Question 1185236
{{{drawing(300,300,-.5,2.2,-.5,1, triangle(0,0,1.732,0,1.732,.5),
rectangle(1.632,0,1.732,.055),
locate(0,0,F), locate(1.732,0,D),locate(1.732,.58,E),locate(.35,.14,30^o)
)}}}<pre>
Since ∆DEF is a 30-6-90 right triangle

{{{DE/FE}}}{{{""=""}}}{{{1/2}}}

{{{FE}}}{{{""=""}}}{{{2*DE}}}

{{{FE + DE}}}{{{""=""}}}{{{12}}}

Substitute 2DE for FE

{{{2*DE + DE}}}{{{""=""}}}{{{12}}}
{{{3*DE}}}{{{""=""}}}{{{12}}}
{{{DE}}}{{{""=""}}}{{{4}}}

Since ∆DEF is a 30-6-90 right triangle

{{{DF/DE}}}{{{""=""}}}{{{sqrt(3)/1}}}

{{{DF/4}}}{{{""=""}}}{{{sqrt(3)/1}}}

{{{DF}}}{{{""=""}}}{{{4sqrt(3)}}}

Edwin</pre>