Question 1067049
If you draw the diagonals,
you realize that each diagonal
divides the quadrilatetal into two triangles,
and the segments connecting the midpoints
of the sides of ABCD
are the midsegments of those triangles.
As such, they are parallel to one or another diagonal,
and half as long as that diagonal.
So, the new quadrilateral is a parallelogram,
and it's perimeter is the sum of the lengths
of the diagonal of ABCD.
{{{drawing(400,400,-6,6,-4,8,grid(1),
green(triangle(4,5,1,5.5,3.5,1.5)),
green(triangle(3,-2,3.5,1.5,-1,-0.5)),
green(triangle(-5,1,-1,-0.5,-3.5,3.5)),
green(triangle(-2,6,1,5.5,-3.5,3.5)),
red(line(-2,6,4,5)),
red(line(4,5,3,-2)),
red(line(3,-2,-5,1)),
red(line(-5,1,-2,6)),
green(line(-2,6,3,-2)),
green(line(4,5,-5,1))
)}}}
The lengths of the diagonals are
{{{AC=sqrt((3-(-2))^2+(6-(-2))^2)=sqrt(5^2+8^2)=sqrt(25+64)=89}}} and
{{{BD=sqrt((4-(-5))^2+(5-1)^2)=sqrt(9^2+4^2)=sqrt(81+16)=sqrt(97)}}} ,
so the perimeter of EFGH is
{{{highlight(sqrt(89)+sqrt(97))}}}