2023 AMC 12B Problem 3

Attempt Problem 3 of the 2023 AMC 12B 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 2023 AMC 12B solutions, or check the answer key.

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

3

4

5

right triangle is inscribed in circle

A,

and a

5

12

13

right triangle is inscribed in circle

B.

What is the ratio of the area of circle

A

to the area of circle

B?

\dfrac{9}{25}

\dfrac{1}{9}

\dfrac{1}{5}

\dfrac{25}{169}

\dfrac{4}{25}

A right triangle inscribed in a circle has its hypotenuse as a diameter

The circles have diameters

5

and

13;

the area ratio is the square of the ratio of radii

The hypotenuse of an inscribed right triangle is a diameter, so circle

A

has diameter

5

and circle

B

has diameter

13.

The ratio of areas is

\left(\tfrac{5}{13}\right)^2= \tfrac{25}{169}.

Thus, the correct answer is D.

Problem 3 in other years