Question

23. In the given figure 12.127 PQRS is a cyclic
quadrilateral. PQ and SR produced meet at T.
S
R
T
Q
P
Fig. 12.127
(i) Prove ATPS – ATRQ
(ii) Find SP if TP = 18 cm, RQ = 4 cm and TR = 6
cm
(iii) Find the area of quadrilateral PQRS if area of
APTS=27 cm?
(ICSE 2016)​

Answer 1

Answer:

We know that

TR−TS=TQ.TP

⇒TPTR=TSTQ

Also ∠RTQ=∠STP

⇒△TPS∼△TRQ

ii) So, TRTP=RQSP

⇒618=4SP

⇒SP=12 cm

iii) ar(△QTR)ar(△PTS)=(TRTP)2=(618)2=9

⇒ar(△QTR)27=9⇒ar(△QTR)=3 cm2

ar(PQRS)=ar(△PTS)−ar(△QTR)=27−3=24 cm2