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?
+129849 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
+121 
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:
\(\frac{4}{\frac{3}{16}}=\frac{64}{3}\)
