To determine the radius of the circle given the arc length and the central angle, we can make use of the relationship between the arc length, the radius, and the central angle measured in radians. Let's follow these steps:
1. Convert the Central Angle to Radians:
The central angle is given in degrees. To apply the arc length formula, we must first convert the angle to radians.
The formula to convert degrees to radians is:
[tex]\[
\text{angle in radians} = \text{angle in degrees} \times \left(\frac{\pi}{180}\right)
\][/tex]
Therefore,
[tex]\[
65^\circ = 65 \times \left(\frac{\pi}{180}\right) \text{ radians}
\][/tex]
Performing the multiplication:
[tex]\[
65 \times \left(\frac{\pi}{180}\right) = \frac{65\pi}{180} \text{ radians}
\][/tex]
Simplifying the fraction:
[tex]\[
\frac{65\pi}{180} = \frac{13\pi}{36} \text{ radians}
\][/tex]
2. Use the Arc Length Formula:
The arc length [tex]\( s \)[/tex] is related to the radius [tex]\( r \)[/tex] and the central angle [tex]\( \theta \)[/tex] (in radians) by the formula:
[tex]\[
s = r \theta
\][/tex]
We are given the arc length [tex]\( s = \frac{26}{9} \pi \)[/tex] centimeters, and the central angle [tex]\( \theta = \frac{13\pi}{36} \)[/tex] radians.
Substituting these values into the formula, we have:
[tex]\[
\frac{26}{9} \pi = r \times \frac{13\pi}{36}
\][/tex]
3. Solve for the Radius:
Now we solve for [tex]\( r \)[/tex]:
[tex]\[
\frac{26}{9} \pi = r \times \frac{13\pi}{36}
\][/tex]
To isolate [tex]\( r \)[/tex], divide both sides of the equation by [tex]\( \frac{13\pi}{36} \)[/tex]:
[tex]\[
r = \frac{\frac{26}{9} \pi}{\frac{13\pi}{36}}
\][/tex]
Simplify the division:
[tex]\[
r = \frac{\frac{26}{9} \pi}{\frac{13\pi}{36}} = \frac{26}{9} \times \frac{36}{13}
\][/tex]
Cancelling [tex]\(\pi\)[/tex] and simplifying the fractions:
[tex]\[
r = \frac{26 \times 36}{9 \times 13} = \frac{936}{117} = 8
\][/tex]
Thus, the radius [tex]\( r \)[/tex] of the circle is:
[tex]\[
r = 8 \text{ cm}
\][/tex]
So, the correct answer is [tex]\( \boxed{8 \text{ cm}} \)[/tex].
