To solve the problem of finding the radius of a circle given the arc length and the central angle, we can follow these steps:
1. Identify the given values:
- Arc length ([tex]\(L\)[/tex]) is [tex]\(\frac{26}{9} \pi\)[/tex] cm.
- Central angle ([tex]\(\theta\)[/tex]) is 65°.
2. Convert the central angle from degrees to radians:
- The relationship between degrees and radians is [tex]\( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \)[/tex].
- So, [tex]\( \theta_{\text{radians}} = 65 \times \frac{\pi}{180} \)[/tex].
3. Calculate the central angle in radians:
- [tex]\(\theta_{\text{radians}} = \frac{65\pi}{180} \approx 1.1344640137963142 \)[/tex] radians.
4. Use the formula for the length of an arc:
- The formula for the arc length is [tex]\( L = r \times \theta \)[/tex] where [tex]\(r\)[/tex] is the radius and [tex]\(\theta\)[/tex] is in radians.
- Rearrange the formula to solve for [tex]\(r\)[/tex]: [tex]\( r = \frac{L}{\theta} \)[/tex].
5. Substitute the known values into the formula:
- [tex]\( r = \frac{\frac{26}{9} \pi}{1.1344640137963142} \)[/tex].
6. Calculate the radius:
- The result of this calculation gives [tex]\( r = 8 \)[/tex].
Therefore, the length of the circle's radius is [tex]\(\boxed{8 \, \text{cm}}\)[/tex].
