The edge length of a regular tetrahedron $ABCD$ is $2.$ $M$ and $N$ are the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find $MN.$ [asy] size(150); import three; currentprojection = orthographic(-0.6,3,2); triple A,B,C,D,O,M,NN; O=(0,0,0); D=(0,0,sqrt(8)); A=(2,0,0); B=(-1,sqrt(3),0); C=(-1,-sqrt(3),0); M=(B+C)/2; NN=(A+D)/2; draw(D--A--B--D--C--B); draw(A--C,dashed); draw(M--NN,dotted); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,N); label("$M$",M,E); label("$N$",NN,NW); [/asy]

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

Image: It won't load but it's a regular tetrahedron and MN is a diagonal from the midpoint of a side to the midpoint of another side that is on the opposite side and on the base.

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Guest Apr 5, 2022