10,11  In space, a parallelogram $A B C D$ and a plane $M$ are given. The distances from points $A, B$, and $C$ to the plane $M$ are $a, b$, and $c$ respectively. Find the distance $d$ from vertex $D$ to the plane $M$.
What is the distance from the center of the parallelogram to the plane $M$?

## Solution

We can assume that the plane $M$ is perpendicular to the x-axis. The sums of the x-coordinates of opposite vertices of the parallelogram are equal (this is twice the x-coordinate of the center $O$ of the parallelogram). The x-coordinates of the points may differ from the distances by their signs. Therefore, the answer depends on the position of the vertices of the parallelogram relative to the plane $M$. For example, if points $A, B$, and $C$ have positive x-coordinates, then the x-coordinate of point $D$ is $a-b+c$.

## Answer

$d=|a \pm c \pm b|$.