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This problem trains for: AMC-10, AMC-12.

In a square ABCD with side length 3 draw the segment AE with E cutting CD in a ratio of DE:EC=2:1. Draw the segment EF perpendicular to AE with F on BC. Draw the segment FG, perpendicular to EF and with G on AB. The area of the figure AEFG can be written as an irreducible fraction m/n. What is the sum m+n?

The figure AEFG is a right angle trapezoid.

A quick round of angle chasing reveals the following similarities:

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figure

As a result, we can make a plan of deriving the area in question using encasement: subtract the sum of the areas of the similar right triangles from the area of the square.

Compute the lengths of the segments:

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figure

The areas of the right angle triangles are:

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And the area of the figure AEFG is:

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