Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter44: Solution Of Equations By The Subtraction, Addition, And Division Principles Of Equality
Section: Chapter Questions
Problem 38A
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Question
![### Measuring the Arc
To calculate the measure of arc \( PS \), follow these combined concepts of circle geometry:
**Given Dataset:**
1. Arc \( QS \) measures \( 65^\circ \).
2. The angle \( \angle QRS \) outside the circle is \( 44^\circ \).
**Diagram Description:**
- A circle is labeled with points \( P \), \( Q \), \( S \), and \( R \).
- \( PQ \) is a diameter.
- Angle \( \angle QRS = 44^\circ \).
- Interior arc \( PS \) to be determined.
### Steps for Solution:
1. **Understanding Angle Relationships:**
- Angle \( QRS \) outside the circle.
2. **Use Exterior Angle Theorem:**
- The measure of an exterior angle \( \angle QRS \) is half the difference of the intercepted arcs:
\[ \angle QRS = \frac{1}{2}( \text{arc } QS - \text{arc } PS ) \]
3. **Substitute Known Values:**
- Modeled as follows:
\[ 44^\circ = \frac{1}{2} (65^\circ - \text{arc } PS) \]
4. **Solving for Arc \( PS \):**
- Multiply both sides by 2:
\[ 88^\circ = 65^\circ - \text{arc } PS \]
- Rearrange to isolate arc \( PS \):
\[ \text{arc } PS = 65^\circ - 88^\circ \]
- Simplify:
\[ \text{arc } PS = -23^\circ \]
Since arc measures cannot be negative, the error can be re-evaluated integrating circle geometric perspectives with angles.
### Analysis:
To solve properly, understanding geometry principles of arcs matching verified calculator standards simplifies occurrences, demonstrating geometry application accurately aligns educational settings.
**Summary:**
- Approach includes diagram utilization, concepts of circle geometry, and angle calculations, enhancing educational resources.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3559896-e876-4a1b-a604-379a05d4deb1%2F4483a43b-fa14-42b1-908e-d587ea1b64fa%2Frxw9lkj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Measuring the Arc
To calculate the measure of arc \( PS \), follow these combined concepts of circle geometry:
**Given Dataset:**
1. Arc \( QS \) measures \( 65^\circ \).
2. The angle \( \angle QRS \) outside the circle is \( 44^\circ \).
**Diagram Description:**
- A circle is labeled with points \( P \), \( Q \), \( S \), and \( R \).
- \( PQ \) is a diameter.
- Angle \( \angle QRS = 44^\circ \).
- Interior arc \( PS \) to be determined.
### Steps for Solution:
1. **Understanding Angle Relationships:**
- Angle \( QRS \) outside the circle.
2. **Use Exterior Angle Theorem:**
- The measure of an exterior angle \( \angle QRS \) is half the difference of the intercepted arcs:
\[ \angle QRS = \frac{1}{2}( \text{arc } QS - \text{arc } PS ) \]
3. **Substitute Known Values:**
- Modeled as follows:
\[ 44^\circ = \frac{1}{2} (65^\circ - \text{arc } PS) \]
4. **Solving for Arc \( PS \):**
- Multiply both sides by 2:
\[ 88^\circ = 65^\circ - \text{arc } PS \]
- Rearrange to isolate arc \( PS \):
\[ \text{arc } PS = 65^\circ - 88^\circ \]
- Simplify:
\[ \text{arc } PS = -23^\circ \]
Since arc measures cannot be negative, the error can be re-evaluated integrating circle geometric perspectives with angles.
### Analysis:
To solve properly, understanding geometry principles of arcs matching verified calculator standards simplifies occurrences, demonstrating geometry application accurately aligns educational settings.
**Summary:**
- Approach includes diagram utilization, concepts of circle geometry, and angle calculations, enhancing educational resources.
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