2021 Fall AMC 12A Problem 11

Problem 11 of 25IntermediateGeometry

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?

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Solution

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,

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.