Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![### Circle Measurements and Calculations
#### Problem 7: Arc Length of \( \overset{\frown}{AB} \)
- **Diagram:** This diagram represents a circle with center \( P \) and radius \( 8 \) ft. The angle \( \angle APB \) is \( 45^\circ \).
- **Objective:** Calculate the arc length of \( \overset{\frown}{AB} \).
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#### Problem 8: Measure of \( \overset{\frown}{DE} \)
- **Diagram:** This diagram illustrates a circle with center \( Q \) and radius \( 10 \) in. The arc \( \overset{\frown}{DE} \) measures \( 8.73 \) in.
- **Objective:** Determine the measure of \( \overset{\frown}{DE} \).
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#### Problem 9: Circumference of Circle \( \odot C \)
- **Diagram:** This diagram shows a circle with center \( C \) and radius \( 7.5 \) m. The angle \( \angle GCF \) is \( 76^\circ \).
- **Objective:** Find the circumference of circle \( \odot C \).
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### Analysis and Computations
#### Arc Length of \( \overset{\frown}{AB} \)
To calculate the arc length \( \overset{\frown}{AB} \):
\[ \text{Arc Length} = \theta \cdot r \]
where \( \theta \) is the central angle in radians and \( r \) is the radius.
Given:
\[ \theta = 45^\circ = \frac{45}{360} \cdot 2\pi = \frac{\pi}{4} \text{ radians} \]
\[ r = 8 \text{ ft} \]
\[ \text{Arc Length} = \frac{\pi}{4} \cdot 8 = 2\pi \text{ ft} \]
#### Measure of \( \overset{\frown}{DE} \)
To find the measure of arc \( \overset{\frown}{DE} \):
\[ \text{Measure of Arc} = \frac{\text{Arc Length}}{2\pi r} \cdot 360^\circ \]
Given:
\[ \text{Arc Length} = 8.73 \text{ in}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42779bdc-66e8-47e8-8a59-0cd69b39b8c3%2F5145ce27-8b15-42fd-b2e5-86d5a281c9be%2F91ft56u_processed.jpeg&w=3840&q=75)
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