2021 AMC 10A Spring Problem 13

Question

What is the volume of tetrahedron

ABCD

with edge lengths

AB = 2,

AC = 3,

AD = 4,

BC = \sqrt{13},

BD = 2\sqrt{5},

and

CD = 5?

3

2\sqrt{3}

4

3\sqrt{3}

6

Try placing

A

at the origin with

AB,AC,AD

along perpendicular axes

Check that the given opposite edge lengths match this rectangular-corner model

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Place

A=(0,0,0),

B=(2,0,0),

C=(0,3,0),

and

D=(0,0,4).

Then

\begin{aligned} BC &= \sqrt{2^2+3^2}=\sqrt{13}, \\ BD &= \sqrt{2^2+4^2}=2\sqrt5, \\ CD &= \sqrt{3^2+4^2}=5, \end{aligned}

so this coordinate model matches all the given edge lengths. The tetrahedron is a rectangular-corner tetrahedron with perpendicular edge lengths

2,3,4

from

A,

so its volume is

\frac16(2)(3)(4)=4.

Thus, C is the correct answer.

Problem 13 in other years