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
![### Problem Statement:
In circle F with \( m \angle EFG = 78^\circ \) and \( EF = 4 \) units, find the length of arc EG. Round to the nearest hundredth.
### Diagram Description:
The image contains a circle with center \( F \). Points \( E \) and \( G \) lie on the circumference of the circle, and line segment \( EF \) is a radius of the circle. The central angle \( \angle EFG \) measures \( 78^\circ \).
### Steps to Solve:
1. **Find the Radius (r):**
Given:
\[
EF = 4 \text{ units}
\]
Since \( EF \) is the radius of the circle, \( r = 4 \) units.
2. **Calculate the Arc Length:**
The formula for the length of an arc (\( L \)) is:
\[
L = r \theta
\]
where \( \theta \) is the central angle in radians.
Convert the angle from degrees to radians:
\[
\theta = 78^\circ \times \left( \frac{\pi}{180^\circ} \right) = \frac{78 \pi}{180} = \frac{13 \pi}{30} \text{ radians}
\]
Substitute \( r \) and \( \theta \) into the arc length formula:
\[
L = 4 \times \frac{13 \pi}{30}
\]
Simplify:
\[
L = \frac{52 \pi}{30} = \frac{26 \pi}{15}
\]
3. **Convert to Decimal:**
\[
L \approx \frac{26 \times 3.14159}{15} \approx 5.44 \text{ units}
\]
Thus, the length of arc \( EG \) is approximately \( 5.44 \) units when rounded to the nearest hundredth.
### Answer:
\[ \boxed{5.44} \]
Feel free to use the provided steps and the diagram to enhance your understanding and solve similar problems involving circles and arc lengths.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f20aeac-3658-4125-82ac-c012d50c949f%2Fdf61eeea-edd1-4346-ac48-0ee70ca24584%2Frsdwuote_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
In circle F with \( m \angle EFG = 78^\circ \) and \( EF = 4 \) units, find the length of arc EG. Round to the nearest hundredth.
### Diagram Description:
The image contains a circle with center \( F \). Points \( E \) and \( G \) lie on the circumference of the circle, and line segment \( EF \) is a radius of the circle. The central angle \( \angle EFG \) measures \( 78^\circ \).
### Steps to Solve:
1. **Find the Radius (r):**
Given:
\[
EF = 4 \text{ units}
\]
Since \( EF \) is the radius of the circle, \( r = 4 \) units.
2. **Calculate the Arc Length:**
The formula for the length of an arc (\( L \)) is:
\[
L = r \theta
\]
where \( \theta \) is the central angle in radians.
Convert the angle from degrees to radians:
\[
\theta = 78^\circ \times \left( \frac{\pi}{180^\circ} \right) = \frac{78 \pi}{180} = \frac{13 \pi}{30} \text{ radians}
\]
Substitute \( r \) and \( \theta \) into the arc length formula:
\[
L = 4 \times \frac{13 \pi}{30}
\]
Simplify:
\[
L = \frac{52 \pi}{30} = \frac{26 \pi}{15}
\]
3. **Convert to Decimal:**
\[
L \approx \frac{26 \times 3.14159}{15} \approx 5.44 \text{ units}
\]
Thus, the length of arc \( EG \) is approximately \( 5.44 \) units when rounded to the nearest hundredth.
### Answer:
\[ \boxed{5.44} \]
Feel free to use the provided steps and the diagram to enhance your understanding and solve similar problems involving circles and arc lengths.
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