19) Let $A B C$ be an equilateral triangle and $D E F$ another equilateral triangle inscribed in it with $A B$ perpendicular to $E D$. The ratio of the areas of $A B C$ and $D E F$ is 
(A) $\sqrt{3}$
(B) 2
(C) $\frac{5}{2}$
(D) 3
(E) $3 \sqrt{2}$.
19) The answer is (D).

Consider the triangles $A D E$ and $B E F$. They are equal since $D E=E F, D \widehat{A} E=E \widehat{B} F, B \widehat{E} F=180^{\circ}-90^{\circ}-$ $60^{\circ}=30^{\circ}=A \widehat{D} E$.

It follows that $A B=A E+E B=A E+A D=A E+2 \cdot A E=3 \cdot A E$ and consequently

$$
A B=3 \cdot A E=3 \cdot \frac{1}{\sqrt{3}} \cdot D E=\sqrt{3} \cdot D E
$$

and thus the ratio of the areas of the two triangles is 3.