help

Well since the yellow is equilateral inside another equilateral, we can confirm that logically all three blue triangles are congruent.

The orange triangle and the red triangle are similar.

Using one of the properties of similar triangles, we know that the ratio of the side length of the orange triangle to the side length of the red triangle is 2:3

Because: (2/3)^2 = 4/9 = 20/45 

Now we found the ratio and that the blue triangles are congruent, we know that the side length of the orange triangle to the side length of the blue triangle is 2: (2 + 3) = 2:5

That means the AREA must be (according to one of the similarity properties) 4:25

So we calculate the area of the blue triangle and find its 125. ( we also can find it using the red triangle.)

Help! Im stuck too! I got the blue triangle as 125, can someone work off from here?

Using the Law of Sines   ih the lower left  blue triangle, we have

sin  (a)  / s  =  sin (60) / S       (1) 

Where  a  is the angle formed by the side of the yellow triangle  and the base of this triangle

And S is the  side of the  yellow triangle

And s  is the  side of the orange triangle with an area of 20

And using the Law of Sines  in  the lower right blue triangle, we have

sin ( 120 - a)  / (1.5s)  =  sin (60) / S     (2)

Where  (180 - 60 - a)  =   (120 - a)   is the angle formed  by  the side of the  yellow triangle  and the  base of this triangle

And S is the side of the yellow triangle

And 1.5s  is the side of the red triangle

So....equating (1) and (2), we have

sin (a) / s   =    sin (120 - a) / (1.5 s)

sin (a)    =   sin (120 - a) / (1.5)

sin (a)  =  [sin(120) cos(a)  -  sin (a)cos (120) ]  / (3/2)

(3/2)sin (a)   = (√3 / 2 )cos (a)  +   sin (a) (1/2)

3sin (a)  = √3 cos (a)  +  sin a

2sin (a)  =  √3 cos (a)

sin (a) / cos(a)  =  √3/2

tan (a)  = √3/2

So  sin (a)   =    √ 3  / [ √ [ 3 + 4 ]  =  √3 / √7

And we  can find s   as

20  = (√3/4)s^2

80 / √3  = s^2

s  = √ [ 80 / √3  ] 

And we can find the side  of the yellow triangle as 

s / sin (a)  = S  / sin (60)

√ [ 80 / √3  ]  / [ √3 / √7 ] =   S / [√3/2 ]

S =  (√3/2) * √ [ 80 / √3  ]  / [ √3 / √7 ]  =      2√35 / [⁴√3 ]

So....the area  of the  yellow equilateral  triangle is

(√3 / 4)S^2  =

(√3/ 4)  *  (2√35)^2 / √3  =

(1/4) 140   =

140 / 4  =

35  units^2

  Gold  triangle  

Area         >  A = 20 u² 

Side         >  X  = ?                      X = sqrt [(A/√3)*4]        X = 6.796 

  Red  triangle  

Area         >  B = 45 u²

Side         >   Y = ?                       Y = sqrt [(B/√3)*4]         Y = 10.194 

 Blue (large) triangle  

Area         >  C = ?                       C = (Z/2)² * √3               C = 125 u²  

Side         >   Z = ?                        Z = X + Y                    Z = 16.99     

 Blue (small) triangle 

Side         >  a = 6.796

Side         >  b = 10.194

Side         >  c = ?                        c² = sqrt[a² + b² - 2ab*cos(q)]     c = 8.99  

Angle       >  q = 60°

Semiperimeter > s = ?                       s = (a + b + c)/2             s = 12.99 

Area         >  A_b = ?                      A_b = sqrt[ s(s-a)(s-b)(s-c)]     A_b = 30 u² 

 Yellow triangle 

Area         >  D = ?                      D = C - 3*A_b             D = 35 u²