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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F056e647e-6927-46de-b6aa-8f2eb320dc65%2F9564241c-ca6a-476b-9e1b-647809db74e2%2F1k9db3n_processed.jpeg&w=3840&q=75)
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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