Difference between revisions of "DVI exam"
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| + | ==2022 221 problem 7== | ||
| + | The volume of a triangular prism <math>ABCA'B'C'</math> with base <math>ABC</math> and side edges <math>AA', BB', CC'</math> is equal to <math>72.</math> Find the volume of the tetrahedron <math>DEFG,</math> where <math>D</math> is the center of the face <math>ABC'A', E</math> is the point of intersection of the medians of <math>\triangle A'B'C', F</math> is the midpoint of the edge <math>AC</math> and <math>G</math> is the middle of the edge <math>BC.</math> | ||
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| + | Solution | ||
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==2022 222 problem 7== | ==2022 222 problem 7== | ||
[[File:MSU 2022 2 7.png|400px|right]] | [[File:MSU 2022 2 7.png|400px|right]] | ||
Revision as of 03:57, 28 January 2024
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)
