Let s=arc length, r=radius, and  = central angle (in radians)



We set up the problem with these two constraints:

The area of the given sector is (9*12)/2 = 54 

The perimeter is 33in





So, 

    2r + s  =  33    (1)

     rs/2   =  54    (2)   

and arc-length relates to r and  by:

       *r = s     (3)





Equations (1) and (2) can be reduced (by substitution) to one equation in variable r:

    2r + (108/r) = 33

    2r - 33 + 108/r = 0



Muliply by r:

    2r^2 - 33r + 108 = 0



The LHS factors to:

    (2r-9)(r-12) = 0    



Solutions:  r=9/2   and r=12  (discard r=12, as that is the already-known circle)



Plug in r=(9/2)in (=4.5in) into (1) to get s=24in, and then use these values for r and s in (3) to get  rad  (=5.3 rad, to one tenth)





NOTICE that the radius of the first circle multiplied by 2 is the arc length found in the 2nd circle, and the arc length from the first circle divided by 2 is the radius of the 2nd circle.  



 

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