If we cut a prism with an oblique plane , find the sum of distances of vertices from base given that...

Question

We cut a prism having a 6-gon regular polygon with side length 1 as base (see the following figure).If the distances of the vertices of the oblique cutter plane from the base are $2,3,x,y,11,z$ calculate $x+y+z$.

I really have no idea for this!

Answer 1

Let $ABCDEF$ be the base hexagon, and let $A'B'C'D'E'F'$ be the oblique hexagon where $$AA'=2,\quad BB'=3,\quad CC'=x,\quad DD'=y,\quad EE'=11,\quad FF'=z$$

Let $M$ be the intersection point of the oblique hexagon with the line parpendicular to the base passing through the center of the base hexagon.

Since the base is a regular hexagon, we can see that the midpoint of $A'D'$, the midpoint of $B'E'$, and the midpoint of $C'F'$ exist at $M$.

It follows from this that $$\dfrac{2+y}{2}=\dfrac{3+11}{2}=\dfrac{x+z}{2}\implies \color{red}{x+y+z=26}$$

Answer 2

Let's position the base hexagon on the $XY$ plane as shown below:

Assuming the vertice heights are ordered counterclockwise, we can give the three vertices in the cutter plane with known heights the coordinates $A' = (\frac 1 2, -\frac{\sqrt 3}{2}, 2)$, $B' = (1, 0, 3)$ and $C' = (-1, 0, 11)$. We can then find the equation of the plane passing through these three points, which is $$4 \sqrt{3}x - 6y + \sqrt{3}z = 7 \sqrt{3}$$

The missing vertice heights can then be found by finding the intersection between the plane above and the lines passing through each vertice which are perpendicular to the $XY$ plane. The equations for these lines are: $$D: x=\frac 1 2, y= \frac{\sqrt 3}{2}, z=t$$ $$E: x=-\frac 1 2, y= \frac{\sqrt 3}{2}, z=t$$ $$F: x=-\frac 1 2, y= -\frac{\sqrt 3}{2}, z=t$$

Inserting the values for each line into the equation for the plane and solving for $t$ we find that $D' = (\frac 1 2, \frac{\sqrt 3}{2}, 8)$, $E' = (-\frac 1 2, \frac{\sqrt 3}{2}, 12)$ and $F' = (-\frac 1 2, -\frac{\sqrt 3}{2}, 6)$. I.e., in the terms of your question, $x=8, y=12, z=6$. Hence $x+y+z=26$.

Below is a figure showing the resulting shape: