We know that in a triangle the line segment joining the mid points of two sides, is parallel to the third side and measures half of the third side.
In the triangle OAB, EF is the line joining the mid points of OA and OB. Hence EF = 1/2 AB.
In the triangle OBC, FG is the line joining the midpoints of OB and OC.
hence, FG = 1/2 BC
similarly, GH = 1/2 CD and HE = 1/2 DA
perimeter of the quadrilateral EFGH = 1/2 AB + 1/2 BC + 1/2 CD + 1/2 DA
= 1/2 * perimeter of parallelogram ABCD.
The ratio = 1/2
we also note that EFGH is a parallelogram and has area 1/4 th of ABCD.