Question

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 $OABC?$

$\sqrt{85}$

$\sqrt{90}$

$\sqrt{95}$

$10$

$\sqrt{105}$

Answer: C

Concepts:3D geometry system of equations volume

Difficulty rating: 1910

Solution:

Let $A = (a, 0, 0),$ $B = (0, b, 0),$ $C = (0, 0, c).$ Assigning the sides, $\begin{aligned} a^2 + b^2 &= 25, \\ b^2 + c^2 &= 36, \\ a^2 + c^2 &= 49. \end{aligned}$

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

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

Thus, C is the correct answer.

2008 AMC 12A Problem 18

Answer: C

Difficulty rating: 1910

Solution:

Problem 18 in Other Years