Question 1210154
<pre>

My drawing is not quite to scale, because GCDE doesn't quite
look like a rhombus.

I labeled two more points than the drawing on the site, H and I. 

{{{drawing(400,10000/17,-4,13,-12,13, 

locate(1.9,11.5,14),
locate(5,9,6),
locate(6,4.5,4),
locate(7.5,3,7),
locate(2.1,0,8),
locate(-.45,2.5,8),
locate(-.7,-4,17),

locate(-2.5,10.5,A),
locate(6.5,12.3,B),
locate(6.5,6.5,C),
locate(11.3,1.5,D),
locate(5.2,0,E),
locate(0,-10.6,F),
locate(-.5,5.5,G),
locate(-.5,0.5,H),
locate(5,2.9,I),




green(line(5.135135135,11.189189189,6,6)),
line(0,-10.625,0,5),
line(-2,10,6.780487805,11.46341463),
line(6.780487805,11.46341463,6,6),
line(6,6,0,5),
line(0,5,5,0),
line(5,0,11,1),
line(0,0,5,0),
line(6,6,5,0),
line(0,-10.625,5,0),
line(11,1,6,6),
line(0,5,11,1),
line(0,5,-2,10)

   )}}}

Area of rhombus GCDE is half the product of the diagonals.

{{{expr(1/2)*DG*CE = expr(1/2)(2*7)(2*4)=56}}}

Area of triangle EFG is half the base times height

{{{expr(1/2)FG*HE = expr(1/2)(FH+HG)*HE=expr(1/2)*(17+8)*(8)=100}}}
 
Now all we need is the area of trapezoid ABCG.

We must find CG. 

The diagonals of a rhombus are perpendicular, and therefore triangle CIG
is a right triangle. GI = DI = 7, and IC = 4, so

{{{CG = sqrt(GI^2+IC^2)=sqrt(7^2+4^2)=sqrt(49+16)=sqrt(65)}}}

The area of a trapezoid is the average base times the height:

The average base of trapezoid ABCG is {{{(AB+CG)/2 =(14+sqrt(65))/2}}}

The height of trapezoid is 6, so the area of trapezoid ABCG is

{{{6*expr((14+sqrt(65))/2)=3(14+sqrt(65))=42+3sqrt(65)}}}

So we add the three areas together:

Area of trapezoid ABCG = {{{3(14+sqrt(65))}}}

Area of rhombus GCDE = 56

Area of triangle EFG = 100

Answer = {{{42+3sqrt(65))+56+100}}}{{{""=""|<pre>

My drawing is not quite to scale, because GCDE doesn't quite
look like a rhombus.

{{{drawing(400,10000/17,-4,13,-12,13, 

locate(1.9,11.5,14),
locate(5,9,6),
locate(6,4.5,4),
locate(7.5,3,7),
locate(2.1,0,8),
locate(-.45,2.5,8),
locate(-.7,-4,17),

locate(-2.5,10.5,A),
locate(6.5,12.3,B),
locate(6.5,6.5,C),
locate(11.3,1.5,D),
locate(5.2,0,E),
locate(0,-10.6,F),
locate(-.5,5.5,G),
locate(-.5,0.5,H),
locate(5,2.9,I),




green(line(5.135135135,11.189189189,6,6)),
line(0,-10.625,0,5),
line(-2,10,6.780487805,11.46341463),
line(6.780487805,11.46341463,6,6),
line(6,6,0,5),
line(0,5,5,0),
line(5,0,11,1),
line(0,0,5,0),
line(6,6,5,0),
line(0,-10.625,5,0),
line(11,1,6,6),
line(0,5,11,1),
line(0,5,-2,10)

   )}}}

Area of rhombus GCDE is half the product of the diagonals.

{{{expr(1/2)*DG*CE = expr(1/2)(2*7)(2*4)=56}}}

Area of triangle EFG is half the base times height

{{{expr(1/2)FG*HE = expr(1/2)(FH+HG)*HE=expr(1/2)*(17+8)*(8)=100}}}
 
Now all we need is the area of trapezoid ABCG.

We must find CG. 

The diagonals of a rhombus are perpendicular, and therefore triangle CIG
is a right triangle. GI = DI = 7, and IC = 4, so

{{{CG = sqrt(GI^2+IC^2)=sqrt(7^2+4^2)=sqrt(49+16)=sqrt(65)}}}

The area of a trapezoid is the average base times the height:

The average base of trapezoid ABCG is {{{(AB+CG)/2 =(14+sqrt(65))/2}}}

The height of trapezoid is 6, so the area of trapezoid ABCG is

{{{6*expr((14+sqrt(65))/2)=3(14+sqrt(65))=42+3sqrt(65)}}}

So we add the three areas together:

Area of trapezoid ABCG = {{{42+3sqrt(65)}}}

Area of rhombus GCDE = 56

Area of triangle EFG = 100

Answer = {{{42+3sqrt(65)+56+100 = 198 + 3sqrt(65)}}} cm<sup>2</sup>

That's approximately 222.2 cm<sup>2</sup> 
 
Edwin</pre>