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.
In the
![**Problem**
In the circle below, segment \( AB \) is a diameter. If the length of arc \( ACB \) is \( 6\pi \), what is the length of the radius of the circle?
**Diagram Explanation**
- A circle is shown with a diameter labeled \( AB \).
- Three points are marked on the circumference: \( A \), \( B \), and \( C \).
- The diameter \( AB \) connects two points on the circle, passing through the center.
- An arc is indicated by \( \overset{\frown}{ACB} \), covering part of the circle's circumference from point \( A \) to point \( B \) through point \( C \).
**Calculation & Conclusion**
To solve for the radius, we use the relationship between the arc length and the circumference of the circle. The given arc length \( \overset{\frown}{ACB} = 6\pi \), which is the length of a semicircle (half of the circle’s circumference).
Thus, the entire circumference of the circle would be:
\[ 2 \times 6\pi = 12\pi \]
The circumference \( C \) of a circle is also given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius. Setting the two expressions for the circumference equal gives:
\[ 2\pi r = 12\pi \]
Solving for \( r \):
\[ r = \frac{12\pi}{2\pi} = 6 \]
So, the radius of the circle is \( \boxed{6} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5cae965-fbbf-45bc-b876-726779e5ae34%2F27aef100-eee8-46ad-899a-c2c7be7238d2%2F51phndb_processed.jpeg&w=3840&q=75)
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