The length of an arc of a circle is [tex] \frac{26}{9} \pi [/tex] centimeters, and the measure of the corresponding central angle is [tex] 65^{\circ} [/tex]. What is the length of the circle's radius?

A. 8 cm
B. 4 cm
C. 2 cm
D. 16 cm



Answer :

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