2008 AMC 12A Problems/Problem 18

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 (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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2008 AMC 12A Problems/Problem 18

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