To find the radius of a circle given the arc length and the central angle, we can follow these steps:
### Step 1: Understand the relationship between the arc length, radius, and central angle
The formula that relates 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]
### Step 2: Convert the central angle from degrees to radians
The central angle is given as [tex]\( 65^\circ \)[/tex]. To use the formula, we need to convert the angle to radians. The conversion factor from degrees to radians is:
[tex]\[ \theta = \theta_{\text{degrees}} \times \frac{\pi}{180^\circ} \][/tex]
So,
[tex]\[ \theta = 65^\circ \times \frac{\pi}{180^\circ} = \frac{65\pi}{180} = \frac{13\pi}{36} \][/tex]
### Step 3: Substitute the known values into the formula
The arc length [tex]\( L \)[/tex] is given as [tex]\( \frac{26}{9} \pi \)[/tex] cm, and we now have the central angle [tex]\( \theta \)[/tex] in radians, [tex]\( \frac{13\pi}{36} \)[/tex].
Using the formula [tex]\( L = r \theta \)[/tex]:
[tex]\[ \frac{26}{9} \pi = r \times \frac{13\pi}{36} \][/tex]
### Step 4: Solve for the radius [tex]\( r \)[/tex]
First, cancel out the common factor [tex]\( \pi \)[/tex] from both sides of the equation:
[tex]\[ \frac{26}{9} = r \times \frac{13}{36} \][/tex]
Next, solve for [tex]\( r \)[/tex] by multiplying both sides of the equation by the reciprocal of [tex]\( \frac{13}{36} \)[/tex]:
[tex]\[ r = \left( \frac{26}{9} \right) \times \left( \frac{36}{13} \right) \][/tex]
Simplify the right-hand side:
[tex]\[ r = \frac{26 \times 36}{9 \times 13} \][/tex]
[tex]\[ r = \frac{936}{117} \][/tex]
[tex]\[ r = 8 \][/tex]
Thus, the radius of the circle is [tex]\( 8 \)[/tex] cm.
Therefore, the correct answer is:
[tex]\[ \boxed{8 \text{ cm}} \][/tex]
