In the given figure, ABCD is a quadrilateral in which ∠D=90°. A circle C (O,r) touches the sides AB, BC, CD and DA at P,Q,R,S respectively, If BC =38 cm, CD=25 cm and BP=27 cm, find the value of r.

r is the radius which is OR = r

Consider quadrilateral DROS

⇒ ∠RDS = 90° …given

⇒ ∠DRO = 90° …radius is perpendicular to the tangent

⇒ DR = DS …tangents drawn from the same point are equal

As the adjacent angles are 90° and adjacent sides are same hence DROS is a square

Hence OR = DR = r …(i)

As tangents drawn from the same point are equal

BQ and BP are tangents drawn from B

⇒ BQ = BP
⇒ BQ = 27 cm …BP is 27 cm given

From figure

⇒ BC = BQ + QC

⇒ 38 = 27 + QC …BC is 38 cm given

⇒ QC = 11 cm

CQ and CR are tangents drawn from C

⇒ CQ = CR …tangents from same point

⇒ CR = 11 cm

Again from figure

⇒ CD = CR + RD

⇒ 25 = 11 + r …CD is 25 given and RD = r from (i)

⇒ r = 14 cm

Hence r radius is 14 cm