2008 AMC 12A Problems/Problem 18

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Problem

A triangle $\triangle ABC$ with sides $5$, $6$, $7$ is placed in the three-dimensional plane with one vertex on the positive $x$ axis, one on the positive $y$ axis, and one on the positive $z$ axis. Let $O$ be the origin. What is the volume if $OABC$?

$\textbf{(A)}\ \sqrt{85} \qquad \textbf{(B)}\ \sqrt{90} \qquad \textbf{(C)}\ \sqrt{95} \qquad \textbf{(D)}\ 10 \qquad  \textbf{(E)}\ \sqrt{105}$

Solution

WLOG, we let $AB$ go between the $x$ and $y$ axes, $BC$ between $y$ and $z$ axes$,$CA$between$z$and$x$axes. Let$x,y,z$be$OA,OB,OC,$respectively. By the [[Pythagorean Theorem]],$x^2+y^2=25$,$y^2+z^2=36$,$z^2+x^2=49$. Thus,$x^2 = 30$,$y^2 = 19$, and$z^2 = 6$. Thus the volume of the tetraehdron is$\frac{\sqrt{30\cdot 19\cdot 6}}{6}=\sqrt{95}\Rightarrow \boxed{C}$.


2008 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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All AMC 12 Problems and Solutions