2008 AMC 12A Problem 18

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}

Let the vertices be

(a, 0, 0),

(0, b, 0),

(0, 0, c)

and write the three side-length equations

Adding the three equations gives

a^2 + b^2 + c^2,

and the volume is

\tfrac{1}{6}abc

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.

Problem 18 in other years