2002 AIME II Problems/Problem 2: Difference between revisions

Revision as of 12:35, 25 June 2015

Three vertices of a cube are $P=(7,12,10)$ , $Q=(8,8,1)$ , and $R=(11,3,9)$ . What is the surface area of the cube?

$PQ=\sqrt{(8-7)^2+(8-12)^2+(1-10)^2}=\sqrt{98}$

$PR=\sqrt{(11-7)^2+(3-12)^2+(9-10)^2}=\sqrt{98}$

$QR=\sqrt{(11-8)^2+(3-8)^2+(9-1)^2}=\sqrt{98}$

So, $PQR$ is an equilateral triangle. Let the side of the cube be $a$ .

$a\sqrt{2}=\sqrt{98}$

So, $a=7$ , and hence the surface area is $6a^2=\box{294}$ (Error compiling LaTeX. Unknown error_msg).

@@ Line 13: / Line 13: @@
 <math>a\sqrt{2}=\sqrt{98}</math>
-So, <math>a=7</math>, and hence the surface area=<math>6a^2=294</math>.
+So, <math>a=7</math>, and hence the surface area is <math>6a^2=\box{294}</math>.
 == See also ==
 {{AIME box|year=2002|n=II|num-b=1|num-a=3}}
 {{MAA Notice}}

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