2002 AMC 12A Problem 7

If an arc of

45^\circ

on circle

A

has the same length as an arc of

30^\circ

on circle

B,

then the ratio of the area of circle

A

to the area of circle

B

Show solution

Solution

Equal arc lengths give

\dfrac{45}{360}\cdot 2\pi R_A = \dfrac{30}{360}\cdot 2\pi R_B,

\dfrac{R_A}{R_B} = \dfrac{30}{45} = \dfrac{2}{3}.

The ratio of areas is

\left(\dfrac{R_A}{R_B}\right)^2 = \dfrac{4}{9}.

Thus, the correct answer is A.