SOLUTION: ABCDEF is a regular hexagon, and PQR is an equilateral triangle. Find the ratio of the area of the triangle to the area of the hexagon. https://ibb.co/BtQsnFj

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Question 1209089: ABCDEF is a regular hexagon, and PQR is an equilateral triangle. Find the ratio of the area of the triangle to the area of the hexagon.
https://ibb.co/BtQsnFj
Answer by ikleyn(52792)   (Show Source): You can put this solution on YOUR website!
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ABCDEF is a regular hexagon, and PQR is an equilateral triangle.
Find the ratio of the area of the triangle to the area of the hexagon.
https://ibb.co/BtQsnFj
~~~~~~~~~~~~~~~~~ 

Let r be the side of the hexagon ABCDEF.


Then the area of the hexagon is the area of 6 equilateral triangles with the side "r"

    area of the hexagon = .


Regarding the triangle PQR, its side is, OBVIOUSLY, arithmetic mean of r and 2r
(its side is the mid-line of the trapezoid CDEB).

So, the side of the triangle PQR is  a =  = 1.5r.


Then the area of the triangle PQR is   = .


Thus the ratio of the area of the triangle to the area of the hexagon is

     =  =  =  = .    ANSWER

Solved, with complete explanations.



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