$17 \cdot 190$ Four line segments have lengths of $1, 9, x,$ and $5$ (where $x$ is a positive real number), and they are used to form two similar right triangles, with $AB$ and $CD$ being two of these line segments (as shown in the figure). The number of possible values for $x$ is
(A) 1.
(B) 3.
(C) 6.
(D) 9.
(Chinese Hebei Province Junior High School Mathematics Competition, 1995)
[Solution] Obviously, $AB$ is the longest, with a length of 9 or $x$.
If $AB=x, CD=9$, then $x=\sqrt{117}=3 \sqrt{13}$;
If $AB=x, CD=5$, then $x=\sqrt{125}=5 \sqrt{5}$;
If $AB=x, CD=1$, then $x=\sqrt{197}$;
If $AB=9, CD=x$, then $x=\sqrt{45}=3 \sqrt{5}$;
If $AB=9, CD=5$, then $x=\sqrt{81-25}-1=2 \sqrt{14}-1$;
If $AB=9, CD=1$, then $x=\sqrt{81-1}-5=4 \sqrt{5}-5$. Therefore, the answer is $(C)$.