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:**
In circle \( P \) with \( m \angle NPQ = 42^\circ \) and \( NP = 19 \) units, find the area of sector \( NPQ \). Round to the nearest hundredth.
**Diagram Explanation:**
The diagram provided depicts a circle \( P \) with three points marked: \( N \), \( P \) (the center of the circle), and \( Q \). There is a sector \( NPQ \) within the circle subtended by the central angle \( N \hat{P}Q = 42^\circ \). The radius \( NP \) of the circle is given as 19 units.
**Solution:**
To find the area of the sector \( NPQ \), we use the formula for the area of a sector of a circle:
\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
Where:
- \( \theta \) = the central angle in degrees.
- \( r \) = the radius of the circle.
Given:
- \( \theta = 42^\circ \)
- \( r = 19 \) units
Plugging in the values:
\[ \text{Area of sector} = \frac{42}{360} \times \pi \times 19^2 \]
\[ \text{Area of sector} = \frac{42}{360} \times \pi \times 361 \]
\[ \text{Area of sector} = \frac{42}{360} \times \pi \times 361 \]
\[ \text{Area of sector} = \frac{42}{360} \times 1134.114947 \]
\[ \text{Area of sector} \approx 132.065048 \]
Rounded to the nearest hundredth, the area of sector \( NPQ \) is approximately \( 132.07 \) square units.
**Answer:** \( 132.07 \) square units.
**Interface for Answer Submission:**
Below the diagram and problem statement, there is a text input box for the user to enter their answer. There is also a "Submit Answer" button for submitting the response.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffceb4de4-226d-4fc7-ac8f-83def6c327b8%2F0a656943-3bc3-4dd7-b8a6-473d20c76811%2Fwrakz7i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
In circle \( P \) with \( m \angle NPQ = 42^\circ \) and \( NP = 19 \) units, find the area of sector \( NPQ \). Round to the nearest hundredth.
**Diagram Explanation:**
The diagram provided depicts a circle \( P \) with three points marked: \( N \), \( P \) (the center of the circle), and \( Q \). There is a sector \( NPQ \) within the circle subtended by the central angle \( N \hat{P}Q = 42^\circ \). The radius \( NP \) of the circle is given as 19 units.
**Solution:**
To find the area of the sector \( NPQ \), we use the formula for the area of a sector of a circle:
\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]
Where:
- \( \theta \) = the central angle in degrees.
- \( r \) = the radius of the circle.
Given:
- \( \theta = 42^\circ \)
- \( r = 19 \) units
Plugging in the values:
\[ \text{Area of sector} = \frac{42}{360} \times \pi \times 19^2 \]
\[ \text{Area of sector} = \frac{42}{360} \times \pi \times 361 \]
\[ \text{Area of sector} = \frac{42}{360} \times \pi \times 361 \]
\[ \text{Area of sector} = \frac{42}{360} \times 1134.114947 \]
\[ \text{Area of sector} \approx 132.065048 \]
Rounded to the nearest hundredth, the area of sector \( NPQ \) is approximately \( 132.07 \) square units.
**Answer:** \( 132.07 \) square units.
**Interface for Answer Submission:**
Below the diagram and problem statement, there is a text input box for the user to enter their answer. There is also a "Submit Answer" button for submitting the response.
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