The point of intersection of the diagonals of a rectangle is 4 cm further away from the smaller side than from the larger side of the rectangle. The perimeter of the rectangle is equal to 56 cm. Find the lengths of the sides of the rectangle.

we have that
If you draw a diagram, and draw perpendiculars from the intersection to the sides of the rectangle, you will see they are the mid point of each side. 

see the attached figure

If the mid point of the smaller side is ---------> x.
The mid point of the larger side will be -----------> (x + 4) 
The full length if the smaller side will be 2x and the length of the larger side will be 2(x + 4) = 2x + 8 

The perimeter = 2 times the length + 2 times the width. 
2(2x + 8) + 4x = 56 
4x + 16 + 4x = 56 
8x = 40 
x = 5 
Therefore 
length = 2x + 8 = 18cm 
width = 2x = 10cm

the answer is
the lengths of the sides of the rectangle are
length = 18cm 
width = 10cm