If an arc of 45° on circle A has the same length as an arc of 30° on circle B, then the ratio of the area of circle A to the area of circle B is

2002 AMC 12A Problem 7

Attempt Problem 7 of the 2002 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

\dfrac{4}{9}

\dfrac{2}{3}

\dfrac{5}{6}

\dfrac{3}{2}

\dfrac{9}{4}

Equal arc lengths:

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

Find

\dfrac{R_A}{R_B};

the area ratio is its square

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.