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120° B F 45° 30% C The circle with center F is divided into sectors, In circle F. EB is a diameter. The length of FB is 3 units. Select the correct expression that represents the arc length of AED 11T 4 137 4

120° B F 45° 30% C The circle with center F is divided into sectors, In circle F. EB is a diameter. The length of FB is 3 units. Select the correct expression that represents the arc length of AED 11T 4 137 4

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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### Educational Content: Sector Angles and Arc Lengths #### Question 5 of 5 **Problem Statement:** The circle with center F is divided into sectors. In circle F, \( EB \) is a diameter. The length of \( EB \) is 3 units. Select the correct expression that represents the arc length of \( \overset{\frown}{AED} \). **Options:** - \( \frac{11\pi}{4} \) - \( \frac{13\pi}{4} \) - \( 4 \) - \( 4\pi \) **Diagram Explanation:** The diagram depicts a circle with center F. The circle is divided into various sectors by radii and other line segments. Notably, the following angles are marked: - Angle \( \angle AEB \) at 120° - Angle \( \angle EFB \) at 45° - Angle \( \angle FDC \) at 30° The line segment \( EB \) is a diameter of the circle. The given diameter \( EB = 3 \) units. To find the arc length \( \overset{\frown}{AED} \), the sum of the central angles involved needs to be considered, and then the formula for the arc length can be applied. To calculate the arc length: 1. Identify the total angle \( \angle AFB + \angle BFD + \angle DFE = 120° + 45° + 30° = 195° \). 2. Convert this angle to radians because arc length calculations generally involve radians: \( 195° \times \left(\frac{\pi}{180}\right) \). Now the arc length \( s \) can be calculated using the formula \( s = r \theta \), where \( r \) is the radius and \( \theta \) is the angle in radians. **Working through this:** - Diameter \( EB = 3 \) units, so the radius \( r = \frac{3}{2} \) units. - Convert \( 195° \) to radians: \( 195° \times \left(\frac{\pi}{180}\right) = \frac{195}{180}\pi = \frac{13\pi}{12} \). Thus, the arc length \( s \): \( s = \left(\frac{3}{2}\right) \times \left
Transcribed Image Text:### Educational Content: Sector Angles and Arc Lengths #### Question 5 of 5 **Problem Statement:** The circle with center F is divided into sectors. In circle F, \( EB \) is a diameter. The length of \( EB \) is 3 units. Select the correct expression that represents the arc length of \( \overset{\frown}{AED} \). **Options:** - \( \frac{11\pi}{4} \) - \( \frac{13\pi}{4} \) - \( 4 \) - \( 4\pi \) **Diagram Explanation:** The diagram depicts a circle with center F. The circle is divided into various sectors by radii and other line segments. Notably, the following angles are marked: - Angle \( \angle AEB \) at 120° - Angle \( \angle EFB \) at 45° - Angle \( \angle FDC \) at 30° The line segment \( EB \) is a diameter of the circle. The given diameter \( EB = 3 \) units. To find the arc length \( \overset{\frown}{AED} \), the sum of the central angles involved needs to be considered, and then the formula for the arc length can be applied. To calculate the arc length: 1. Identify the total angle \( \angle AFB + \angle BFD + \angle DFE = 120° + 45° + 30° = 195° \). 2. Convert this angle to radians because arc length calculations generally involve radians: \( 195° \times \left(\frac{\pi}{180}\right) \). Now the arc length \( s \) can be calculated using the formula \( s = r \theta \), where \( r \) is the radius and \( \theta \) is the angle in radians. **Working through this:** - Diameter \( EB = 3 \) units, so the radius \( r = \frac{3}{2} \) units. - Convert \( 195° \) to radians: \( 195° \times \left(\frac{\pi}{180}\right) = \frac{195}{180}\pi = \frac{13\pi}{12} \). Thus, the arc length \( s \): \( s = \left(\frac{3}{2}\right) \times \left
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