UPSC CDS 1 2021 Math Paper

let the circle touches the side of quadrilateral AB, BC, CD and DA at P, Q, R and S.
AB=AP+PB=9
BC=BQ+QC=8
CD=CR+RD=12
DA=DS+SA=?
we know that the tangents from a point to the circle are equal. So,
AP=AS
BP=BQ
CQ=CR
DR=DS
Now,
DA=DS+SA
=DR+AP
=(12−CR)+(9−PB)
=21−(CQ+BQ)
=21−8
=13