So EGA angle EGF angle =90 degrees.
So the triangle EGF is a right triangle.
So ef 2 = eg 2+fg 2.
Because the rectangular ABCD is folded along EF
So CE=PE
CF=PF
AB=CD
AD = BC
Angle ABC= angle BAD= angle BCD= angle ADC=90 degrees.
AB parallel CD
Parallel BC
So angle ABC= angle PAD=90 degrees.
Angle EGF= angle BAD=90 degrees
So AB is parallel to EF.
So the quadrilateral ABEG is a parallelogram.
So AB=EG
BE=AG
In the right triangle EBP, the angle ABC=90 degrees.
So PE 2 = Pb 2+Be 2.
Because AB=4 AP=2
PB=AB+AP
So PB=6
EG=4
Because AD=8
So BC=8
So BE=BC-CE=8-PE.
So PE 2 = 6 2+(8-PE) 2.
So AG=BE=7/4.
In the right triangle PAF, the angle PAF=90 degrees.
So pf 2 = AP 2+AF 2.
In a right triangle CDF, the angle ADC=90 degrees.
So cf 2 = df 2+CD 2
So AP 2+AF 2 = CD 2+DF 2.
Because DF=AD-AF=8-AF
AB=CD=4
So af 2+4 = 16+(8-af) 2.
In AF= 19/4.
Because FG=AF-AG=3
So EF= radical sign (3 2+4 2) = 5)
So EF=5