Track / Stage 5 / 126 of 400 #726 of 1964

Problem 726

AIME late

Geometry Difficulty 5.3 Find the answer

$2+$ [Coordinate Method in Space (Miscellaneous)] 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 $M$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

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

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

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.