CPhill

Sorry...see my corrected answer  !!!

Ratio of Perimeters   =   7: 9   =  (scale factor)

Ratio  of Areas  =  (scale factor)^2 =  (7 : 9 )^2  =  49 : 81

Area of  Red Hexagon  = (Area of Large Hexagon)  * ( scale  factor)^2   =

                                           (210)  * ( 49/81)  ≈   127  in^2

Ratio of side  lengths    Ratio  of Perimeters    Ratio of Areas

(Scale Factor)                (Scale Factor)            (Square of Scale Factor )

        7 : 12                           7 : 12                        49 : 144

       32 : 44                         32: 44                       1024 : 20736

The area  of the larger circle is  pi * radius^2  = pi * 6^2  = 36 pi  (1)

The radii of the two smaller   circles  = 3

So...the area of these two circles   is   2 * pi * (3)^2  =  18 pi    (2)

So...the shaded area is  (1) - (2)   =  [ 36 - 18 ] pi  =   18 pi  [ units^2 ]

Draw  radius OC

And since  OC = OA, then angles OCA  and OAC  are equal 

Then angle  COA  =  180   - 18  -18 = 180  - 36  = 144°

And by symmetry, angle BOC = 144°

So angle AOB  = 360  - 144 - 144  = 360 - 288  = 72°

Assuming that these are similar figures...and that the side of the larger triangle with a length of 6  corresponds to the side of the smaller  triangle with a length of 2

The  scale factor of the larger triangle to the smaller  is just  the above ratio  = 6/2  = 3/1

This is the linear scale factor for  any dimension....all corresponding sides of the larger figure to the smaller  wil have this ratio...also...the height of the larger triangle to the smaller wioll also have this ratio

Perimeter is a linear measure...so ...the ratio of the perimeters will just  be  the same  ratio    3 : 1  

The ratio of the areas will be  the square of the scale factor  = (3/1)^2  = 9/1  = 9 : 1

So...the area of the the blue triangle will be :

Area of the smaller  triangle * (scale factor)^2  =

2  * (3/1)^2  =

2 * 9  =

18 ft^2

To  see this last part a litle more clearly

Note that the area of the smaller triangle  = (1/2) base * height  = (1/2) b * h

But since each dimension of the larger triangle is 3 times that of the smaller...its area  is

(1/2) ( 3* base of smaller triangle) ( 3 * height of smaller triangle)  =

(1/2) (3 b) (3 h)  = (1/2) 9 bh  =  9 [ (1/2) bh ] =   9 times [ the area of the smaller triangle]

Does that make some sense ???

The area of the larger circle  will be  4 pi

So...the  shaded area  is  =  (5/12) * 4 pi  =  20/12 pi  =  ( 5/3 ) pi 

Let  the measure  of  the smaller angle   ADC  = theta  (in rads)    

And let the measure of the larger angle ADC  = [2pi - theta ]  in rads

So...the  area of the sector subtended by the large circle  is (1/2) 4^2 * θ =  2θ

And let the area of the sector subtended by the larger circle be  (1/2) * 1^2 * (2pi - θ) =

(1/2) (2pi - θ)  

So we have that

2θ  +  (1/2)(2pi - θ)  =  (5/3)pi

2θ + pi - θ/2  =  (5/3)pi

(3/2)θ =  2/3 pi

θ = (2/3)(2/3) pi  =  (4/9) pi  

So...the smaller angle ADC  =  (4/9)pi * 180 / pi  =   20 * 4   = 80°

Proof

Area of  shaded sector of lager  circle  = 2(4/9) pi  =  8/9 pi

Area of shaded  sector of smaller circle = (1/2) (2pi - (4/9) pi)  = (1/2) (14/9) pi = (7/9) pi

Combined areas =  (8/9 + 7/9) pi  =  (15/9) pi  =  (5/3) pi

We have an equilateral  triangle...

The area of this triangle  is  3(1/2) * 8^2  * sin (120°) =  3 (32) * √3 / 2  =  3 * 16 * √3  =

48√3  cm^ 2     (1)

The area of the circle  = pi * 8^2  =  64 pi   cm^2     (2)

So...the shaded area  is  (2)  - (1)  =  [ 64 pi  - 48√3 ] cm^2   =   "A"

Let  a  >  c  >  b

Then

l a - b l  = a  - b

l b - c l  = c - b    and

l c - a l  = a  - c

So

l a - b l + l b - c l +  l c - a l    = 20      

a - b + c - b + a - c  = 20    simplify

2a  - 2b  = 20

a - b  =10

So...the max  value  for   l a - b l    = 10   

You will need to  draw  two circles...each labelled with a radius of 4

Then...we will have  a rectangle labelled with a height of 8  and  a width of 2pi *4 = 8 pi units

It will look something  like this :