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
![Certainly! Below is the transcription of the text and the detailed explanation of the diagram.
---
### 4. Solve for \( x \).
#### Diagram Explanation:
The diagram depicts a circle with center \( I \) and points \( E, F, G, H \) positioned on the circumference.
- **Angle \(\angle EIF = 68^\circ\)**
- **Angle \(\angle EGH = (9x - 5)^\circ\)**
The objective is to solve for \( x \).
**Details of the Diagram:**
- Point \( E \) and point \( F \) are endpoints of a chord within the circle.
- Point \( G \) is situated on the circumference, and \( \angle EGH \) involves points \( E \), \( G \), and \( H \).
- The circle is intersected in such a way that creates the two given angles.
### Problem-Solving Steps:
1. Given that \( \angle EIF = 68^\circ \) is an inscribed angle.
2. Because \( \angle EGH \) subtends the same arc \( \overarc{EH} \) as \( \angle EIF \), the angles have a relationship due to the inscribed angle theorem.
The inscribed angle theorem states that if two inscribed angles of a circle intercept the same arc, then the angles are equal.
Thus,
\[
\angle EGH = \angle EIF
\]
Substituting the values,
\[
(9x - 5)^\circ = 68^\circ
\]
Solving for \( x \):
\[
9x - 5 = 68
\]
\[
9x = 68 + 5
\]
\[
9x = 73
\]
\[
x = \frac{73}{9}
\]
\[
x = \frac{73}{9} = 8.11 \text{ (approx)}
\]
### Conclusion:
The value of \( x \) is approximately \( 8.11 \).
---
This text and diagram explanation can help students understand how to solve for \( x \) in the given geometric context.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6a56ab44-8a54-4bc1-8e02-32dc70f20879%2Fe3241881-2044-43ff-a3f4-42a0e41f1135%2Fg8gyon_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Below is the transcription of the text and the detailed explanation of the diagram.
---
### 4. Solve for \( x \).
#### Diagram Explanation:
The diagram depicts a circle with center \( I \) and points \( E, F, G, H \) positioned on the circumference.
- **Angle \(\angle EIF = 68^\circ\)**
- **Angle \(\angle EGH = (9x - 5)^\circ\)**
The objective is to solve for \( x \).
**Details of the Diagram:**
- Point \( E \) and point \( F \) are endpoints of a chord within the circle.
- Point \( G \) is situated on the circumference, and \( \angle EGH \) involves points \( E \), \( G \), and \( H \).
- The circle is intersected in such a way that creates the two given angles.
### Problem-Solving Steps:
1. Given that \( \angle EIF = 68^\circ \) is an inscribed angle.
2. Because \( \angle EGH \) subtends the same arc \( \overarc{EH} \) as \( \angle EIF \), the angles have a relationship due to the inscribed angle theorem.
The inscribed angle theorem states that if two inscribed angles of a circle intercept the same arc, then the angles are equal.
Thus,
\[
\angle EGH = \angle EIF
\]
Substituting the values,
\[
(9x - 5)^\circ = 68^\circ
\]
Solving for \( x \):
\[
9x - 5 = 68
\]
\[
9x = 68 + 5
\]
\[
9x = 73
\]
\[
x = \frac{73}{9}
\]
\[
x = \frac{73}{9} = 8.11 \text{ (approx)}
\]
### Conclusion:
The value of \( x \) is approximately \( 8.11 \).
---
This text and diagram explanation can help students understand how to solve for \( x \) in the given geometric context.
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