In figure 3, a circle touches all the four sides of a quadrilateral ABCD whose sides AB = 6 cm, BC = 9cm and CD = 8 cm. find the length of side AD.

As we know, tangents drawn from an external point to a circle are equal

PC = CQ (i) [Tangents from point C]

PD = DS (ii) [Tangents from point D]

AR = AS (iii) [Tangents from point A]

RB = BQ (iv) [Tangents from point B

Adding all four equations

PC + PD + AR + RB = CQ + DS + AS + BQ

⇒ (PC + PD) + (AR + RB) = (CQ + BQ) + (DS + AS)

⇒ AD = CD + AB – BC

⇒ AD = 8 + 6 – 9

⇒ AD = 5cm