1996 AMC 12 Problem 11

Question

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

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

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

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.

Problem 11 in other years