If you were told to draw a rectangle along the lines of a sheet of graph paper such that its area is 40 squares, you could choose rectangles measuring 8x5, 10x4, 20x2 or 40x1.

For two of these, 8x5 or 10x4, you would find that you could draw a diagonal across the rectangle that would pass through exactly 12 squares.

What is the smallest number of squares that could be the area of *three* different rectangles whose diagonals pass through the same number of squares? How many squares does this diagonal pass through?

No, you are wrong. Please read the question probably. It is mentioned that 'what is the smallest number of squares that could be the area of THREE DIFFERENT RECTANGLES whose diagonals..........' As it is meant for three different rectangles, there must be three sets of rectangles to be formed.

When the question mentions 3 sets of different rectangles, it does not mention that it should not be the imaginative rectangles and so, it is justifiable to draw the four diagonal lines from one corner of the square to another.

It is erroneous to assume that a square is the solution. This is due to a square is a square and not rectangle.