Area of a square inside a square created by connecting point-opposite midpoint

Solution 1:

By moving small triangles we can make $5$ equal small squares.

Solution 2:

By drawing lines along and parallel to the segments within the square, a grid overlapping the original square is produced. If $A$ represents the area of the shaded square, then $9A$ represents the area of the circumscribing square with sides parallel to the shaded square

It is easy to see a right triangle on each side of the original square with its hypotenuse on the side of the original square, each with an area of $\frac{2\cdot A}{2} = A$. Thus the total area of the original square is $5A$, or the ratio of the area of the shaded square to the original square is $1/5$.