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
100%
Here is a diagram of a circle centered at point \( P \). The circle has four points on its circumference labeled \( T \), \( Q \), \( R \), and \( S \).
- The arc \( TS \) has a central angle of \( x + 23^\circ \).
- The arc \( SQ \) has a central angle of \( x + 29^\circ \).
- The arc \( QR \) has a central angle of \( x + 16^\circ \).
- The arc \( RT \) has a central angle of \( 77^\circ \).
### To find the value of \( x \):
1. Recognize that the sum of all central angles in a circle is \( 360^\circ \).
2. Set up the equation combining all angles:
\[
(x + 23^\circ) + (x + 29^\circ) + (x + 16^\circ) + 77^\circ = 360^\circ
\]
3. Simplify and solve for \( x \):
\[
x + 23 + x + 29 + x + 16 + 77 = 360
\]
\[
3x + 145 = 360
\]
\[
3x = 360 - 145
\]
\[
3x = 215
\]
\[
x = \frac{215}{3}
\]
\[
x = 71.67^\circ
\]
### Therefore, the value of \( x \) is approximately \( 71.67^\circ \).
\[ x = \boxed{71.67} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2912ac0f-6502-438a-afff-f22364def2fb%2F14d821b7-a71b-45d9-80bf-7de8cdb14e58%2Fqt254m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### What is the value of \( x \)?
Here is a diagram of a circle centered at point \( P \). The circle has four points on its circumference labeled \( T \), \( Q \), \( R \), and \( S \).
- The arc \( TS \) has a central angle of \( x + 23^\circ \).
- The arc \( SQ \) has a central angle of \( x + 29^\circ \).
- The arc \( QR \) has a central angle of \( x + 16^\circ \).
- The arc \( RT \) has a central angle of \( 77^\circ \).
### To find the value of \( x \):
1. Recognize that the sum of all central angles in a circle is \( 360^\circ \).
2. Set up the equation combining all angles:
\[
(x + 23^\circ) + (x + 29^\circ) + (x + 16^\circ) + 77^\circ = 360^\circ
\]
3. Simplify and solve for \( x \):
\[
x + 23 + x + 29 + x + 16 + 77 = 360
\]
\[
3x + 145 = 360
\]
\[
3x = 360 - 145
\]
\[
3x = 215
\]
\[
x = \frac{215}{3}
\]
\[
x = 71.67^\circ
\]
### Therefore, the value of \( x \) is approximately \( 71.67^\circ \).
\[ x = \boxed{71.67} \]
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