Problem:
Which of the following sets could NOT be the lengths of the external diagonals of a right rectangular prism [a "box"]? (An external diagonal is a diagonal of one of the rectangular faces of the box.)
Answer Choices:
A. {4,5,6}
B. {4,5,7}
C. {4,6,7}
D. {5,6,7}
E. {5,7,8}
Solution:
Let a,b and c, with a≤b≤c, be the lengths of the edges of the box; and let p,q and r, with p≤q≤r, be the lengths of its external diagonals. The Pythagorean Theorem implies that
p2=a2+b2,q2=a2+c2 and r2=b2+c2.
It follows that r2=p2+q2−2a2<p2+q2 is a necessary condition for a set {p,q,r} to represent the lengths of the diagonals. Only choice (B) fails this test. The other four choices do correspond to actual prisms because the condition r2<p2+q2 is also sufficient. To see this, just solve the equations for a,b and c:
a2=2p2+q2−r2,b2=2p2−q2+r2 and c2=2−p2+q2+r2.