Problem:
Tetrahedron ABCD has AB=5,AC=3,BC=4,BD=4,AD=3, and CD=512​2​. What is the volume of the tetrahedron?
Answer Choices:
A. 32​
B. 25​
C. 524​
D. 33​
E. 524​2​
Solution:
Triangles ABC and ABD are 3−4−5 right triangles with area 6. Let CE be the altitude of △ABC. Then CE=512​. Likewise in △ABD, DE=512​. Triangle CDE has sides 512​,512​, and 512​2​, so it is an isosceles right triangle with right angle CED. Therefore DE is the altitude of the tetrahedron to base ABC. The tetrahedron's volume is 31​⋅6⋅512​=(C)524​​.
