To determine the radius of the circle, let's follow these steps:
1. Understand the relationship between the arc length, radius, and central angle:
- The formula that relates the arc length ([tex]\(s\)[/tex]), radius ([tex]\(r\)[/tex]), and central angle in radians ([tex]\(\theta\)[/tex]) is:
[tex]\[
s = r \cdot \theta
\][/tex]
2. Convert the central angle from degrees to radians:
- The central angle is given as [tex]\(65^\circ\)[/tex]. To convert degrees to radians, we use the conversion factor [tex]\(\pi\)[/tex] radians = [tex]\(180^\circ\)[/tex]:
[tex]\[
\theta = 65^\circ \times \frac{\pi \text{ radians}}{180^\circ} = \frac{65\pi}{180} = \frac{13\pi}{36} \text{ radians}
\][/tex]
3. Plug the known values into the formula:
- Here, the length of the arc ([tex]\(s\)[/tex]) is [tex]\(\frac{26}{9} \pi\)[/tex] cm, and [tex]\(\theta\)[/tex] (in radians) is [tex]\(\frac{13\pi}{36}\)[/tex]:
[tex]\[
\frac{26}{9} \pi = r \cdot \frac{13\pi}{36}
\][/tex]
4. Solve for the radius ([tex]\(r\)[/tex]):
- First, we can simplify the equation by dividing both sides by [tex]\(\pi\)[/tex]:
[tex]\[
\frac{26}{9} = r \cdot \frac{13}{36}
\][/tex]
- To isolate [tex]\(r\)[/tex], multiply both sides by [tex]\(\frac{36}{13}\)[/tex]:
[tex]\[
r = \frac{26}{9} \times \frac{36}{13} = \frac{26 \times 36}{9 \times 13}
\][/tex]
- Simplifying this further:
[tex]\[
r = \frac{936}{117} = 8 \text{ cm}
\][/tex]
Therefore, the radius of the circle is [tex]\(\boxed{8 \text{ cm}}\)[/tex].
