The letter box

The rectangle area is B[1+√(B/(B-A))].

The rectangle side lengths are √X and √Y. The key is that all triangles are similar. It is then a matter of expressing their area in terms of the rectangle area T, as shown in the diagram.

From the lower half of the rectangle we get 1+T/2X+T/2Y=T/2B, whereas the upper half gives 1+T/2X+T/2Y+T/2(X+Y)=T/2A. Subtracting these equations gives 1/(X+Y)=1/A-1/B, which leads to T2(1/X+1/Y)=AB/(B-A).

Adding the two equations and using the above leads to T2/B-2T-AB/(B-A)=0. This is a quadratic equation in T, which can be solved to find the desired answer.

 6) Math Art

Math Art sent by Rafael Araujo