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In the image above, the green, blue, and orange triangles are all equilateral triangles. How many times is the area the leftmost green triangle of that of the rightmost orange triangle?

Dec 18, 2019

#1
+106519
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Call the side  of the  largest green triangle, S

Then the side  of the  largest  orange triangle  = (√3/2) * S

And the side  of  the second largest green triangle  =  (√3/4) * S

And the side  of the next largest orange triangle is    (3/8) * S

And the side  of the  last green triangle is   (3/16) * S

And the side of the last orange triangle  =   (3/32)√3 * S

So....the ratio  of the area  of the  largest  green triangle     to the  smallest orange  triangle  =

S^2                                            1                   32^2          1024

______________  =          __________  =   _____   =    _____

[ (3/32)√3* S] ^2                   27/ 32^2             27               27

Dec 19, 2019
#2
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The equilateral triangles seem to be constructed as in the diagram above:

they are constructed on the three sides of a 30-60-90 right triangle. The leftmost green one on the hypothenuse, the leftmost orange one on the longer leg, and the next green one in the sequence on the shorter leg. So the ratio of the sides of the triangles in the green-orange sequence, I think, is as follows:

The sides are in the ratio $$2:\sqrt{3}:1:\frac{\sqrt{3}}{2} :\frac{1}{2}:\frac{\sqrt{3}}{4}$$. So the ratio of the area of the leftmost green to that of the  rightmost

orange should be:

$$\frac{4}{\frac{3}{16}}=\frac{64}{3}$$

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Dec 19, 2019