1. ## 3D prism problem

The cross section of a right prism is an equilateral triangle. The rectangular face ABCD of the prism lies on the plane $x+y+5z=\sqrt{2}$, where A and B are the points $(0,\sqrt{2},0)$ and $(\sqrt{2},0,0)$ respectively. EF is the edge in which the other two rectangular faces ADEF and BCEF meet.
Prove that the equation of the plane containing A, B, F is
$5x+5y-2z=5\sqrt{2}$.
If the origin lies inside the prism, determine the equations of the line EF.

I have proven the first part, but I can't get the equations of EF.
F(x,y,z)
AF=BF from this x=y
AB=AF=BF=2
Perpendicular distance of F from plane ABCD= $\sqrt 3=AF(\cos \theta)$
Where $\theta$ is the angle of elevation of F from A.
$\cos\theta=[(\frac{1}{3\sqrt 3})(\frac{x-\sqrt 2}{\surd (x^2-2\sqrt 2x+2+y^2+z^2)})+(\frac{1}{3\sqrt 3})(\frac{y}{\surd (x^2-2\sqrt 2x+2+y^2+z^2)})+(\frac{5}{3\sqrt 3})(\frac{z}{\surd(x^2-2\sqrt 2x+2+y^2+z^2)})]$
Fill in the values for the perpendicular distance.
Then we have $9=\frac{2x-2\sqrt 2+2y+10z}{\surd (x^2-2\sqrt 2x+2+y^2+z^2)}$
BF= $\surd (x^2-2\sqrt 2x+2+y^2+z^2)=2$
So
$9=\frac{2x-2\sqrt 2+2y+10z}{2}$
$9=x-\sqrt 2+y+5z$
$5x+5y=45+5\sqrt 2-25z=5\sqrt 2+2z$ from the plane ABF
$z=\frac{5}{3}$
now x=y
$2x=9+\sqrt 2-5(\frac{5}{3})$
$x=y=\frac{1}{3}+\frac{\sqrt 2}{2}$
So $F(\frac{1}{3}+\frac{\sqrt 2}{2}, \frac{1}{3}+\frac{\sqrt 2}{2}, \frac{5}{3})$
The direction ratios of the line are 5:5:-2 since it is perpendicular to the plane ABF
the equations for the line I arrive at are
$-2x=-2y=5z-9-\sqrt 2$

the answer is supposed to be $-2x=-2y=5z+9-\sqrt 2$
I can't find where I'm wrong.
Thanks!

2. A unit vector perpendicular to ABCD is $\frac1{3\sqrt3}(1,1,5)$. The midpoint of AB is $P = \bigl(\frac{\sqrt2}2,\frac{\sqrt2}2,0\bigr)$. The point F is at a distance $\sqrt3$ from P along this vector. Therefore F is the point $\bigl(\frac{\sqrt2}2\pm\frac13,\frac{\sqrt2}2\pm\f rac13,\pm\frac53\bigr)$. Notice the plus/minus signs. They are there because F could be on either side of the plane ABCD. But we are told that the origin lies inside the prism. Therefore F and the origin must lie in the same half-space $x+y+5z<\sqrt{2}$ or $x+y+5z>\sqrt{2}$. The origin clearly lies in the first of these, hence so does F, which means that we must take the minus signs. Thus $F = \bigl(\frac{\sqrt2}2-\frac13,\frac{\sqrt2}2-\frac13,-\frac53\bigr)$.

That should account for the change of sign from –9 to 9 in the final result.