PQRS is a cyclic quadrilateral in which PQ = x cm, QR = 16.8 cm, RS = 14 cm, PS = 25.2 cm, and PR bisects QS. What is the value of x ?

21

We know that , Angles made by an chord on the circumference of a circle are same.

Angles made  by chord SP

⇒ ∠SQP = ∠SRO  = ∠OQP

Similarly , Angles made  by chord QR

⇒ ∠QPR = ∠QSR = ∠OSR

In triangle POQ and SOR,

∠POQ = ∠SOR

∠OQP = ∠SRO

∠QPR = ∠OSR

⇒ Triangle POQ ∼ SOR

We know that ,

\(\frac{OQ}{OR}\) = \(\frac{PQ}{SR}\)

( Gievn :- PQ = x cm )

\(\frac{OQ}{OR}\) = \(\frac{x}{14}\)

It is gievn that OQ = OS

⇒ \(\frac{OS}{OR}\) = \(\frac{x}{14}\)   -----(1)

Similarly triangle POS ∼ QOR

\(\frac{OS}{OR}\) = \(\frac{PS}{QR}\)

\(\frac{OS}{OR}\) = \(\frac{25.2}{16.8}\)   ----(2)

On comparing equation 1 and 2

\(\frac{x}{14}\) = \(\frac{25.2}{16.8}\)

x = \(\frac{25.2}{16.8}\) x 14

= 21 cm