Construct a right triangle of area 8 out of three other right triangles with areas 1, 3 and 4.

I was puzzled by this when seeing it in the queue for the first time; there seemed to be no way three right triangles of the given areas would make a fourth of the given area.

I drew a 4 x 4 grid and divided it across a major diagonal yielding a right isosceles. The result is something akin to:

  A      D'
   | \ E
D | / \
   | .   \
 B|     . \ C  

The dotted line represents the boundary between 2 triangles.

The construction.
Construct a right isosceles triangle with "legs" of 4 units.  Let the sides be AB, BC and the hypotenuse AC.

Bisect AB labelled D.  Construct the mirror D' of point D on the opposite side of AC.  Join DD' and DC.  

Triangle AED has area of 1 unit, DCB has 4 and ECD has 3 units.

Posted by brianjn on 2008-12-07 18:51:43