MathDB

In the figure,

\triangle ABC

has

\angle A =45^{\circ}

and

\angle B =30^{\circ}

. A line

DE

, with

D

AB

and

\angle ADE =60^{\circ}

, divides

\triangle ABC

into two pieces of equal area. (Note: the figure may not be accurate; perhaps

E

is on

CB

instead of

AC

.) The ratio

\frac{AD}{AB}

is [asy] size((220)); draw((0,0)--(20,0)--(7,6)--cycle); draw((6,6)--(10,-1)); label("A", (0,0), W); label("B", (20,0), E); label("C", (7,6), NE); label("D", (9.5,-1), W); label("E", (5.9, 6.1), SW); label("

45^{\circ}

", (2.5,.5)); label("

60^{\circ}

", (7.8,.5)); label("

30^{\circ}

", (16.5,.5)); [/asy]

<span class='latex-bold'>(A)</span>\ \frac{1}{\sqrt{2}} \qquad<span class='latex-bold'>(B)</span>\ \frac{2}{2+\sqrt{2}} \qquad<span class='latex-bold'>(C)</span>\ \frac{1}{\sqrt{3}} \qquad<span class='latex-bold'>(D)</span>\ \frac{1}{\sqrt[3]{6}} \qquad<span class='latex-bold'>(E)</span>\ \frac{1}{\sqrt[4]{12}}

Problem Statement