Certainly! Let's solve the problem step by step.
We are given:
- The length of the arc, [tex]\( \text{arc length} = \frac{26}{9} \pi \)[/tex] centimeters.
- The central angle [tex]\( \theta = 65^\circ \)[/tex].
We need to find the length of the circle's radius.
### Step 1: Convert the central angle from degrees to radians
The formula to convert degrees to radians is:
[tex]\[ \theta (\text{radians}) = \theta (\text{degrees}) \times \frac{\pi}{180} \][/tex]
Plugging in the given central angle:
[tex]\[ \theta (\text{radians}) = 65^\circ \times \frac{\pi}{180} = \frac{65\pi}{180} \][/tex]
Simplify the fraction:
[tex]\[ \theta (\text{radians}) = \frac{65\pi}{180} = \frac{13\pi}{36} \approx 1.1344640137963142 \][/tex]
### Step 2: Use the formula for the arc length
The formula for the arc length of a circle is:
[tex]\[ \text{arc length} = r \theta \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( \theta \)[/tex] is the central angle in radians.
### Step 3: Rearrange the formula to solve for radius [tex]\( r \)[/tex]
[tex]\[ r = \frac{\text{arc length}}{\theta} \][/tex]
Plug in the given values:
[tex]\[ r = \frac{\frac{26}{9} \pi}{\frac{13 \pi}{36}} \][/tex]
### Step 4: Simplify the expression
First, notice that [tex]\( \pi \)[/tex] in the numerator and denominator cancels out:
[tex]\[ r = \frac{\frac{26}{9}}{\frac{13}{36}} \][/tex]
Now, to divide by a fraction, multiply by its reciprocal:
[tex]\[ r = \frac{26}{9} \times \frac{36}{13} = \frac{26 \times 36}{9 \times 13} \][/tex]
Simplify the fraction:
[tex]\[ r = \frac{936}{117} \][/tex]
Further simplification:
[tex]\[ r = 8 \][/tex]
### Conclusion
So, the length of the circle's radius is:
[tex]\[ r = 8 \text{ cm} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{8 \text{ cm}} \][/tex]
