P is the midpoint of line BD. AP = BP = 4, line AP is perpendicular to line BD, line BD is perpendicular to line DC, line

Not too bad, ACG  !!!

Note that...if AP  = BP....then triangle APB is an  isosceles right triangle

So.....AB  =  sqrt (4^2 + 4^2)  =  sqrt (32)   =  sqrt (16 * 2) =  4sqrt (2)

If  P  is the midpoint of BD....then   BP  = 4  and BD  = 8

Note that BDC is a right angle....and since angle B is bisected, then  CBD  = 45°  .....so  since BDC  = 90°......then , by default  angle BCD  = 45°

So...triangle BDC  is also isosceles with BD  = DC  = 8

So......BC  = sqrt  (8^2 + 8^2)  =  sqrt (128)  = sqrt (64* 2)  =  8 sqrt (2)

So....the perimeter is

BA + AP + PD + DC + BC =

4sqrt (2) + 4 + 4 + 8   + 8sqrt (2)  =

16  + 12sqrt (2) units  ≈   32.97  units