<p>Triangle <img alt="$ABC$" height="13" src="https://latex.artofproblemsolving.com/e/2/a/e2a559986ed5a0ffc5654bd367c29dfc92913c36.png" width="42" />, with sides of length <img alt="$5$" height="12" src="https://latex.artofproblemsolving.com/7/9/0/79069377f91364c2f87a64e5f9f562a091c8a6c1.png" width="8" />, <img alt="$6$" height="12" src="https://latex.artofproblemsolving.com/6/0/1/601a7806cbfad68196c43a4665871f8c3186802e.png" width="8" />, and <img alt="$7$" height="12" src="https://latex.artofproblemsolving.com/e/0/a/e0a0db32027a732ac57d37ef2ae9bb150f65b108.png" width="9" />, has one <a href="https://artofproblemsolving.com/wiki/index.php/Vertex" title="Vertex">vertex</a> on the positive <img alt="$x$" height="10" src="https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" width="12" />-axis, one on the positive <img alt="$y$" height="13" src="https://latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png" width="11" />-axis, and one on the positive <img alt="$z$" height="10" src="https://latex.artofproblemsolving.com/b/1/3/b13f21416d84e13708696f34dea81026cda583c9.png" width="11" />-axis. Let <img alt="$O$" height="12" src="https://latex.artofproblemsolving.com/5/1/d/51da37d984564162c87710ca27bea422f657fb73.png" width="13" /> be the <a href="https://artofproblemsolving.com/wiki/index.php/Origin" title="Origin">origin</a>. What is the volume of <a href="https://artofproblemsolving.com/wiki/index.php/Tetrahedron" title="Tetrahedron">tetrahedron</a> <img alt="$OABC$" height="13" src="https://latex.artofproblemsolving.com/b/9/1/b91381b367f9753ded66ab981c7451f2cb117578.png" width="56" />?</p> <p><img alt="$\mathrm{(A)}\ \sqrt{85}\qquad\mathrm{(B)}\ \sqrt{90}\qquad\mathrm{(C)}\ \sqrt{95}\qquad\mathrm{(D)}\ 10\qquad\mathrm{(E)}\ \sqrt{105}$" height="21" src="https://latex.artofproblemsolving.com/9/a/9/9a94bcd981db19acc94bb9f2886ff3adf2cd330b.png" width="458" /></p>

Question No: 322

Triangle $ABC$ , with sides of length $5$ , $6$ , and $7$ , has 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 of tetrahedron $OABC$ ?

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

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Tags: Geometry

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