Question

考虑两个同心圆，半径分别为 $17$ 和 $19$ 。较大的圆有一条弦，其中一半位于较小圆内。这条较大圆中的弦长是多少？

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?

$12\sqrt{2}$

$10\sqrt{3}$

$\sqrt{17 \cdot 19}$

$18$

$8\sqrt{6}$

答案：E

知识点：弦勾股定理

难度评级：1590

解答：

设这条弦到共同圆心的距离为 $d$ 。它的总长度为 $2\sqrt{361 - d^2}$ ，位于较小圆内的部分长度为 $2\sqrt{289 - d^2}$ 。

因为弦的一半位于小圆内， $2\sqrt{289 - d^2} = \tfrac{1}{2}\cdot 2\sqrt{361 - d^2}$ 。平方得 $4(289 - d^2) = 361 - d^2$ ，所以 $3d^2 = 795$ ， $d^2 = 265$ 。

弦长为 $2\sqrt{361 - 265} = 2\sqrt{96} = 8\sqrt{6}$ 。

所以正确答案是 E。

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 第 11 题

答案：E

难度评级：1590

解答：

其他年份的第 11 题