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The edge length of a regular tetrahedron ABCD is 2. M and N are the midpoints of BC and AD, respectively. Find angle AMD.

 Apr 23, 2022
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Point  N  is irrelevant here

 

Since   triangle  ABC  is equliateral with sides of 2,  its  height  = the slant height of the tetrahedron  = sqrt (3) = AM

 

Call the center of  the  base  E   

 

The distance  across  the base  also  =  sqrt 3 .....so 1/2  of this distance  = sqrt (3) / 2

 

And  AME forms a right triangle  with   leg   ME  =  sqrt (3) / 2   and  hypotenuse AM  =  sqrt (3)

 

Then   cos  AME   = cos AMD  =  ME / AM  =     [sqrt (3) / 2 ]   /sqrt (3)  =   1/2

 

So

 

arccos  (1/2)  =  measure of AMD   =  60°

 

cool cool cool

 Apr 23, 2022

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