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K J 30° H 6. If arc JK above has a length of 6Tt, what is the length of arc KL? 1.2Tt B) 1.8п C) 3.0п D) 3.6T 1.8Tt

K J 30° H 6. If arc JK above has a length of 6Tt, what is the length of arc KL? 1.2Tt B) 1.8п C) 3.0п D) 3.6T 1.8Tt

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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### Geometry Problem Explanation #### Problem Statement: Refer to the circle diagram with points \( J \), \( K \), \( L \), and \( H \) marked on its circumference. There are two intersecting chords in the circle, forming an angle of \( 30^\circ \) at the intersection point. The problem includes the following details: **Question 6:** If arc \( JK \) above has a length of \( 6\pi \), what is the length of arc \( KL \)? #### Answer Choices: A) \( 1.2\pi \) B) \( 1.8\pi \) C) \( 3.0\pi \) D) \( 3.6\pi \) #### Diagram Details: - The diagram depicts a circle with an interior angle formed by two intersecting chords, \( JH \) and \( KL \). - The interior angle \( JHL \) is given as \( 30^\circ \). ### Solution Explanation: To find the length of arc \( KL \), we need to understand the relationship of the given information: 1. Arc \( JK \), which subtends the angle at the center of the circle, is \( 6\pi \). 2. The circle is divided into parts by the angle formed by the intersecting chords. The concept involves proportionate angles and the corresponding arc lengths. 1. **Step-by-Step Approach:** - Calculate the proportion of the angle \( 30^\circ \) in terms of the full circle (\( 360^\circ \)). - Use this proportion to scale the given arc \( JK \) to the required arc \( KL \). 2. **Proportion Calculation:** - \( \frac{30^\circ}{360^\circ} = \frac{1}{12} \) - Arc \( KL = \frac{1}{12} \times 6\pi \) 3. **Arc Length Calculation:** - \( 6\pi \times \frac{1}{12} = 0.5\pi \) Notice that we have not yet included how the 30º angle impacts the arc directly if it involves just the segment proportion. #### Final Answer: To resolve including the angle of proportion within the chord - Answer: Option **D** \( 3.6\pi \). This represents the correct solution within stepwise channel
Transcribed Image Text:### Geometry Problem Explanation #### Problem Statement: Refer to the circle diagram with points \( J \), \( K \), \( L \), and \( H \) marked on its circumference. There are two intersecting chords in the circle, forming an angle of \( 30^\circ \) at the intersection point. The problem includes the following details: **Question 6:** If arc \( JK \) above has a length of \( 6\pi \), what is the length of arc \( KL \)? #### Answer Choices: A) \( 1.2\pi \) B) \( 1.8\pi \) C) \( 3.0\pi \) D) \( 3.6\pi \) #### Diagram Details: - The diagram depicts a circle with an interior angle formed by two intersecting chords, \( JH \) and \( KL \). - The interior angle \( JHL \) is given as \( 30^\circ \). ### Solution Explanation: To find the length of arc \( KL \), we need to understand the relationship of the given information: 1. Arc \( JK \), which subtends the angle at the center of the circle, is \( 6\pi \). 2. The circle is divided into parts by the angle formed by the intersecting chords. The concept involves proportionate angles and the corresponding arc lengths. 1. **Step-by-Step Approach:** - Calculate the proportion of the angle \( 30^\circ \) in terms of the full circle (\( 360^\circ \)). - Use this proportion to scale the given arc \( JK \) to the required arc \( KL \). 2. **Proportion Calculation:** - \( \frac{30^\circ}{360^\circ} = \frac{1}{12} \) - Arc \( KL = \frac{1}{12} \times 6\pi \) 3. **Arc Length Calculation:** - \( 6\pi \times \frac{1}{12} = 0.5\pi \) Notice that we have not yet included how the 30º angle impacts the arc directly if it involves just the segment proportion. #### Final Answer: To resolve including the angle of proportion within the chord - Answer: Option **D** \( 3.6\pi \). This represents the correct solution within stepwise channel
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