. 18.54 If the perimeter of rectangle $ABCD$ is 20 inches, then the minimum value of diagonal $AC$ (in inches) is
(A) 0.
(B) $\sqrt{50}$
(C) 10.
(D) $\sqrt{200}$.
(E) None of the above.
(18th American High School Mathematics Examination, 1967)
[Solution] Let the lengths of the sides and the diagonal of the rectangle be $a, b, c$, and the perimeter be $p$, then
by
$$
\begin{aligned}
& p=2(a+b) . \\
c^{2} & =a^{2}+b^{2}=(a+b)^{2}-2 a b \\
= & \frac{p^{2}}{4}-2 a b .
\end{aligned}
$$

Given $p$, when $a b$ is maximized, $c^{2}$ is minimized. By $2 \sqrt{a b} \leqslant a+b$, equality holds if and only if $a=b$. This means: among rectangles with a given perimeter, the diagonal of a square is the shortest. Therefore, the answer is $(B)$.