$19 \cdot 37$ In a circle with radius 1, there is an inscribed polygon. If all its sides are greater than 1 and less than $\sqrt{2}$, then the number of sides of this polygon must be
(A) 7 .
(B) 6 .
(C) 5 .
(D) 4 .
(China Junior High School Mathematics League, 1992)
[Solution] If the number of sides of the polygon that satisfies the condition is greater than or equal to 6, then the central angle opposite at least one side is no more than $60^{\circ}$.

As shown in the right figure, let the length of this side be $a$, and the central angle it subtends be $\alpha$. Then, by $\beta \geqslant \alpha$, we know $a \leqslant 1$.

If the number of sides of the polygon is 4, then the central angle opposite at least one side is no less than $90^{\circ}$.
By the cosine rule, it can be proven that the length of this side is no less than $\sqrt{2}$.
Therefore, the answer is $(C)$.