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.
### Diagram Description
In the provided diagram, there is a circle centered at point O with a radius of 4 units. The points A and B are on the circumference of the circle, and the angle subtended by the arc AB at the center O is 120 degrees.
### Answer Choices
- [ ] 5
- [ ] 6
- [ ] 8
- [ ] 9
### Solution Explanation
To find the arc length of AB, we use the formula for the arc length in a circle:
\[ \text{Arc Length} = \theta \times r \]
where:
- \(\theta\) is the angle in radians.
- \(r\) is the radius of the circle.
First, convert the central angle from degrees to radians:
\[ \theta = 120^\circ \times \left(\frac{\pi \text{ radians}}{180^\circ}\right) = \frac{2\pi}{3} \text{ radians} \]
Next, use the arc length formula with the given radius \( r = 4 \):
\[ \text{Arc Length} = \frac{2\pi}{3} \times 4 = \frac{8\pi}{3} \]
\[ \text{Arc Length} \approx 8.38 \]
Rounding to the nearest integer, the arc length is approximately 8 units.
Therefore, the answer is:
- [ ] 5
- [ ] 6
- [x] 8
- [ ] 9](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf730a62-54d5-4728-9031-15f18154a0bf%2F715c49cf-ae76-46c9-a10e-d1638bc23402%2F5xravpp_processed.png&w=3840&q=75)
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