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
![### Find the Measure of Arc ST
In this section, we'll explore how to find the measure of arc ST in circle Z using the given information.
#### Given:
1. \( VZ = ZW \)
2. \( SV = 21 \)
3. \( m\angle UT = 112° \)
#### Diagram Explanation:
The provided image includes a circle labeled Z, which contains the following points arranged clockwise: S, V, Z, W, T, and U. There are two marked equal line segments, \( VZ \) and \( ZW \), and point T is located on the circle on the RT tangent which creates an inscribed angle with chord SU. An inscribed angle, \( \angle UT \), measures 112°.
To find the measure of arc ST:
- Recognize that \( VZ = ZW \) indicates these segments are equal.
- The measure of an inscribed angle, such as \( \angle UT \), is half the measure of the arc it intercepts.
Therefore:
\[ m\angle UT = \frac{1}{2} m(\overset{\frown}{ST}) \]
\[ 112° = \frac{1}{2} m(\overset{\frown}{ST}) \]
\[ m(\overset{\frown}{ST}) = 2 \times 112° \]
\[ m(\overset{\frown}{ST}) = 224° \]
#### Conclusion:
The measure of arc ST is 224°.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2994886d-ee4f-443d-b178-240cdf3d6c71%2F87116b03-0571-44e3-816e-02f64cc0d597%2Fdvjxr9p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Find the Measure of Arc ST
In this section, we'll explore how to find the measure of arc ST in circle Z using the given information.
#### Given:
1. \( VZ = ZW \)
2. \( SV = 21 \)
3. \( m\angle UT = 112° \)
#### Diagram Explanation:
The provided image includes a circle labeled Z, which contains the following points arranged clockwise: S, V, Z, W, T, and U. There are two marked equal line segments, \( VZ \) and \( ZW \), and point T is located on the circle on the RT tangent which creates an inscribed angle with chord SU. An inscribed angle, \( \angle UT \), measures 112°.
To find the measure of arc ST:
- Recognize that \( VZ = ZW \) indicates these segments are equal.
- The measure of an inscribed angle, such as \( \angle UT \), is half the measure of the arc it intercepts.
Therefore:
\[ m\angle UT = \frac{1}{2} m(\overset{\frown}{ST}) \]
\[ 112° = \frac{1}{2} m(\overset{\frown}{ST}) \]
\[ m(\overset{\frown}{ST}) = 2 \times 112° \]
\[ m(\overset{\frown}{ST}) = 224° \]
#### Conclusion:
The measure of arc ST is 224°.
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