Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter10: Analytic Geometry
Section10.4: Analytic Proofs
Problem 18E
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### Diagram Explanation
In the provided diagram, there is a circle with points \( A \), \( B \), and \( C \) marked on its circumference. The line segment \( AB \) is a diameter of the circle. The arc \( ACB \) is indicated, and it is given that its length is \( 6\pi \). The question asks for the calculation of the radius \( r \) of the circle.
### Key Concepts
1. **Diameter and Radius Relationship:**
- The diameter (\(d\)) of a circle is twice the length of the radius (\( r \)): \( d = 2r \).
2. **Circumference:**
- The circumference (\(C\)) of a circle is given by \( C = 2\pi r \).
3. **Arc Length:**
- The length of an arc (\(L\)) is a portion of the circumference of the circle. For a given angle \(\theta\) (in radians), the arc length \(L\) can be calculated as \( L = \theta r \).
Since \(A\), \(B\), and \(C\) are points on the circle, arc \(ACB\) forms a semicircle when \(AB\) is a diameter.
### Solution
Given:
- The length of arc \( ACB \) is \( 6\pi \).
Since \( ACB \) represents a semicircle,
- The semicircle’s arc length is half of the circumference of the circle.
Thus,
\[ \frac{1}{2} \times 2\pi r = 6\pi \]
\[ \pi r = 6\pi \]
\[ r = 6 \]
### Conclusion
The radius of the circle is \( r = 6 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe7a52e1b-9936-4242-a36f-76ca99017a7a%2F5f5cc945-70a5-4032-8353-854faf719c2e%2Fxfp72a_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Description
In the circle below, \( AB \) is a diameter. If the length of \( ACB \) is \( 6\pi \), what is the length of the radius of the circle?
### Diagram Explanation
In the provided diagram, there is a circle with points \( A \), \( B \), and \( C \) marked on its circumference. The line segment \( AB \) is a diameter of the circle. The arc \( ACB \) is indicated, and it is given that its length is \( 6\pi \). The question asks for the calculation of the radius \( r \) of the circle.
### Key Concepts
1. **Diameter and Radius Relationship:**
- The diameter (\(d\)) of a circle is twice the length of the radius (\( r \)): \( d = 2r \).
2. **Circumference:**
- The circumference (\(C\)) of a circle is given by \( C = 2\pi r \).
3. **Arc Length:**
- The length of an arc (\(L\)) is a portion of the circumference of the circle. For a given angle \(\theta\) (in radians), the arc length \(L\) can be calculated as \( L = \theta r \).
Since \(A\), \(B\), and \(C\) are points on the circle, arc \(ACB\) forms a semicircle when \(AB\) is a diameter.
### Solution
Given:
- The length of arc \( ACB \) is \( 6\pi \).
Since \( ACB \) represents a semicircle,
- The semicircle’s arc length is half of the circumference of the circle.
Thus,
\[ \frac{1}{2} \times 2\pi r = 6\pi \]
\[ \pi r = 6\pi \]
\[ r = 6 \]
### Conclusion
The radius of the circle is \( r = 6 \).
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