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

Given that the sides of the quadrilateral ABCD are as follows: S

AB = 6 cm, BC = 9 cm, CD = 8 cm.

Let P, Q, R, S be the points where the tangent touches the sides AD, AB, BC, and CD respectively. P R

From a point outside of a circle, two tangents to the circle are always equal. Q

Therefore, DP = DS

AP = AQ

BR = BQ

CR = CS

Thus, adding the left sides and right sides separately and equating them, we get:

(DP + AP) + (BR + CR) = (DS + CS) + (AQ + BQ)

⇒ AD + BC = CD + AB

⇒ AD = CD + AB - BC

= 8 + 6 – 9

= 5

∴ AD = 5 cm.