SOLUTION: ABCG is a trapezoid. GCDE is a rhombus. GEF is half of a kite. Solve for the area of the composite figure. https://ibb.co/4RsFBYjz

Algebra ->  Surface-area -> SOLUTION: ABCG is a trapezoid. GCDE is a rhombus. GEF is half of a kite. Solve for the area of the composite figure. https://ibb.co/4RsFBYjz      Log On


   

Question 1210154: ABCG is a trapezoid.
GCDE is a rhombus.
GEF is half of a kite.
Solve for the area of the composite figure.
https://ibb.co/4RsFBYjz
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

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. 



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

expr%281%2F2%29%2ADG%2ACE+=+expr%281%2F2%29%282%2A7%29%282%2A4%29=56

Area of triangle EFG is half the base times height


 
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%28GI%5E2%2BIC%5E2%29=sqrt%287%5E2%2B4%5E2%29=sqrt%2849%2B16%29=sqrt%2865%29

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

The average base of trapezoid ABCG is %28AB%2BCG%29%2F2+=%2814%2Bsqrt%2865%29%29%2F2

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

6%2Aexpr%28%2814%2Bsqrt%2865%29%29%2F2%29=3%2814%2Bsqrt%2865%29%29=42%2B3sqrt%2865%29

So we add the three areas together:

Area of trapezoid ABCG = 3%2814%2Bsqrt%2865%29%29

Area of rhombus GCDE = 56

Area of triangle EFG = 100

Answer = 42%2B3sqrt%2865%29%29%2B56%2B100

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

expr%281%2F2%29%2ADG%2ACE+=+expr%281%2F2%29%282%2A7%29%282%2A4%29=56

Area of triangle EFG is half the base times height


 
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%28GI%5E2%2BIC%5E2%29=sqrt%287%5E2%2B4%5E2%29=sqrt%2849%2B16%29=sqrt%2865%29

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

The average base of trapezoid ABCG is %28AB%2BCG%29%2F2+=%2814%2Bsqrt%2865%29%29%2F2

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

6%2Aexpr%28%2814%2Bsqrt%2865%29%29%2F2%29=3%2814%2Bsqrt%2865%29%29=42%2B3sqrt%2865%29

So we add the three areas together:

Area of trapezoid ABCG = 42%2B3sqrt%2865%29

Area of rhombus GCDE = 56

Area of triangle EFG = 100

Answer = 42%2B3sqrt%2865%29%2B56%2B100+=+198+%2B+3sqrt%2865%29 cm2

That's approximately 222.2 cm2 
 
Edwin