SOLUTION: http://img.geocaching.com/cache/large/8f671ade-1846-4e62-b2d7-2c73512ae1c8.gif Three big one-inch squares (S3) are joined corner-to-corner to make an equilateral triangle. Insid

Algebra ->  Triangles -> SOLUTION: http://img.geocaching.com/cache/large/8f671ade-1846-4e62-b2d7-2c73512ae1c8.gif Three big one-inch squares (S3) are joined corner-to-corner to make an equilateral triangle. Insid      Log On


   

Question 620349: http://img.geocaching.com/cache/large/8f671ade-1846-4e62-b2d7-2c73512ae1c8.gif
Three big one-inch squares (S3) are joined corner-to-corner to make an equilateral triangle. Inside that triangle, three little squares (S1) are fit into a similar shape. Finally, the line segment from the little square to the corner of the large triangle is used to form the medium square (S2) as shown.
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What is the ratio of areas of the medium square to the little square (S2:S1)?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I see three little kites in the corners of the triangle.
If you split one of them along the long diagonal, you get two congruent 30-60-90 right triangles.
The ratio of short leg to long leg in those triangles (and in all 30-60-90 right triangles) is 1%2Fsqrt%283%29 .
(It is usually written as tan%2830%5Eo%29=sqrt%283%29%2F3 just for elegance, but it works better for me as 1%2Fsqrt%283%29 in this case).
That is the ratio of the side of the small square to the side of the medium square.
The ratio of their areas is the square of the ratio of their sides, or highlight%281%2F3%29.