Let ABCD be a parallelogram and let AA', BB', CC', and DD' be parallel rays in space on the same side of the plane determined by ABCD. If AA'=10, BB'=8, CC'=18, DD'=22, and M and N are the midpoints of A'C' and B'D', respectively, then MN=

1997 AMC 12 Problem 25

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25.

Let

ABCD

be a parallelogram and let

\overrightarrow{AA'},

\overrightarrow{BB'},

\overrightarrow{CC'},

and

\overrightarrow{DD'}

be parallel rays in space on the same side of the plane determined by

ABCD.

AA'=10,

BB'=8,

CC'=18,

DD'=22,

and

M

and

N

are the midpoints of

A'C'

and

B'D',

respectively, then

MN=

0

1

2

3

4

Use vectors and the parallelogram identity

A+C=B+D

The planar components of the two midpoints coincide; compare only their ray-direction components

Choose the common ray direction as a unit vector

u.

Then

\begin{aligned} M&=\frac{A+C}{2}+14u,\\ N&=\frac{B+D}{2}+15u. \end{aligned}

Since

A+C=B+D

for a parallelogram,

N-M=u,

MN=1.

The correct answer is B.