To find the radius of the circle given the length of an arc and the measure of the corresponding central angle, we can follow these steps:
1. Identify the given values:
- Arc length, [tex]\( L = \frac{26}{9} \pi \)[/tex] centimeters.
- Central angle, [tex]\( \theta = 65^\circ \)[/tex].
2. Convert the central angle from degrees to radians:
The formula to convert degrees to radians is [tex]\( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \)[/tex].
Using the given central angle:
[tex]\[
\theta_{radians} = 65^\circ \times \frac{\pi}{180}
\][/tex]
3. Simplify the conversion:
[tex]\[
\theta_{radians} = \frac{65 \pi}{180} = \frac{65}{180} \pi = \frac{13}{36} \pi \text{ radians}
\][/tex]
4. Use the arc length formula:
The formula for the arc length of a circle is:
[tex]\[
L = r \theta_{radians}
\][/tex]
where [tex]\( r \)[/tex] is the radius of the circle.
5. Substitute the known values into the arc length formula:
[tex]\[
\frac{26}{9} \pi = r \times \frac{13}{36} \pi
\][/tex]
6. Isolate [tex]\( r \)[/tex] by dividing both sides of the equation by [tex]\( \frac{13}{36} \pi \)[/tex]:
[tex]\[
r = \frac{\frac{26}{9} \pi}{\frac{13}{36} \pi}
\][/tex]
7. Cancel out [tex]\(\pi\)[/tex] from both the numerator and the denominator:
[tex]\[
r = \frac{\frac{26}{9}}{\frac{13}{36}}
\][/tex]
8. Simplify the fraction:
[tex]\[
r = \frac{26}{9} \times \frac{36}{13} = \frac{26 \times 36}{9 \times 13}
\][/tex]
9. Calculate the numerator and the denominator:
[tex]\[
\frac{26 \times 36}{9 \times 13} = \frac{936}{117}
\][/tex]
10. Simplify [tex]\(\frac{936}{117}\)[/tex] further:
Divide both the numerator and the denominator by their greatest common divisor (which is 13):
[tex]\[
\frac{936 \div 13}{117 \div 13} = \frac{72}{9} = 8
\][/tex]
Hence, the radius of the circle is:
[tex]\[
\boxed{8 \text{ cm}}
\][/tex]
