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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. [tex]\(2 \, \text{cm}\)[/tex]

B. [tex]\(4 \, \text{cm}\)[/tex]

C. [tex]\(8 \, \text{cm}\)[/tex]

D. [tex]\(16 \, \text{cm}\)[/tex]



Answer :

GinnyAnswer
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]

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