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 Description
**Objective:**
Find the value of \( x \) and the length of segment \( \overline{HG} \).
**Diagram Explanation:**
- The diagram shows a circle with a chord \( \overline{HG} \) intersected by a radius \( \overline{PG} \).
- Point \( H \), point \( G \), and point \( P \) are marked on the circle.
- \( \overline{HG} \) is divided into two segments: \( \overline{HP} \) and \( \overline{PG} \).
- \( \overline{HP} \) is labeled with the expression \( 3x - 2 \).
- \( \overline{PG} \) is labeled with the expression \( x + 8 \).
- There is a right angle at point \( P \) indicating that \( \overline{PG} \) is perpendicular to \( \overline{HG} \).
**Equations:**
Use the property of the segments of chords intersected inside the circle:
\[ \text{Length}\ of\ \overline{HP} + \text{Length}\ of\ \overline{PG} = \overline{HG} \]
### Steps to Solve
1. **Find the value of \( x \):**
- Use the Pythagorean Theorem, since \( \overline{PG} \) is perpendicular to \( \overline{HG} \).
- \((3x - 2) + (x + 8) = \overline{HG} \)
2. **Calculate the length of segment \( \overline{HG} \):**
- Substitute the value of \( x \) back into \( 3x - 2 \) and \( x + 8 \).
- Add the results to find \( \overline{HG} \).
Insert the solution values into:
- \( x = \) ______
- The length of segment \( \overline{HG} \) is ______
Use these steps to complete the required calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe4e618c1-0f79-4e06-a44a-13c0a5facb61%2Ffc156fe4-7f1f-4951-89f3-b6aaadda3cae%2Fqec2ia2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Description
**Objective:**
Find the value of \( x \) and the length of segment \( \overline{HG} \).
**Diagram Explanation:**
- The diagram shows a circle with a chord \( \overline{HG} \) intersected by a radius \( \overline{PG} \).
- Point \( H \), point \( G \), and point \( P \) are marked on the circle.
- \( \overline{HG} \) is divided into two segments: \( \overline{HP} \) and \( \overline{PG} \).
- \( \overline{HP} \) is labeled with the expression \( 3x - 2 \).
- \( \overline{PG} \) is labeled with the expression \( x + 8 \).
- There is a right angle at point \( P \) indicating that \( \overline{PG} \) is perpendicular to \( \overline{HG} \).
**Equations:**
Use the property of the segments of chords intersected inside the circle:
\[ \text{Length}\ of\ \overline{HP} + \text{Length}\ of\ \overline{PG} = \overline{HG} \]
### Steps to Solve
1. **Find the value of \( x \):**
- Use the Pythagorean Theorem, since \( \overline{PG} \) is perpendicular to \( \overline{HG} \).
- \((3x - 2) + (x + 8) = \overline{HG} \)
2. **Calculate the length of segment \( \overline{HG} \):**
- Substitute the value of \( x \) back into \( 3x - 2 \) and \( x + 8 \).
- Add the results to find \( \overline{HG} \).
Insert the solution values into:
- \( x = \) ______
- The length of segment \( \overline{HG} \) is ______
Use these steps to complete the required calculations.
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