2008 AMC 10A Problems/Problem 21
Problem
A cube with side length is sliced by a plane that passes through two diagonally opposite vertices and and the midpoints and of two opposite edges not containing or , as shown. What is the area of quadrilateral ?
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 , it follows that is a rhombus. The area of the rhombus can be computed by the formula , where are the diagonals of the rhombus (or of a kite in general). has the same length as a face diagonal, or . is a space diagonal, with length . Thus .
See also
2008 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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All AMC 10 Problems and Solutions |