Ratio of Areas in a Square

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Given: a square with side length and diagonal . Place point on and let . Connect to divide the square into four areas. Given , calculate the ratio of areas. For example, let .

Contributed by: Abraham Gadalla (August 2011)
Open content licensed under CC BY-NC-SA



The key here is that the height of equals the height of , so the ratio of the areas between those two triangles is .

Hence, .

Similarly, comparing areas, . Therefore |.

Substituting gives .

At this point, one can calculate the height of as .

Then and .

Also, .

After simplification, the ratio of the four areas is: .

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