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A square is divided into triangles and quadrilaterals as shown. What fraction of the square area is covered by the shaded triangle?

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Solution

The shaded fraction is 1/15.

Suppose a unit square. The shaded triangle AFH is the difference of triangles AFB and AHB. These have the same base of 1 and heights as indicated x and y.

Lets start with AFB. It is seen to be similar to GFE by comparing angles. Now their heights are x and 1-x, respectively and from their bases these are seen to have the proportion 4 to 1. Hence x=4(1-x). It follows that x=4/5.

In the same way y can be calculated as 2/3. The required fraction is (x-y)/2=1/15.

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