To determine the length of the radius of a circle given the length of an arc and the measure of the corresponding central angle, follow these steps:
1. Understand the given data:
- The length of the arc ([tex]\(L\)[/tex]) is given as [tex]\(\frac{26}{9} \pi\)[/tex] centimeters.
- The measure of the central angle ([tex]\(\theta\)[/tex]) is given as [tex]\(65^\circ\)[/tex].
2. Convert the central angle from degrees to radians:
- The formula to convert degrees to radians is: [tex]\(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)[/tex].
- Apply this formula to the central angle:
[tex]\[
\theta_{rad} = 65^\circ \times \frac{\pi}{180}
\][/tex]
[tex]\[
\theta_{rad} \approx 1.1344640137963142 \, \text{radians}
\][/tex]
3. Use the formula for the length of an arc:
- The formula for the length of an arc is:
[tex]\[
L = r \times \theta
\][/tex]
where [tex]\(L\)[/tex] is the arc length, [tex]\(r\)[/tex] is the radius, and [tex]\(\theta\)[/tex] is the central angle in radians.
4. Rearrange the formula to solve for [tex]\(r\)[/tex]:
- Isolate [tex]\(r\)[/tex] on one side of the equation:
[tex]\[
r = \frac{L}{\theta}
\][/tex]
- Substitute the known values ([tex]\(L = \frac{26}{9} \pi\)[/tex] and [tex]\(\theta_{rad} = 1.1344640137963142\)[/tex]):
[tex]\[
r = \frac{\frac{26}{9} \pi}{1.1344640137963142}
\][/tex]
5. Calculate the radius:
- Performing the division:
[tex]\[
r \approx 8.0 \, \text{cm}
\][/tex]
Thus, the length of the circle's radius is [tex]\(8 \, \text{cm}\)[/tex].
Final answer:
A. 8 cm
