A piece of paper has the precise shape of a triangle with the three side lengths AB, AC and BC being respectively 2, 4/3 and √10/3. The paper is folded along a line perpendicular to the side AB.

Determine the largest possible overlapped area.

In a construction in Geometers' Sketchpad, call the point on AB where the perpendicular is erected K, and its intersection with AC, call L. Initially as K moves away from A, the area of overlap is just the area of triangle ALK and is growing. When K is far enough from A, A' (the reflection of A) will coincide with B. As K continues farther past B, the area becomes a quadrilateral ALMB as LA' intersects CB in point M.

Initially after this passing of A' beyond B, the gains in area outweigh the losses, until LM = MA', where the maximal area sought is found.

GSP finds the area as 0.29792 cm^2. Compare to the area of the whole triangle: .64550 cm^2.

Posted by Charlie on 2014-02-11 21:34:08