2001 AMC 12 Problems/Problem 15

Problem

An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)

$\text{(A) }\frac {1}{2} \sqrt {3} \qquad \text{(B) }1 \qquad \text{(C) }\sqrt {2} \qquad \text{(D) }\frac {3}{2} \qquad \text{(E) }2$

Solution

Given any path on the surface, we can unfold the surface into a plane to get a path of the same length in the plane. Consider the net of a tetrahedron in the picture below. A pair of opposite points is marked by dots. It is obvious that in the plane the shortest path is just a segment that connects these two points. Its length is the same as the length of the tetrahedron's edge, i.e. $\boxed{1}$ .

$[asy] unitsize(2cm); defaultpen(0.8); pair A=(0,0), B=(1,0), C=rotate(60)*B, D1=B+C-A, D2=A+C-B, D3=A+B-C; draw(A--B--C--cycle); draw(D1--B--C--cycle); draw(D2--A--C--cycle); draw(D3--B--A--cycle); dot( A+0.5*(C-A) ); dot( B+0.5*(C-A) ); draw( ( A+0.5*(C-A) ) -- ( B+0.5*(C-A) ), dashed ); [/asy]$

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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All AMC 12 Problems and Solutions

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2001 AMC 12 Problems/Problem 15

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