Wolfram Demonstrations Project

Given: a square

with side length

and diagonal

. Place point

and let

. Connect

to divide the square into four areas. Given

, calculate the ratio of areas. For example, let

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

Then

and

Also,

After simplification, the ratio of the four areas is:

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Ratio of Areas in a Square