Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3785ad64-93eb-4008-8832-b47a1a40c0c9%2Ff702a67c-83de-47fd-bbe1-2f6af2f3f116%2Ffj5d2t8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### 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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