As shown in the figure, in the square ABCD, point P is a point on the diagonal AC, PE ⊥ AB, PF ⊥ BC, and the perpendicular feet are points E and f respectively. If the perimeter of the square ABCD is 8cm, then the perimeter of the quadrilateral EBFP is______ cm．

As shown in the figure, in the square ABCD, point P is a point on the diagonal AC, PE ⊥ AB, PF ⊥ BC, and the perpendicular feet are points E and f respectively. If the perimeter of the square ABCD is 8cm, then the perimeter of the quadrilateral EBFP is______ cm．

According to the meaning of the question, the quadrilateral EBFP is a rectangle, so BF = PE, PF = be, and the point P is on the diagonal AC, ∠ BAC = 45 °, so AE = PE. The perimeter of the square ABCD is 8cm, so the side length AB = 2cm, so the perimeter of the quadrilateral EBFP is be + EP + pf + BF = be + AE + pf + AE = 2Ab = 4cm

In square ABCD, P is a point on diagonal AC, PE ⊥ AB is in E, PF ⊥ BC is in E. try to guess the quantitative and positional relationship between EF and PD, and give the proof

EF = PD
∵ ABCD is a square, ∵ EB ⊥ FB, PE ⊥ EB, PF ⊥ FB, ∵ bepf is a rectangle, ∵ EF = Pb
∵ ABCD is a square, ∵ BC = DC, ∵ BCP = DCP = 45 ° and CP = CP, ≌ BCP ≌ DCP,
∴PB＝PD.
From EF = Pb, Pb = PD, EF = PD

The square ABCD. P is perpendicular to the point (not the midpoint) PE on the diagonal AC AB.PF Vertical BC. Connect EF and PD. Explain PD = EF
It's urgent``

Make PM vertical CD through P, PN vertical ad, because AC is a square diagonal, so PM = PF, PE = PN
Because of the vertical relationship of auxiliary lines, PNDM is rectangular, so PN = DM, so PE = PN = DM
Because PM = PF, PE = PN = DM, angle PMD = angle FPE = 90 degrees, PMD and FPE are identical
So PD = EF