prove that circle $ A $ with center $ (0, 4) $ and radius $ 4 $ is similar to circle $ B $ with center $ (-2, -7) $ and radius $ 6 $.#### Explanation:To prove that two circles are similar, we need to show that one can be transformed into the other using a combination of scaling (resizing) and translation (repositioning).1. **Circle Centers**: - Circle $ A $ has its center at $ (0, 4) $. - Circle $ B $ has its center at $ (-2, -7) $.2. **Radius**: - Circle $ A $ has a radius of $ 4 $. - Circle $ B $ has a radius of $ 6 $.### Steps for Proof:1. **Scaling**: - The ratio of the radii of Circle $ B $ to Circle $ A $ is $ \frac{6}{4} = 1.5 $. - This ratio indicates that Circle $ B $ is a scaled version of Circle $ A $ by a factor of 1.5.2. **Translation**: - To match the centers, Circle $ A $ needs to be translated such that its center moves from $ (0, 4) $ to $ (-2, -7) $. - The movement required is $ \Delta x = -2 - 0 = -2 $ and $ \Delta y = -7 - 4 = -11 $.By applying these transformations: 1. Scale Circle $ A $ by a factor of 1.5. 2. Translate the scaled Circle $ A $ by $ \Delta x = -2 $ and $ \Delta y = -11 $.Circle $ A $ can be transformed into Circle $ B $, proving that they are similar.### Conclusion:Circle $ A $ with center $ (0, 4) $ and radius $ 4 $ is similar to Circle $ B $ with center $ (-2, -7) $ and radius $ 6 $, due to the possibility of transforming one into the other through scaling and translation.

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i. Prove that circle A with center (0, 4) and radius 4 is similar to circle B with center (-2, -7) and radius 6.

Elementary Geometry For College Students, 7e

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ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

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Question

### Proving Circle Similarity

#### Problem Statement:

Prove that circle $ A $ with center $ (0, 4) $ and radius $ 4 $ is similar to circle $ B $ with center $ (-2, -7) $ and radius $ 6 $.

#### Explanation:

To prove that two circles are similar, we need to show that one can be transformed into the other using a combination of scaling (resizing) and translation (repositioning).

1. **Circle Centers**:
- Circle $ A $ has its center at $ (0, 4) $.
- Circle $ B $ has its center at $ (-2, -7) $.

2. **Radius**:
- Circle $ A $ has a radius of $ 4 $.
- Circle $ B $ has a radius of $ 6 $.

### Steps for Proof:

1. **Scaling**:
- The ratio of the radii of Circle $ B $ to Circle $ A $ is $ \frac{6}{4} = 1.5 $.
- This ratio indicates that Circle $ B $ is a scaled version of Circle $ A $ by a factor of 1.5.

2. **Translation**:
- To match the centers, Circle $ A $ needs to be translated such that its center moves from $ (0, 4) $ to $ (-2, -7) $.
- The movement required is $ \Delta x = -2 - 0 = -2 $ and $ \Delta y = -7 - 4 = -11 $.

By applying these transformations:
1. Scale Circle $ A $ by a factor of 1.5.
2. Translate the scaled Circle $ A $ by $ \Delta x = -2 $ and $ \Delta y = -11 $.

Circle $ A $ can be transformed into Circle $ B $, proving that they are similar.

### Conclusion:

Circle $ A $ with center $ (0, 4) $ and radius $ 4 $ is similar to Circle $ B $ with center $ (-2, -7) $ and radius $ 6 $, due to the possibility of transforming one into the other through scaling and translation.

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