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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Topic Video
Question
![### Find the length of arc \( \overset{\frown}{XYZ} \). Leave your answer in terms of \( \pi \).
#### Description of the Diagram:
The diagram shows a circle with center at an unlabeled point and radius marked as 6 meters (m). The circle is intersected by a line segment from the center to a point on the circumference, yet another from the center forming a 90-degree angle with the first segment, and an arc subtended by these points.
The key points are labeled as:
- \(X\), \(Y\) inside the circle
- \(P\) on the circumference of the circle
#### Illustration Explained:
- A circle with radius 6 meters.
- \(X\) and \(Y\) are two points on the circumference that form a right angle (90 degrees) at the center.
- The arc \( \overset{\frown}{XYZ} \) represents a quarter of the circle.
#### Objective:
Calculate the length of the arc \( \overset{\frown}{XYZ} \).
#### Multiple Choice Answers:
- \(\quad\) \(3\pi \text{ m}\)
- \(\quad\) \(18\pi \text{ m}\)
- \(\quad\) \(540\pi \text{ m}\)
- \(\quad\) \(9\pi \text{ m}\)
---
**Solution Explanation:**
To determine the length of the arc \( \overset{\frown}{XYZ} \), we follow these steps:
1. The circumference of the entire circle is \(2\pi r\), where \(r\) is the radius of the circle.
2. Calculate the total circumference:
\[
\text{Circumference} = 2\pi \times 6 \text{ m} = 12\pi \text{ m}
\]
3. Since \( \overset{\frown}{XYZ} \) is a quarter of the circle (90 degrees is a quarter of 360 degrees):
\[
\text{Arc length of } \overset{\frown}{XYZ} = \frac{1}{4} \times \text{Circumference}
\]
4. Thus:
\[
\text{Arc length} = \frac{1}{4} \times 12\pi \text{ m} = 3\pi](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe59874f9-f705-461a-94a3-07d771601ab4%2Fa0298f4e-f1f4-4276-b782-b1ab377121d1%2Fe755a3vi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Find the length of arc \( \overset{\frown}{XYZ} \). Leave your answer in terms of \( \pi \).
#### Description of the Diagram:
The diagram shows a circle with center at an unlabeled point and radius marked as 6 meters (m). The circle is intersected by a line segment from the center to a point on the circumference, yet another from the center forming a 90-degree angle with the first segment, and an arc subtended by these points.
The key points are labeled as:
- \(X\), \(Y\) inside the circle
- \(P\) on the circumference of the circle
#### Illustration Explained:
- A circle with radius 6 meters.
- \(X\) and \(Y\) are two points on the circumference that form a right angle (90 degrees) at the center.
- The arc \( \overset{\frown}{XYZ} \) represents a quarter of the circle.
#### Objective:
Calculate the length of the arc \( \overset{\frown}{XYZ} \).
#### Multiple Choice Answers:
- \(\quad\) \(3\pi \text{ m}\)
- \(\quad\) \(18\pi \text{ m}\)
- \(\quad\) \(540\pi \text{ m}\)
- \(\quad\) \(9\pi \text{ m}\)
---
**Solution Explanation:**
To determine the length of the arc \( \overset{\frown}{XYZ} \), we follow these steps:
1. The circumference of the entire circle is \(2\pi r\), where \(r\) is the radius of the circle.
2. Calculate the total circumference:
\[
\text{Circumference} = 2\pi \times 6 \text{ m} = 12\pi \text{ m}
\]
3. Since \( \overset{\frown}{XYZ} \) is a quarter of the circle (90 degrees is a quarter of 360 degrees):
\[
\text{Arc length of } \overset{\frown}{XYZ} = \frac{1}{4} \times \text{Circumference}
\]
4. Thus:
\[
\text{Arc length} = \frac{1}{4} \times 12\pi \text{ m} = 3\pi
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