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?

$12\sqrt{2}$

$10\sqrt{3}$

$\sqrt{17 \cdot 19}$

$18$

$8\sqrt{6}$

Answer: E

Concepts:chord Pythagorean Theorem

Difficulty rating: 1590

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

2021 AMC 12A Fall Problem 11

Answer: E

Difficulty rating: 1590

Solution:

Problem 11 in Other Years