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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![**Problem Statement**:
Find the value of x.
**Diagram Explanation**:
In the provided diagram, we have a circle with several lines intersecting inside it, forming different geometric shapes. Specifically, the circle is divided into multiple segments by line segments that intersect at various points inside and on the circle.
- The diagram shows two prominent colored angular sectors inside the circle.
- The first angular sector (shaded in purple) on the left side is labeled as \((4x + 9)^\circ\).
- The second angular sector (shaded in cyan) on the right side is labeled as \((7x - 6)^\circ\).
This is a classic example of solving for variables in geometry problems involving circles and angles.
**Solution**:
To find the value of \( x \), we recognize that the sum of the angles subtended by the same arc at the circumference of a circle is equal. Since both angles add up to give the entire angle at the circumference of a common arc:
\[
(4x + 9)^\circ + (7x - 6)^\circ = 180^\circ
\]
Combine and solve for \( x \):
\[
4x + 9 + 7x - 6 = 180
\]
Combine like terms:
\[
11x + 3 = 180
\]
Subtract 3 from both sides:
\[
11x = 177
\]
Divide by 11:
\[
x = 16.09
\]
This value of \( x \) can then be used to check and verify the individual angles if needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70379ec6-c51e-43ca-871d-f3a8cf6849c2%2Ff7b7072c-5fa3-4d7f-b958-6a51c7cf2d78%2Fqhb9s1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**:
Find the value of x.
**Diagram Explanation**:
In the provided diagram, we have a circle with several lines intersecting inside it, forming different geometric shapes. Specifically, the circle is divided into multiple segments by line segments that intersect at various points inside and on the circle.
- The diagram shows two prominent colored angular sectors inside the circle.
- The first angular sector (shaded in purple) on the left side is labeled as \((4x + 9)^\circ\).
- The second angular sector (shaded in cyan) on the right side is labeled as \((7x - 6)^\circ\).
This is a classic example of solving for variables in geometry problems involving circles and angles.
**Solution**:
To find the value of \( x \), we recognize that the sum of the angles subtended by the same arc at the circumference of a circle is equal. Since both angles add up to give the entire angle at the circumference of a common arc:
\[
(4x + 9)^\circ + (7x - 6)^\circ = 180^\circ
\]
Combine and solve for \( x \):
\[
4x + 9 + 7x - 6 = 180
\]
Combine like terms:
\[
11x + 3 = 180
\]
Subtract 3 from both sides:
\[
11x = 177
\]
Divide by 11:
\[
x = 16.09
\]
This value of \( x \) can then be used to check and verify the individual angles if needed.
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