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2014 AIME II Problems/Problem 3: Difference between revisions

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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>.  


[[2014 AIME II Problems/Problem 3|Solution]]
<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 19:56, 27 March 2014

Problem

A rectangle has sides of length $a$ 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 $a$ 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 $a$ parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find $a^2$.

[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

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