Question 1189025
<pre>
I'll pirate Greenestamps picture

<br>
{{{drawing(400,400,-1,7,-1,7
,line(0,0,4.9,0),line(0,0,0,4.9),line(4.9,0,0,4.9)
,line(1.79,0,2.45,2.45),line(0,1.79,2.45,2.45),line(1.79,0,0,1.79),
locate(-.1,0,B), locate(4.9,0,C),locate(-.1,5.2,A),locate(2.5,2.7,F), locate(-.27,2,D),locate(1.7,0,E),locate(3.6,1.6,2sqrt(3)),locate(.9,.47,45^o),
locate(1.4,.8,60^o),locate(1.9,.47,75^o), locate(4,.47,45^o)


)}}}

By the law of sines,

{{{EF/sin(C)}}}{{{""=""}}}{{{FC/sin("<FEC")}}}

{{{EF/sin(45^o)}}}{{{""=""}}}{{{2sqrt(3)/sin(75^o)}}}

{{{EF*sin(75^o)}}}{{{""=""}}}{{{2sqrt(3)sin(45^o)}}}

We know sin(45<sup>o</sup>), we must find sin(75<sup>o</sup>).

{{{sin(75^o)}}}{{{""=""}}}{{{sin(45^o+30^o)}}}{{{""=""}}}{{{sin(45^o)cos(30^o)+cos(45^o)sin(30^o)}}}{{{""=""}}}
{{{(sqrt(2)/2)(sqrt(3)/2)+(sqrt(2)/2)(1/2)}}}{{{""=""}}}{{{sqrt(6)/4+sqrt(2)/4}}}{{{""=""}}}{{{(sqrt(6)+sqrt(2))/4}}}

{{{EF*((sqrt(6)+sqrt(2))/4))}}}{{{""=""}}}{{{2sqrt(3)*(sqrt(2)/2)}}}

{{{EF*(sqrt(6)+sqrt(2))/4))}}}{{{""=""}}}{{{2sqrt(3)(sqrt(2)/2)}}}

{{{EF*((sqrt(6)+sqrt(2))/4))}}}{{{""=""}}}{{{sqrt(6)}}}

{{{EF*((sqrt(6)+sqrt(2))))}}}{{{""=""}}}{{{4*sqrt(6)}}}

{{{EF}}}{{{""=""}}}{{{4sqrt(6)/((sqrt(6)+sqrt(2))))}}}

Rationalize the denominator:

{{{EF}}}{{{""=""}}}{{{(4sqrt(6))/( (sqrt(6)+sqrt(2) ))}}}{{{""*""}}}{{{(sqrt(6)-sqrt(2))/(sqrt(6)-sqrt(2))}}}

{{{EF}}}{{{""=""}}}{{{(  4sqrt(6)(sqrt(6)-sqrt(2)))/(6-2)}}}

{{{EF}}}{{{""=""}}}{{{(4sqrt(6)(sqrt(6)-sqrt(2)))/(4)}}}

{{{EF}}}{{{""=""}}}{{{sqrt(6)(sqrt(6)-sqrt(2))}}}

{{{EF}}}{{{""=""}}}{{{6-sqrt(12)}}}

{{{EF}}}{{{""=""}}}{{{6-sqrt(4*3)}}}

{{{EF}}}{{{""=""}}}{{{6-2sqrt(3)}}}{{{""=""}}}{{{2(3-sqrt(3))}}}

Since ΔDEF is equilateral, its perimeter is 3 times its side EF.

Perimeter of ΔDEF = {{{6(3-sqrt(3))}}}, choice A)

Edwin</pre>