¶ 1988 AHSME Problem 23
Problem:
The six edges of tetrahedron measure , , , , and units. If the length of edge is , then the length of edge is
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Consider the edge of length and the two triangular faces of which share this edge. For both of these triangles, the other two sides must have lengths differing by less than , for otherwise the Triangle Inequality would be violated. Of the numbers given, only the pairs and satisfy this requirement, leaving the edge of length as the one opposite to that of length of . Consequently, we must have one of the two arrangements pictured below:
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The arrangement on the left is impossible because fails to satisfy the Triangle Inequality. This leaves the arrangement on the right, in which .
The reader may wish to verify that the second arrangement is possible by constructing a physical model - or by showing mathematically that it is constructible.