2021 AMC 12A Fall Problem 11

Question

Consider two concentric circles of radius

17

and

19.

The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle?

12\sqrt{2}

10\sqrt{3}

\sqrt{17 \cdot 19}

18

8\sqrt{6}

Let the chord be at distance

d

from the center. Its half-length is

\sqrt{19^2 - d^2},

and the part inside the small circle has half-length

\sqrt{17^2 - d^2}

“Half lies inside” means the inside portion equals half the whole chord:

2\sqrt{17^2 - d^2} = \sqrt{19^2 - d^2}

Let the chord lie at distance

d

from the common center. Its total length is

2\sqrt{361 - d^2},

and the portion inside the smaller circle has length

2\sqrt{289 - d^2}.

Since half the chord lies inside,

2\sqrt{289 - d^2} = \tfrac{1}{2}\cdot 2\sqrt{361 - d^2}.

Squaring gives

4(289 - d^2) = 361 - d^2,

so

3d^2 = 795

and

d^2 = 265.

The chord length is

2\sqrt{361 - 265} = 2\sqrt{96} = 8\sqrt{6}.

Thus, the correct answer is E.

Problem 11 in other years