1997 AHSME Problems/Problem 25

Problem

Let $ABCD$ be a parallelogram and let $\overrightarrow{AA^\prime}$ , $\overrightarrow{BB^\prime}$ , $\overrightarrow{CC^\prime}$ , and $\overrightarrow{DD^\prime}$ be parallel rays in space on the same side of the plane determined by $ABCD$ . If $AA^\prime = 10$ , $BB^\prime = 8$ , $CC^\prime = 18$ , and $DD^\prime = 22$ and $M$ and $N$ are the midpoints of $A^\primeC^\prime$ (Error compiling LaTeX. Unknown error_msg) and $B^\primeD^\prime$ (Error compiling LaTeX. Unknown error_msg), respectively, then $MN =$

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

Solution

Let $ABCD$ be a unit square with $A(0,0,0)$ , $B(0,1,0)$ , $C(1,1,0)$ , and $D(1,0,0)$ . Assume that the rays go in the +z direction. In this case, $A^\prime(0,0,10)$ , $B^\prime(0,1,8)$ , $C^\prime(1,1,18)$ , and $D^\prime(1,0,22)$ . Finding the midpoints of $A^\primeC^\prime$ (Error compiling LaTeX. Unknown error_msg) and $B^\primeD^\prime$ (Error compiling LaTeX. Unknown error_msg) gives $M(\frac{1}{2}, \frac{1}{2}, 14)$ and $M(\frac{1}{2}, \frac{1}{2}, 15)$ . The distance $MN$ is $15 - 14 = 1$ , and the answer is $\boxed{B}$ .

While this solution is not a proof, the problem asks for a value $MN$ that is supposed to be an invariant. Additional assumptions that are consistient with the problem's premise should not change the answer. The assumptions made in this solution are that $ABCD$ is a square, and that all rays $XX^\prime$ are perpendicular to the plane of the square. These assumptions are consistient with all facts of the problem. Additionally, the assumptions allow for a quick solution to the problem.

1997 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Problem 26
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1997 AHSME Problems/Problem 25

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