2008 AMC 12A Problem 18

Below is the professionally curated solution for Problem 18 of the 2008 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 12A solutions, or check the answer key.

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Concepts:3D geometry system of equations volume

Difficulty rating: 1910

18.

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 $OABC?$

$\sqrt{85}$

$\sqrt{90}$

$\sqrt{95}$

$10$

$\sqrt{105}$

Solution:

Let $A = (a, 0, 0),$ $B = (0, b, 0),$ $C = (0, 0, c).$ Assigning the sides, $a^2 + b^2 = 25, \quad b^2 + c^2 = 36, \quad a^2 + c^2 = 49.$

Adding gives $a^2 + b^2 + c^2 = 55,$ so $a^2 = 19,$ $b^2 = 6,$ and $c^2 = 30.$

The volume is $\dfrac{1}{6}abc = \dfrac{1}{6}\sqrt{19 \cdot 6 \cdot 30} = \dfrac{1}{6}\sqrt{3420} = \sqrt{95}.$

Thus, C is the correct answer.

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