To determine the radius of a circle given the length of an arc and the angle subtended by that arc at the center, we can follow these steps:
1. Understand the Given Information:
- The arc length [tex]\( L \)[/tex] is [tex]\(\frac{26}{9} \pi\)[/tex] centimeters.
- The central angle [tex]\( \theta \)[/tex] subtended by the arc is [tex]\( 65^\circ \)[/tex].
2. Convert the Central Angle from Degrees to Radians:
Since the formula for the arc length involves the angle in radians, we need to convert [tex]\( 65^\circ \)[/tex] to radians.
[tex]\[
\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}
\][/tex]
[tex]\[
\theta_{\text{radians}} = 65 \times \frac{\pi}{180}
\][/tex]
[tex]\[
\theta_{\text{radians}} = \frac{65\pi}{180}
\][/tex]
Simplifying [tex]\(\frac{65}{180}\)[/tex]:
[tex]\[
\frac{65\pi}{180} = \frac{13\pi}{36}
\][/tex]
Therefore,
[tex]\[
\theta_{\text{radians}} \approx 1.1344640137963142
\][/tex]
3. Use the Formula for the Arc Length to Find the Radius:
The formula relating the arc length [tex]\( L \)[/tex], the radius [tex]\( r \)[/tex], and the central angle [tex]\( \theta \)[/tex] (in radians) is:
[tex]\[
L = r \theta
\][/tex]
Plug in the known values [tex]\( L = \frac{26}{9} \pi \)[/tex] and [tex]\( \theta = \frac{13\pi}{36} \)[/tex]:
[tex]\[
\frac{26}{9} \pi = r \times \frac{13\pi}{36}
\][/tex]
4. Solve for the Radius [tex]\( r \)[/tex]:
To isolate [tex]\( r \)[/tex], divide both sides by [tex]\( \frac{13\pi}{36} \)[/tex]:
[tex]\[
r = \frac{\frac{26}{9} \pi}{\frac{13\pi}{36}}
\][/tex]
Simplify the fraction:
[tex]\[
r = \frac{\frac{26}{9} \pi \times 36}{13\pi}
\][/tex]
[tex]\[
r = \frac{26 \times 36}{9 \times 13}
\][/tex]
Cancel common factors:
[tex]\[
r = \frac{26 \times 4}{13}
\][/tex]
[tex]\[
r = \frac{104}{13}
\][/tex]
[tex]\[
r = 8
\][/tex]
Therefore, the radius of the circle is [tex]\(8 \text{ cm}\)[/tex].
The correct answer is [tex]\( \boxed{8 \text{ cm}} \)[/tex].
