ABCD is a cyclic quadrilateral in which AB = 16.5 cm, BC = x cm, CD = 11 cm, AD = 19.8 cm, and BD is bisected by AC at O. What is the value of x ?

13.2 cm

\(\angle\)ACD = \(\angle\)ABD   (As Angle made by a chord in same segment are equal)

\(\angle\)BDC = \(\angle\)BAC  (As Angle made by a chord in same segment are equal)

So, \(\Delta \)AOB is similar to \(\Delta \)DOC by AA.

\(\frac{AB}{DC}\) = \(\frac{OB}{OC}\)

= \(\frac{OB}{OC}\) = \(\frac{16.5}{11}\)

Given, OB = OD

\(\frac{OD}{OC}\) = \(\frac{16.5}{11}\)     ..(1)

As we know,

\(\Delta \)ADO is similar to \(\Delta \)BCO

\(\frac{OD}{OC}\) = \(\frac{AD}{BC}\)

= \(\frac{OD}{OC}\) = \(\frac{19.8}{X}\)     ..(2)

From eq (1) and eq (2)

\(\frac{19.8}{X}\) = \(\frac{16.5}{11}\)

= X = \(\frac{11\;×\;19.8}{16.5}\)

= X = 13.2 cm.