DVI exam
2022 221 problem 7
The volume of a triangular prism 
with base 
and side edges 
is equal to 
Find the volume of the tetrahedron 
where 
is the center of the face 
is the point of intersection of the medians of 
is the midpoint of the edge 
and 
is the middle of the edge 
Solution
2022 222 problem 7
![\[r = 0.5, h = 3/2, KM = \frac {\sqrt{3}}{2},\]](http://latex.artofproblemsolving.com/1/d/c/1dca112ae88f73b3097142899ae597d9184f3a30.png)
![\[AM = KM \cdot \tan \alpha, BM = \frac {KM}{\tan \alpha},\]](http://latex.artofproblemsolving.com/2/b/c/2bc88738c2efa898feb0c22b93dc8754aa27f8d8.png)
![\[AM + BM = AB \implies\]](http://latex.artofproblemsolving.com/6/6/b/66bd3dd3f4588d2579c1569d2c36491417e20a8c.png)
![\[\tan \alpha + \frac {1}{\tan \alpha} = 2 \sqrt {2} \implies \tan \alpha = \sqrt {2} - 1 \implies \tan 2 \alpha = 1 \implies 2 \alpha = \frac {\pi}{4}.\]](http://latex.artofproblemsolving.com/a/d/9/ad94bbf8346d00722305ab7c3dca276520748cb6.png)
