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.
![### Finding the Arc Length of a Circle Segment
**Problem Statement:**
Find the arc length of the highlighted arc. Solve for \( x \).
**Diagram Explanation:**
The diagram is a circle with a radius of 16 feet. A central angle measuring 150 degrees (represented as \( 150^\circ \)) partially extends from the center to the circumference, indicating the portion of the circle that the arc spans. The highlighted arc spans a part of the circumference and is marked in yellow. The length of this arc is denoted as "x".
**Solution Steps:**
1. **Understand the formula for arc length:**
The formula for the length (\( L \)) of an arc is given by:
\[
L = \theta \times r
\]
where \( \theta \) is the central angle in radians, and \( r \) is the radius of the circle.
2. **Convert the angle from degrees to radians:**
We know:
\[
\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}
\]
Substituting the given angle:
\[
\theta_{radians} = 150^\circ \times \frac{\pi}{180} = \frac{5\pi}{6} \text{ radians}
\]
3. **Substitute the values into the arc length formula:**
Given \( r = 16 \) ft:
\[
L = \left( \frac{5\pi}{6} \right) \times 16
\]
4. **Calculate the arc length:**
\[
L = \frac{5 \times 16 \times \pi}{6} = \frac{80\pi}{6} = \frac{40\pi}{3} \text{ feet}
\]
\[
L \approx 41.89 \text{ feet} \text{ (rounded to two decimal places)}
\]
Thus, the arc length \( x \) is approximately \( 41.89 \) feet.
### Conclusion
By following the steps outlined above, you can determine the arc length of any segment in a circle, provided you have the measure of the central angle and the radius of the circle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5632c3a2-cb94-41fb-b077-6290f79dedec%2F1831d3aa-5e70-4c26-b634-522ed4b2710e%2Fbe4g8jd_processed.jpeg&w=3840&q=75)
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