Finding the length of line-plane intersection 
It is known that an ABCD.EFGH cube has a 12 cm side. If P was an intersection between two lines, BG and FC, and. Q is the intersecting point of segment EP on plane DBFH, find out the length of segment EQ. 
Now I tried to make another plane, which is ACGE in an attempt to find where is Q exactly? So that's the only problem, I can't seem to find out what should I do to find the EQ. I will do the math on my own, thank you!
 A: From $P$ and $Q$ drop perpendiculars $PL$ and $QM$ to the top face of the cube. In square $EFGH$ you can then easily find, by similarity, $EM=(2/3)EL$ and then, again by similarity, $EQ=(2/3)EP$. On the other hand:
$$
EP=\sqrt{EL^2+LP^2}=\sqrt{EF^2+FL^2+LP^2}=\sqrt{12^2+6^2+6^2}=6\sqrt6.
$$
Hence: $EQ=4\sqrt6$.

A: I will use Aretino's picture to describe it. We have to find the x,y and z coordinates of the Q point.Let's find x and y coordinate first from the EFGH face.
So,let's find $EN$ first.for that, I will work on $EFGH$ face only.considering the face as a 2D plane where $E$ is the origin.$EF$ is toward $x$ direction and $EH$ is toward $y$ direction.Now,let's find the coordinates.Here,$E(0,0),F(12,0),G(12,12),H(0,12)$.So,we notice that the equation of $FH$ line is
$$y=-x+12......(1)$$
And the equation of $EL$ line is
$$y=\dfrac{1}{2}x......(2)$$
Solving (1) and (2) we get M(8,4).
So,x and y coordinate of the Q point is 8 and 4.Now we have to calculate z coordinate.So, let's work on the face $ABEF$
The equation of line EP is 
$$x+2y=24.................(3)$$
$$[\text{As, It is passing through E(0,12),P(12,6)}]$$
Emagine the line 
$$x=8..........(4)$$
So,we get (8,8)
So,the z coordinate is 8.[4 from plane EFGH]
Q(8,4,8).
These gives $$EM=\sqrt{8^2+4^2}=4\sqrt{5}$$
Hence, $$EQ=\sqrt{(4\sqrt{5})^2+4^2}=4\sqrt{6}$$
