2002 AMC 10A Problem 7

Question

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

is

\dfrac{4}{9}

\dfrac{2}{3}

\dfrac{5}{6}

\dfrac{3}{2}

\dfrac{9}{4}

Arc length is

\dfrac{\theta}{360}\cdot 2\pi r

Equal arc lengths give the ratio

\dfrac{r_A}{r_B}

Equal arc lengths give

\dfrac{45}{360}\cdot 2\pi r_A=\dfrac{30}{360}\cdot 2\pi r_B,

so

45 r_A=30 r_B

and

\dfrac{r_A}{r_B}=\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.

Problem 7 in other years