In figure (ii) given below, quadrilateral ABCD is circumscribed and AD ⊥ DC; find x if radius of incircle is 10 cm.

Join OR as shown in the figure below:

From figure,

OS = OR (As both are radius of circle.)

SD = OR (As OSDR form a square.)

∴ SD = OS.

From D, DS and DR are the tangents to the circle.

∴ DR = DS = 10 cm. (∵ if two tangents are drawn to a circle from an external point then the tangents have equal lengths.)

From B, BP and BQ are the tangents to the circle.

∴ BQ = BP = 27 cm. (∵ if two tangents are drawn to a circle from an external point then the tangents have equal lengths.)

From figure,

CQ = BC - BQ = 38 - 27 = 11 cm.

Now from C, CQ and CR are the tangents to the circle

CR = CQ = 11 cm.

⇒ DC = x = DR + CR = 10 + 11 = 21 cm.

Hence, the length of x = 21 cm.