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Two concentric circles are centered at point P. The sides of a 45 degree angle at P form an arc on the smaller circle that is the same length as an arc on the larger circle formed by the sides of a 60 degree angle at P. What is the ratio of the area of the smaller circle to the area of the larger circle? Express your answer as a common fraction.

 Nov 17, 2020
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This doesn't seem possible...I believe this should  read "The sides of a 45 degree angle at P form an arc on the larger circle that is the same length as an arc on the smaller circle formed by the sides of a 60 degree angle at P.

 

On the larger circle 45° =  pi/4      ... On the smaller circle 60° = pi/3

Arc length on smaller circle  = r(pi/ 3)      where  r is the radius of the smaller circle

Arc length on larger circle   = R (pi/4)  where R is the radius of the larger  circle

 

And  since the   arc lengths are equal

r (pi/3)  = R (pi/4)

 

r = R (pi/4) / (pi/3)

 

r = (3/4)R

 

Area of  smaller circle  =     pi  ( (3/4)R)^2             9

_________________       _____________  =     ___

Area of larger circle            pi  R^2                       16

 

 

cool cool cool

 Nov 17, 2020

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