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
![### Problem Statement:
**If \(\overline{ED}\) is tangent to circle \(H\), find the value of \(x\).**
### Diagram Explanation:
The provided diagram contains:
1. A circle labeled \(H\).
2. Line segment \(\overline{ED}\) that is tangent to the circle \(H\).
3. Point \(G\) where \(\overline{ED}\) touches the circle.
4. Point \(F\) on the line \(\overline{EG}\), creating a tangent-secant angle.
5. Angles:
- \(\angle EGF\)—marked as \(147°\)
- \(\angle FEH\)—marked as \(45°\)
- \(\angle EFD\)—marked as \((7x - 5)°\)
### Options for the value of \(x\):
A. \( x = 6 \)
B. \( x = 7 \)
C. \( x = 8 \)
D. \( x = 9 \)
### Solution Approach:
1. Since \(\overline{ED}\) is tangent to circle \(H\) at point \(G\), and \(GF\) is a secant, the relationship between the angles can be calculated.
2. Given:
\[
\angle FEH = 45°
\]
\[
\angle EGF = 147°
\]
\[
\angle EFD = (7x - 5)°
\]
3. Using the exterior angle theorem:
\[
\angle FEH + 90° = (7x - 5)°
\]
Since \( \angle EGF = 147° \) and the complete angle through point \(F\) is \(180° - 147° = 33° \), thus:
\[
45° + 33° = (7x - 5)°
\]
Simplifying,
\[
78° = 7x - 5°
\]
Solving for \(x\):
\[
7x = 83
\]
\[
x = \frac{83}{7} \approx 11.86
\]
However, recheck the primarily given options suggesting the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d44061d-5891-4c6f-9f7e-bc7f6369bc06%2F7a1bc2f4-dd54-4d13-8832-ed1caed48ab4%2F7jnajbi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
**If \(\overline{ED}\) is tangent to circle \(H\), find the value of \(x\).**
### Diagram Explanation:
The provided diagram contains:
1. A circle labeled \(H\).
2. Line segment \(\overline{ED}\) that is tangent to the circle \(H\).
3. Point \(G\) where \(\overline{ED}\) touches the circle.
4. Point \(F\) on the line \(\overline{EG}\), creating a tangent-secant angle.
5. Angles:
- \(\angle EGF\)—marked as \(147°\)
- \(\angle FEH\)—marked as \(45°\)
- \(\angle EFD\)—marked as \((7x - 5)°\)
### Options for the value of \(x\):
A. \( x = 6 \)
B. \( x = 7 \)
C. \( x = 8 \)
D. \( x = 9 \)
### Solution Approach:
1. Since \(\overline{ED}\) is tangent to circle \(H\) at point \(G\), and \(GF\) is a secant, the relationship between the angles can be calculated.
2. Given:
\[
\angle FEH = 45°
\]
\[
\angle EGF = 147°
\]
\[
\angle EFD = (7x - 5)°
\]
3. Using the exterior angle theorem:
\[
\angle FEH + 90° = (7x - 5)°
\]
Since \( \angle EGF = 147° \) and the complete angle through point \(F\) is \(180° - 147° = 33° \), thus:
\[
45° + 33° = (7x - 5)°
\]
Simplifying,
\[
78° = 7x - 5°
\]
Solving for \(x\):
\[
7x = 83
\]
\[
x = \frac{83}{7} \approx 11.86
\]
However, recheck the primarily given options suggesting the
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