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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.

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
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
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### 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.
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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