$18 \cdot 113$ If the larger base of an isosceles trapezoid equals the diagonal, and the smaller base equals the height, then the ratio of the smaller base to the larger base is
(A) $\frac{1}{2}$.
(B) $\frac{2}{3}$.
(C) $\frac{3}{4}$.
(D) $\frac{3}{5}$.
(E) $\frac{2}{5}$.
(4th American High School Mathematics Examination, 1953)
[Solution] As shown in the figure, trapezoid $ABCD$, let
$$
AD = AH = s, AC = BC = t,
$$

then $BH = \frac{t-s}{2}$.
In the right triangle $\triangle AHC$, by the Pythagorean theorem, we have
$$
t^{2} = s^{2} + \left(t - \frac{t-s}{2}\right)^{2},
$$

which simplifies to $t^{2} = s^{2} + \left(\frac{t+s}{2}\right)^{2}$ or $t^{2} - s^{2} = \left(\frac{t+s}{4}\right)^{2}$, hence $t - s = \frac{t+s}{4}$ or $\frac{t-s}{t+s} = \frac{1}{4}$, thus $\frac{t}{s} = \frac{5}{3}$. Therefore, the answer is $(D)$.