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E is the midpoint of CF and H is the midpoint of GI. What is the value of b? I 4b D G ww LL

E is the midpoint of CF and H is the midpoint of GI. What is the value of b? I 4b D G ww LL

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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### Midpoint Problem in a Circle Geometry #### Problem Statement E is the midpoint of CF and H is the midpoint of GI. What is the value of \( b \)? #### Diagram Description The diagram displays a circle with several points labeled: C, F, G, I, E, H, and D. Below is a detailed description of the elements in the diagram: - Points C, F, G, I are on the circumference of the circle. - E is marked as the midpoint of CF. - H is marked as the midpoint of GI. - D is an internal point connected to E and H. - Line segments CE and EF are equal to each other as E is the midpoint of CF. - Likewise, line segments GH and HI are equal to each other since H is the midpoint of GI. - There is a green line DE and another green line DH. **Given Lengths:** - Segment HE is labeled as \( b + 18 \). - Segment EC is labeled as \( 4b \). ### Question What is the value of \( b \)? ### Solution Outline 1. **Identify Midpoints:** - \( E \) is the midpoint of \( CF \), which implies \( CE = EF \). - \( H \) is the midpoint of \( GI \), which implies \( GH = HI \). 2. **Using Properties of Midpoints:** - Since \( E \) is the midpoint of \( CF \), the lengths \( CE \) and \( EF \) are both \( 4b \). - Since \( H \) is the midpoint of \( GI \), the lengths \( GH \) and \( HI \) are both \( b + 18 \). ### Calculations Let's solve for \( b \) using the relationship given in the problem: Since both \( CE \) and \( EF \) are \( 4b \): \[ CF = CE + EF = 4b + 4b = 8b \] Similarly, \[ GI = GH + HI = (b + 18) + (b + 18) = 2b + 36 \] For the given arrangement of points to be consistent within the circle's geometrical properties, \( CF \) should equal \( GI \): \[ 8b = 2b + 36 \] Solve for \( b \): \[ 8b - 2b
Transcribed Image Text:### Midpoint Problem in a Circle Geometry #### Problem Statement E is the midpoint of CF and H is the midpoint of GI. What is the value of \( b \)? #### Diagram Description The diagram displays a circle with several points labeled: C, F, G, I, E, H, and D. Below is a detailed description of the elements in the diagram: - Points C, F, G, I are on the circumference of the circle. - E is marked as the midpoint of CF. - H is marked as the midpoint of GI. - D is an internal point connected to E and H. - Line segments CE and EF are equal to each other as E is the midpoint of CF. - Likewise, line segments GH and HI are equal to each other since H is the midpoint of GI. - There is a green line DE and another green line DH. **Given Lengths:** - Segment HE is labeled as \( b + 18 \). - Segment EC is labeled as \( 4b \). ### Question What is the value of \( b \)? ### Solution Outline 1. **Identify Midpoints:** - \( E \) is the midpoint of \( CF \), which implies \( CE = EF \). - \( H \) is the midpoint of \( GI \), which implies \( GH = HI \). 2. **Using Properties of Midpoints:** - Since \( E \) is the midpoint of \( CF \), the lengths \( CE \) and \( EF \) are both \( 4b \). - Since \( H \) is the midpoint of \( GI \), the lengths \( GH \) and \( HI \) are both \( b + 18 \). ### Calculations Let's solve for \( b \) using the relationship given in the problem: Since both \( CE \) and \( EF \) are \( 4b \): \[ CF = CE + EF = 4b + 4b = 8b \] Similarly, \[ GI = GH + HI = (b + 18) + (b + 18) = 2b + 36 \] For the given arrangement of points to be consistent within the circle's geometrical properties, \( CF \) should equal \( GI \): \[ 8b = 2b + 36 \] Solve for \( b \): \[ 8b - 2b
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