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In circle G, m/HGI = 100° and the length of HI = H GH = 45 GH = 15 GH 18 = GH = 3 = G I TT. Find the length of GH.

In circle G, m/HGI = 100° and the length of HI = H GH = 45 GH = 15 GH 18 = GH = 3 = G I TT. Find the length of GH.

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: Finding Segment Length **Problem Statement:** In circle \( G \), \( m\angle HGI = 100^\circ \) and the length of arc \( HI \) is \( \frac{5}{3} \pi \). Find the length of segment \( \overline{GH} \). **Diagram:** Below the problem statement is a diagram of a circle labeled \( G \). Point \( G \) is the center of the circle. Points \( H \) and \( I \) are points on the circumference of the circle connected by lines \( \overline{GH} \) and \( \overline{GI} \). ![Circle Diagram](circle-diagram.png) **Multiple Choice Options:** - \( \bigcirc \) GH = 45 - \( \bigcirc \) GH = 15 - \( \bigcirc \) GH = 18 - \( \bigcirc \) GH = 3 **Explanation:** To solve this problem, use the relationships between angles, arcs, and radii in circles. Calculate the radius of the circle by relating the given arc length and central angle. Once the radius (GH) is found, choose the correct answer from the provided options. ### Analytical Steps: 1. **Arc Length Formula:** \( l = \theta \cdot r \) - Here, \( l = \frac{5}{3} \pi \) and \( \theta = 100^\circ \), convert \( \theta \) into radians ( \( \theta \text{ in radians} = \frac{100^\circ \cdot \pi}{180^\circ} \) ). 2. **Solve for Radius \( r = GH \):** \[ \frac{5}{3}\pi = \left(\frac{100\pi}{180}\right) \cdot r \] 3. **Simplify and Solve:** \[ \frac{5}{3}\pi = \left(\frac{5\pi}{9}\right) \cdot r \implies r = \frac{5}{3} \cdot \frac{9}{5} = 3 \] 4. **Identifying the Correct Option:** - The radius \( \overline{GH} \) is found to be 3, which corresponds to
Transcribed Image Text:### Geometry Problem: Finding Segment Length **Problem Statement:** In circle \( G \), \( m\angle HGI = 100^\circ \) and the length of arc \( HI \) is \( \frac{5}{3} \pi \). Find the length of segment \( \overline{GH} \). **Diagram:** Below the problem statement is a diagram of a circle labeled \( G \). Point \( G \) is the center of the circle. Points \( H \) and \( I \) are points on the circumference of the circle connected by lines \( \overline{GH} \) and \( \overline{GI} \). ![Circle Diagram](circle-diagram.png) **Multiple Choice Options:** - \( \bigcirc \) GH = 45 - \( \bigcirc \) GH = 15 - \( \bigcirc \) GH = 18 - \( \bigcirc \) GH = 3 **Explanation:** To solve this problem, use the relationships between angles, arcs, and radii in circles. Calculate the radius of the circle by relating the given arc length and central angle. Once the radius (GH) is found, choose the correct answer from the provided options. ### Analytical Steps: 1. **Arc Length Formula:** \( l = \theta \cdot r \) - Here, \( l = \frac{5}{3} \pi \) and \( \theta = 100^\circ \), convert \( \theta \) into radians ( \( \theta \text{ in radians} = \frac{100^\circ \cdot \pi}{180^\circ} \) ). 2. **Solve for Radius \( r = GH \):** \[ \frac{5}{3}\pi = \left(\frac{100\pi}{180}\right) \cdot r \] 3. **Simplify and Solve:** \[ \frac{5}{3}\pi = \left(\frac{5\pi}{9}\right) \cdot r \implies r = \frac{5}{3} \cdot \frac{9}{5} = 3 \] 4. **Identifying the Correct Option:** - The radius \( \overline{GH} \) is found to be 3, which corresponds to
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