In the circle below, segment AB is a diameter. If the length of arc is 6 , what is the length of the radius of the circle?
Transcribed Image Text:**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 \( \boxed{6} \).
Two-dimensional figure measured in terms of radius. It is formed by a set of points that are at a constant or fixed distance from a fixed point in the center of the plane. The parts of the circle are circumference, radius, diameter, chord, tangent, secant, arc of a circle, and segment in a circle.
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![**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)
