19.33 A convex $n$-sided polygon has a total of 27 diagonals, then the number of sides of this convex $n$-sided polygon is 
(A) 11.
(B) 10.
(C) 9.
(D) 8.
(E) 7.
(Wuhan, Hubei Province, China Junior High School Mathematics Competition, 1985)
[Solution] The number of diagonals in a convex $n$-sided polygon is $\frac{1}{2} n(n-3)$. According to the problem, we have
$$
\frac{1}{2} n(n-3)=27 \text {, }
$$

which simplifies to
$$
n^{2}-3 n-54=0 \text {. }
$$

Solving this, we get $n=9$ or -3 (discard the negative solution).
Therefore, the answer is (C).