Difference between revisions of "2014 AIME II Problems/Problem 3"
Kevin38017 (talk | contribs) (Created page with "==Problem== A rectangle has sides of length <math>a</math> and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides...") |
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A rectangle has sides of length <math>a</math> and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length <math>a</math> can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length <math>a</math> parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find <math>a^2</math>. | A rectangle has sides of length <math>a</math> and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length <math>a</math> can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length <math>a</math> parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find <math>a^2</math>. | ||
| − | + | <asy> | |
| + | pair A,B,C,D,E,F,R,S,T,X,Y,Z; | ||
| + | dotfactor = 2; | ||
| + | unitsize(.1cm); | ||
| + | A = (0,0); | ||
| + | B = (0,18); | ||
| + | C = (0,36); | ||
| + | // don't look here | ||
| + | D = (12*2.236, 36); | ||
| + | E = (12*2.236, 18); | ||
| + | F = (12*2.236, 0); | ||
| + | draw(A--B--C--D--E--F--cycle); | ||
| + | dot(" ",A,NW); | ||
| + | dot(" ",B,NW); | ||
| + | dot(" ",C,NW); | ||
| + | dot(" ",D,NW); | ||
| + | dot(" ",E,NW); | ||
| + | dot(" ",F,NW); | ||
| + | //don't look here | ||
| + | R = (12*2.236 +22,0); | ||
| + | S = (12*2.236 + 22 - 13.4164,12); | ||
| + | T = (12*2.236 + 22,24); | ||
| + | X = (12*4.472+ 22,24); | ||
| + | Y = (12*4.472+ 22 + 13.4164,12); | ||
| + | Z = (12*4.472+ 22,0); | ||
| + | draw(R--S--T--X--Y--Z--cycle); | ||
| + | dot(" ",R,NW); | ||
| + | dot(" ",S,NW); | ||
| + | dot(" ",T,NW); | ||
| + | dot(" ",X,NW); | ||
| + | dot(" ",Y,NW); | ||
| + | dot(" ",Z,NW); | ||
| + | // sqrt180 = 13.4164 | ||
| + | // sqrt5 = 2.236</asy> | ||
| + | |||
| + | ==Solution== | ||
Revision as of 20:56, 27 March 2014
Problem
A rectangle has sides of length 
and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find 
.
![[asy] pair A,B,C,D,E,F,R,S,T,X,Y,Z; dotfactor = 2; unitsize(.1cm); A = (0,0); B = (0,18); C = (0,36); // don't look here D = (12*2.236, 36); E = (12*2.236, 18); F = (12*2.236, 0); draw(A--B--C--D--E--F--cycle); dot(" ",A,NW); dot(" ",B,NW); dot(" ",C,NW); dot(" ",D,NW); dot(" ",E,NW); dot(" ",F,NW); //don't look here R = (12*2.236 +22,0); S = (12*2.236 + 22 - 13.4164,12); T = (12*2.236 + 22,24); X = (12*4.472+ 22,24); Y = (12*4.472+ 22 + 13.4164,12); Z = (12*4.472+ 22,0); draw(R--S--T--X--Y--Z--cycle); dot(" ",R,NW); dot(" ",S,NW); dot(" ",T,NW); dot(" ",X,NW); dot(" ",Y,NW); dot(" ",Z,NW); // sqrt180 = 13.4164 // sqrt5 = 2.236[/asy]](http://latex.artofproblemsolving.com/9/5/c/95cac62aaac35028476cf23e914b63412df36f70.png)
