Difference between revisions of "2008 AMC 10A Problems/Problem 21"
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==Problem== | ==Problem== | ||
− | A cube with side length <math>1</math> is sliced by a plane that passes through two diagonally opposite vertices <math>A</math> and <math>C</math> and the | + | A [[cube]] with side length <math>1</math> is sliced by a plane that passes through two diagonally opposite vertices <math>A</math> and <math>C</math> and the [[midpoint]]s <math>B</math> and <math>D</math> of two opposite edges not containing <math>A</math> or <math>C</math>, as shown. What is the area of [[quadrilateral]] <math>ABCD</math>? |
− | + | <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> | ||
<math>\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}</math> | <math>\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}</math> | ||
==Solution== | ==Solution== | ||
− | {{ | + | <center><asy> |
+ | import three; | ||
+ | unitsize(3cm); | ||
+ | defaultpen(fontsize(8)+linewidth(0.7)); | ||
+ | currentprojection=obliqueX; | ||
+ | |||
+ | triple 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,NNE); | ||
+ | dot((0.5,1,0.5)); | ||
+ | label("$D$",D,ESE); | ||
+ | dot((0,0,0.5)); | ||
+ | label("$B$",B,NNW); | ||
+ | draw(B--C--A--B--D,linetype("4 4")); | ||
+ | draw(A--D--C); | ||
+ | </asy></center> | ||
+ | Since <math>AB = AD = CB = CD = \sqrt{\left(\frac{1}{2}\right)^2+1^2}</math>, it follows that <math>ABCD</math> is a [[rhombus]]. The area of the rhombus can be computed by the formula <math>A = \frac 12 d_1d_2</math>, where <math>d_1,\,d_2</math> are the diagonals of the rhombus (or of a [[kite]] in general). <math>BD</math> has the same length as a face diagonal, or <math>\sqrt{1^2 + 1^2} = \sqrt{2}</math>. <math>AC</math> is a space diagonal, with length <math>\sqrt{1^2+1^2+1^2} = \sqrt{3}</math>. Thus <math>A = \frac 12 \times \sqrt{2} \times \sqrt{3} = \frac{\sqrt{6}}{2}\ \mathrm{(A)}</math>. | ||
==See also== | ==See also== | ||
{{AMC10 box|year=2008|ab=A|num-b=20|num-a=22}} | {{AMC10 box|year=2008|ab=A|num-b=20|num-a=22}} | ||
+ | |||
+ | [[Category:Introductory Geometry Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 19:42, 4 June 2021
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
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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.