2004 AIME I Problems/Problem 4

Problem

A square has sides of length 2. Set $S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$ . Find $100k$ .

Solution

Without loss of generality, let $(0,0)$ , $(2,0)$ , $(0,2)$ , and $(2,2)$ be the vertices of the square. Suppose the endpoints of the segment lie on the two sides of the square determined by the vertex $(0,0)$ . Let the two endpoints of the segment have coordinates $(x,0)$ and $(0,y)$ . Because the segment has length 2, $x^2+y^2=4$ . Using the midpoint formula, we find that the midpoint of the segment has coordinates $\left(\frac{x}{2},\frac{y}{2}\right)$ . Let $d$ be the distance from $(0,0)$ to $\left(\frac{x}{2},\frac{y}{2}\right)$ . Using the distance formula we see that $d=\sqrt{\left(\frac{x}{2}\right)^2+\left(\frac{y}{2}\right)^2}= \sqrt{\frac{1}{4}\left(x^2+y^2\right)}=\sqrt{\frac{1}{4}(4)}=1$ . Thus the midpoints lying on the sides determined by vertex $(0,0)$ form a quarter-circle with radius 1.

$[asy] size(100); pointpen=black;pathpen = black+linewidth(0.7); pair A=(0,0),B=(2,0),C=(2,2),D=(0,2); D(A--B--C--D--A); picture p; draw(p,CR(A,1));draw(p,CR(B,1));draw(p,CR(C,1));draw(p,CR(D,1)); clip(p,A--B--C--D--cycle); add(p); [/asy]$

The set of all midpoints forms a quarter circle at each corner of the square. The area enclosed by all of the midpoints is $4-4\cdot \left(\frac{\pi}{4}\right)=4-\pi \approx .86$ to the nearest hundredth. Thus $100\cdot k=086$

2004 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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2004 AIME I Problems/Problem 4

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