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
![### Understanding Circle Arc Length
In the diagram, we have a circle with center \( D \). The points \( A \), \( B \), and \( C \) are on the circumference of the circle. The segment \( AC \) is an internal chord of the circle, creating an angle of \( 140^\circ \) at the center, denoted as \( \angle ADC = 140^\circ \). The radius of the circle is 4 meters.
The task is to determine the length of the arc \( ABC \), which is the arc passing through points \( A \), \( B \), and \( C \).
#### Steps to Calculate the Length of Arc \( ABC \)
1. **Understand the central angle**: The central angle \(\angle ADC = 140^\circ\).
2. **Convert the angle to radians**:
\[
\text{Angle in radians} = 140^\circ \times \frac{\pi}{180^\circ} = \frac{140\pi}{180} = \frac{7\pi}{9} \text{ radians}
\]
3. **Arc Length Formula**:
\[
\text{Arc Length} = \text{Radius} \times \text{Angle in Radians}
\]
\[
\text{Arc Length of ABC} = 4 \text{ meters} \times \frac{7\pi}{9}
\]
\[
\text{Arc Length of ABC} = \frac{28\pi}{9} \text{ meters}
\]
So the length of the arc \(ABC\) is:
\[
\frac{28\pi}{9} \text{ meters}
\]
### Conclusion
The length of the arc \(ABC\) in the given circle is, \(\frac{28\pi}{9}\) meters. This calculation helps in understanding how the central angle and radius can be used to determine the arc length in a circle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe489ebd6-5c8e-44bd-8a5f-33d2b46a9596%2Ff8cb657f-d34d-46e4-9a17-f609d2cbfa59%2Fhx3m4dr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Circle Arc Length
In the diagram, we have a circle with center \( D \). The points \( A \), \( B \), and \( C \) are on the circumference of the circle. The segment \( AC \) is an internal chord of the circle, creating an angle of \( 140^\circ \) at the center, denoted as \( \angle ADC = 140^\circ \). The radius of the circle is 4 meters.
The task is to determine the length of the arc \( ABC \), which is the arc passing through points \( A \), \( B \), and \( C \).
#### Steps to Calculate the Length of Arc \( ABC \)
1. **Understand the central angle**: The central angle \(\angle ADC = 140^\circ\).
2. **Convert the angle to radians**:
\[
\text{Angle in radians} = 140^\circ \times \frac{\pi}{180^\circ} = \frac{140\pi}{180} = \frac{7\pi}{9} \text{ radians}
\]
3. **Arc Length Formula**:
\[
\text{Arc Length} = \text{Radius} \times \text{Angle in Radians}
\]
\[
\text{Arc Length of ABC} = 4 \text{ meters} \times \frac{7\pi}{9}
\]
\[
\text{Arc Length of ABC} = \frac{28\pi}{9} \text{ meters}
\]
So the length of the arc \(ABC\) is:
\[
\frac{28\pi}{9} \text{ meters}
\]
### Conclusion
The length of the arc \(ABC\) in the given circle is, \(\frac{28\pi}{9}\) meters. This calculation helps in understanding how the central angle and radius can be used to determine the arc length in a circle.
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