Problem:
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?
Answer Choices:
A. 85
B. 90
C. 95
D. 10
E. 105 Solution:
It may be assumed that A=(a,0,0),B=(0,b,0),C=(0,0,c), AB=5,BC=6, and CA=7. Then
a2+b2=52,b2+c2=62, and a2+c2=72
from which
a2+b2+c2=21(52+62+72)=55
It follows that a=55−62=19,b=55−72=6,c=55−52=30, and the volume of tetrahedron OABC can be expressed as