A3. On the diagonal $B D$ of the rectangle $A B C D$, there are points $E$ and $F$ such that the lines $A E$ and $C F$ are perpendicular to the diagonal $B D$ (see figure). The area of triangle $A B E$ is equal to $\frac{1}{3}$ of the area of rectangle $A B C D$. What is the ratio $|A D|:|A B|$?
(A) $2: \sqrt{5}$
(B) $1: \sqrt{2}$
(C) $2: 3$
(D) $1: \sqrt{3}$
(E) $1: 2$

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A3. Since the area of triangle $A B E$ is equal to $\frac{1}{3}$ of the area of rectangle $A B C D$, and the area of triangle $A B D$ is $\frac{1}{2}$ of the area of rectangle $A B C D$, the area of triangle $D A E$ is equal to $\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$ of the area of rectangle $A B C D$. Therefore, the ratio of the areas of triangles $D A E$ and $A B E$ is $p_{D A E}$ : $p_{A B E}=\frac{1}{6}: \frac{1}{3}=1: 2$. Since these two triangles are similar, the ratio of their corresponding sides is equal to $|A D|:|A B|=\sqrt{1}: \sqrt{2}=1: \sqrt{2}$.