L-Shape Halves

To make an ‘L-shape’, you generally cut a rectangular piece out of one corner of a larger rectangle. The problem of cutting this remaining shape into 2 equal halves is an interesting one, that also solves the problem of how to balance a piece of card of this shape, either on the edge of a desk or on a pencil point ...

This Problem goes beyond that and asks whether it is possible to divide both the L-shape itself and the corner piece removed with a single cut. Read on ...

The Problems :

A         The diagram below shows an L-shaped piece of card. ( The shaded part has been removed from the original larger rectangle )

Can you find a straight cut that divides the L-shape into 2 halves of equal area ?

Are there other ways to do this ? How many ?

B

There are clearly lots of ways to cut Area A into equal halves, but can you find a single  (straight) line to cut both the corner piece and the L-shape into equal pieces at the same time ?

( Hint : If you divide A in two, and ( A + B ) in two, you must have divided B in two as well )

C         Now try your hand at these L-shapes ...

D         If you’ve found a technique for cutting both the L-shapes and their missing corners in two with the same single cut, then you can probably see a way to solve the same problem for these 2 shapes - even though they look so much harder !