Question

Suppose that circles R and S have a central angle measuring 125°. Additionally, circle R has a radius of 2 3 feet and the radius of circle S is 3 4 feet. If the length of the intercepted arc for circle R is 4 9 π feet, what is the length of the intercepted arc for circle S?

Answer 1

Let 
rR--------> radius of the circle R
rS-------> radius of the circle S
LR------> the length of the intercepted arc for circle R
LS------> the length of the intercepted arc for circle S

we have that
rR=2/3 ft
rS=3/4 ft
rR/rS=8/9--------> rS/rR=9/8
LR=(4/9)π ft

we know that
if Both circle R and circle S have a central angle , the ratio of the radius of circle R to the radius of circle S is equals to the ratio of the length of circle R to the length of circle S

rR/rS=LR/LS--------> LS=LR*rS/rR-----> [(4/9)π*9/8]----> (1/2)π ft

the answer is
the length of the intercepted arc for circle S is  (1/2)π ft