If BQ = CR = DS = AP, then the shapes of the 4 pieces are the same. That implies PR and QS are composed of 1 long and 1 small "interior" sides.
And so, PR = QS.
The fact that we know the perimeter is useless...
Fig, can you pls. explain how you go from 'BQ = CR = DS = AP' to 'the shapes of the 4 pieces are the same'
If BQ = CR = DS = AP, then QC = RD = AS = PB, since it is a square. How do you proceed from here?
Well, I can try
As BQ = CR = DS = AP, we know that QC = RD = SA = BP because we have a square with each side thus equal.
Then, at this point, we know that the triangles BQP, QCR, RDS and SAP are right similar triangles.
That implies that PQ=QR=RS=SP.
In addition, each angle of PQRS are rights by the following reasonning:
o angle(PQR) = 180 - angle(BQP) - angle(RQC)
= 180 - (90-angle(BPQ)) - angle(RQC) as BQP is a right triangle
= 90 + angle(BPQ) - angle(RQC)
= 90 + angle(BPQ) - angle(BPQ) as BPQ and RQC are similar triangles
We are thus now with PQRS definied by:
o 4 equal sides
o 4 right angles
It's the definition of a square and that means its diagonals PR and QS are equal.