Imagine a square ABCD with a diagonal BD. Now draw a line EF parallel to BD, such that E lies on BC and F lies on CD. Also length of EF = length of AB. Now Colour the space enclosed by BDFE. Of the square ABCD, what percentage of area is coloured ?

Call the side length of the square s.  Then length of DB is s√2  .
Triangles FEC and DBC are similar, so |DB|/|BC| = |EF|/|EC| and we are told |EF| = |AB| = s. Solve this to find |EC| = s/√2 which by symmetry is also |FC|.
Thus A(ΔFEC) = sē/4,  A(ΔDBC) = sē/2 so A(BDFE) = sē/4  which means 25% of the square is colored.

Posted by Sean on 2005-02-01 14:44:27