Question

给定一个半径为 $2$ 的圆，有许多长度为 $2$ 的线段在各自中点处与圆相切。求所有这些线段组成区域的面积。

Given a circle of radius $2,$ there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.

$\frac{\pi}{4}$

$4-\pi$

$\frac{\pi}{2}$

$\pi$

$2\pi$

答案：D

知识点：geometric loci annulus area

难度评级：1630

小提示：

每条切线段从中点向两个方向各延伸一个单位

Each tangent segment extends one unit in each direction from its midpoint

大提示：

从圆心作直角三角形，求出扫过区域的内、外半径

Use a right triangle from the circle’s center to locate the inner and outer radii of the swept region

解答：

每条线段的中心是距圆心 $2$ 个单位的切点，并沿切线向两个方向各延伸 $1$ 个单位。当切点绕圆转动时，这些线段填满一个内半径为 $2$ 、外半径为 $\sqrt{2^2+1^2}=\sqrt5$ 的圆环。其面积为 $\pi(5-4)=\pi$ ，所以正确答案是 D。

Each segment is centered at a tangency point $2$ units from the circle’s center and extends $1$ unit along the tangent in both directions. As the tangency point rotates, the segments fill the annulus with inner radius $2$ and outer radius $\sqrt{2^2+1^2}=\sqrt5.$ Its area is $\pi(5-4)=\pi,$ so the correct answer is D.

1996 AMC 12 第 11 题

答案：D

难度评级：1630

小提示：

大提示：

解答：

其他年份的第 11 题