$17 \cdot 50$ In $\triangle A B C$, if $\angle A=2 \angle B$, side $b=4, c=5$, then side $a$ equals
(A)6.
(B) 7 .
(C) $3 \sqrt{5}$.
(D) 5 .
(Shaanxi Province, China Junior High School Mathematics Competition, 1997)
[Solution] As shown in the figure, draw the angle bisector $AD$ of $\angle A$. According to the Angle Bisector Theorem, we have
$$
\begin{array}{l} 
C D=\frac{4}{9} a, B D=\frac{5}{9} a . \\
\text { Also } \angle C D A=2 \angle D A B=\angle A . \\
\therefore \quad \triangle A D C \backsim \triangle B A C .
\end{array}
$$

Thus, $\frac{4 a}{9}: 4=4: a$. Therefore, $a^{2}=36$, which gives $a=6$. Hence, the answer is $(A)$.