$13 \cdot 57$ The maximum value of the shortest side of a decagon inscribed in a circle with radius 1 is
(A) $\frac{\sqrt{5}-1}{2}$.
(B) $\frac{\sqrt{3}}{2}$.
(C) $\frac{\sqrt{5}+1}{4}$.
(D) 1 .
(China Beijing High School Mathematics Competition, 1987)
[Solution] If the central angle corresponding to the shortest side of a decagon inscribed in a circle with radius 1 is $\alpha$, then $\alpha \leqslant 36^{\circ}$.
The length of the shortest side $l=2 \sin \frac{\alpha}{2} \leqslant 2 \sin 18^{\circ}$.
Therefore, when $\alpha=36^{\circ}$, the shortest side reaches its maximum value of $2 \sin 18^{\circ}$.
It is easy to calculate this value as $\frac{\sqrt{5}-1}{2}$. Hence, the answer is $(A)$.