SOLUTION: There are two equilateral triangles with different areas, where O and O' are their circumcenters respectively. In the first, the distance OA is 9 cm, and in the second, the distanc

Question 1209088: There are two equilateral triangles with different areas, where O and O' are their circumcenters respectively. In the first, the distance OA is 9 cm, and in the second, the distance O'C' is 4√2 cm. Find the ratio of the area of the first to the area of the second.
https://ibb.co/Cz6tWsL
Answer by ikleyn(52855) (Show Source): You can put this solution on YOUR website!
.
There are two equilateral triangles with different areas, where O and O' are t
heir circumcenters respectively. In the first, the distance OA is 9 cm,
and in the second, the distance O'C' is 4√2 cm.
Find the ratio of the area of the first to the area of the second.
https://ibb.co/Cz6tWsL
~~~~~~~~~~~~~~~~~~~~~~~~~

These equilateral triangles are similar (as any two equilateral triangles,
by three angles). 

Their corresponding similar elements are OC and O'C'.


In the second triangle, we are given  O'C' =  cm.


In the first triangle, we are given OA = 9 cm.



      But triangle OAC is a right-angled 90°-60°-30°-triangle;

      therefore, the hypotenuse OC is twice the leg OA,

      so, the hypotenuse OC is 2*9 = 18 cm.



Thus, the ratio of corresponding linear dimensions in triangles is

    OC        18            9         9*sqrt(2)
   ----- = ----------- = -------- = ----------.
    O'C'    4*sqrt(2)     2*sqrt(2)      4


The ratio of the areas of the second triangle to the first triangle is the square
of the ratio of their corresponding linear elements, i.e.

     =  =  = 10.125.    ANSWER

Solved.

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