The 3-D Coordinate System — Question 8

Question 8

A rectangular box has one vertex at the origin and the opposite vertex $P=(a,b,c)$ in the first octant. Its space diagonal has length 13, its $xy$-face diagonal has length 5, and the $y$-edge is longer than the $x$-edge. If the side lengths are positive integers, determine $P$.

Question 8 – Solution

Strategy Use the face diagonal to determine two integer side lengths, then use the space diagonal for the third.

Base dimensions The positive integer side lengths satisfying $a^2+b^2=25$ are 3 and 4. Since the $y$-edge is longer than the $x$-edge, $a=3$ and $b=4$.

Third dimension The space diagonal gives $$a^2+b^2+c^2=169\quad\Longrightarrow\quad 25+c^2=169,$$ so $c=12$. Therefore $\boxed{P=(3,4,12)}$.

Verification The face diagonal is $\sqrt{3^2+4^2}=5$ and the space diagonal is $\sqrt{3^2+4^2+12^2}=13$.