2008 AMC 10A Problems/Problem 21

Revision as of 11:36, 26 April 2008 by Azjps (talk | contribs) (solution)

Problem

A cube with side length $1$ is sliced by a plane that passes through two diagonally opposite vertices $A$ and $C$ and the midpoints $B$ and $D$ of two opposite edges not containing $A$ or $C$, as shown. What is the area of quadrilateral $ABCD$?

[asy] import three; unitsize(3cm); defaultpen(fontsize(8)+linewidth(0.7)); currentprojection=obliqueX;  draw((0.5,0,0)--(0,0,0)--(0,0,1)--(0,0,0)--(0,1,0),linetype("4 4")); draw((0.5,0,1)--(0,0,1)--(0,1,1)--(0.5,1,1)--(0.5,0,1)--(0.5,0,0)--(0.5,1,0)--(0.5,1,1)); draw((0.5,1,0)--(0,1,0)--(0,1,1)); dot((0.5,0,0)); label("$A$",(0.5,0,0),WSW); dot((0,1,1)); label("$C$",(0,1,1),NE); dot((0.5,1,0.5)); label("$D$",(0.5,1,0.5),ESE); dot((0,0,0.5)); label("$B$",(0,0,0.5),NW);[/asy]

$\mathrm{(A)}\ \frac{\sqrt{6}}{2}\qquad\mathrm{(B)}\ \frac{5}{4}\qquad\mathrm{(C)}\ \sqrt{2}\qquad\mathrm{(D)}\ \frac{5}{8}\qquad\mathrm{(E)}\ \frac{3}{4}$

Solution

import three;
unitsize(3cm);
defaultpen(fontsize(8)+linewidth(0.7));
currentprojection=obliqueX;

pair A=(0.5,0,0),C=(0,1,1),D=(0.5,1,0.5),B=(0,0,0.5);
draw((0.5,0,0)--(0,0,0)--(0,0,1)--(0,0,0)--(0,1,0),linetype("4 4"));
draw((0.5,0,1)--(0,0,1)--(0,1,1)--(0.5,1,1)--(0.5,0,1)--(0.5,0,0)--(0.5,1,0)--(0.5,1,1));
draw((0.5,1,0)--(0,1,0)--(0,1,1));
dot((0.5,0,0));
label("$A$",A,WSW);
dot((0,1,1));
label("$C$",C,NE);
dot((0.5,1,0.5));
label("$D$",D,ESE);
dot((0,0,0.5));
label("$B$",B,NW);
draw(B--C--A--B--D,linetype("4 4"));
draw(A--D--C);
 (Error compiling LaTeX. pair A=(0.5,0,0),C=(0,1,1),D=(0.5,1,0.5),B=(0,0,0.5);
       ^
c61372ada9ce42f9f6218ef0acd0853c5bc235c1.asy: 8.8: cannot cast 'triple' to 'pair'
pair A=(0.5,0,0),C=(0,1,1),D=(0.5,1,0.5),B=(0,0,0.5);
                   ^
c61372ada9ce42f9f6218ef0acd0853c5bc235c1.asy: 8.20: cannot cast 'triple' to 'pair'
pair A=(0.5,0,0),C=(0,1,1),D=(0.5,1,0.5),B=(0,0,0.5);
                             ^
c61372ada9ce42f9f6218ef0acd0853c5bc235c1.asy: 8.30: cannot cast 'triple' to 'pair'
pair A=(0.5,0,0),C=(0,1,1),D=(0.5,1,0.5),B=(0,0,0.5);
                                           ^
c61372ada9ce42f9f6218ef0acd0853c5bc235c1.asy: 8.44: cannot cast 'triple' to 'pair')

Since $AB = AD = CB = CD = \sqrt{.5^2+1^2}$, it follows that $ABCD$ is a rhombus. The area of the rhombus can be computed by the formula $A = \frac 12 d_1d_2$, where $d_1,\,d_2$ are the diagonals of the rhombus (or of a kite in general). $BD$ has the same length as a face diagonal, or $\sqrt{1^2 + 1^2} = \sqrt{2}$. $AC$ is a space diagonal, with length $\sqrt{1^2+1^2+1^2} = \sqrt{3}$. Thus $A = \frac 12 \times \sqrt{2} \times \sqrt{3} = \frac{\sqrt{6}}{2}\ \mathrm{(A)}$.

See also

2008 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
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All AMC 10 Problems and Solutions
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