The Slanted Square

Strategies and Hints

  1. Triangles FAD, BAF, BFD, and CED are similar.

  2. Segment BC is the side of the shaded square. If you know the length of segment BC, you can find the area.

  3. Using the Pythagorean Theorem and properties of similar triangles:

    AD     =   FE = 8   AF = ED = 4

    AB     = CE =    CD = BF =

  4. The figure described in the hint is shown.

  1. The total unshaded area is 12 units, so the unshaded area inside the original
    12-by-12 square is 6 units. The total area of the original square is 10 units, so the fraction of the original square which is shaded is .

Solution

The side of the shaded square can be found from the first three hints by subtracting AB and CD from AD.



Squaring gives the solution, .To find the solution without the side of the shaded square, use the ratio found in the last hint and compute of 144.