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Starting with an equilateral triangle, the corners are cut off to form a regular hexagon.  What is the ratio of the hexagon to the area of the original triangle?

 Jul 1, 2020
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The easiest way would be to draw an equilateral triangle 3 units on each side.

Call the original triangle ABC. 

 

Divide each side into thirds.

Label the two points of trisection of AB as U and V.

Label the two points of trisection of BC as W and X.

Label the two points of trisection of CA as Y and Z.

 

Draw VW, XY and ZU to get hexagon UVWXYZ.

 

Draw UX, VY, and WZ.

They will meet at O, the center of the triangle.

 

You now have 9 congruent trianges, 6 in the hexagon.

The ratio of the hexagon to the area of the original triangle:  6 : 9  =  2 : 3.

 

You can also find the areas of the triangles and get the same answer that way.

 Jul 1, 2020

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