17-111 If the projections of the two legs of a right triangle onto the hypotenuse are in the ratio 1:4, then the ratio of these two legs is
(A) $1: \sqrt{2}$.
(B) $1: 2$.
(C) $1: 3$.
(D) $1: 4$.
(China Sichuan Province Junior High School Mathematics Competition, 1989)
［Solution］As shown in the figure, in the right triangle $\triangle ABC$, $AD \perp BC$,
and $BD: DC=1: 4$.
Let $BD=1$, then $DC=4$.
Also, $AD^{2}=BD \cdot DC=1 \cdot 4=4$, so $AD=2$.
\[
\begin{aligned} \therefore \quad AB^{2} & =AD^{2}+BD^{2}=2^{2}+1^{2}=5, \\ AC^{2} & =AD^{2}+CD^{2}=2^{2}+4^{2}=20 .\end{aligned}
\]
Thus,
\[
\frac{AB}{AC}=\sqrt{\frac{5}{20}}=\frac{1}{2} \text{. }
\]

Therefore, the answer is $(B)$.