Problem
Given the cube $ABCD.EFGH$. The point $M$ is on the edge $AD$ such that $|AM|=2|MD|$. Calculate the tangent of the angle between the planes $BCF$ and $BGM$.
(A) $3 \sqrt 2$
(B) $2 \sqrt 2$
(C) $\frac{3}{2} \sqrt 2$
(D) $\frac{1}{2} \sqrt 2$
(E) $\sqrt 2$
Attempts
Denote each point as $A=(0,0,0)$, $B=(1,0,0)$, $C=(1,1,0)$, $D=(0,1,0)$, $E=(0,0,1)$, $F=(1,0,1)$, $G=(1,1,1)$, $H=(0,1,1)$, and $M=(0,\frac{2}{3},0)$. Using an online software to compute plane equation given through three points, I arrived at $BCF \equiv x=1$ and $BGM \equiv -\frac{2}{3}x-y+z=-\frac{2}{3}$. Using dihedral angle formula, I got $\frac{\sqrt{22}}{11}$, which makes me unable to choose the answer from given the options above.
Without relying much on Euclidean geometry, how can I calculate this distance problem of solid geometry?
Thank you.
The value that you've got, $\frac{\sqrt{22}}{11}$ is the cosine of the angle. Then $$\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=\frac{\sqrt{1-\cos^2\alpha}}{\cos\alpha}\\=\frac{\sqrt{1-\frac{22}{11^2}}}{\frac{\sqrt{22}}{11}}=\sqrt{\frac{99}{22}}=3\frac1{\sqrt{2}}=\frac32\sqrt 2$$