Area of a Cyclic Quadrilateral

Geometry Level pending

P T = 18 , R Q = 4 , T R = 6 PT=18,RQ=4,TR=6 and area of △ P T S = 27. \triangle PTS=27.
What is the area of the Quadrilateral P Q R S ? PQRS?


The answer is 24.

Ayush G Rai by Ayush G Rai , 20, India

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1 solution

Ayush G Rai
Oct 23, 2016

We observe that ∠ S P Q = ∠ T R Q \angle SPQ=\angle TRQ and ∠ R S P = ∠ T Q R \angle RSP=\angle TQR [Interior opposite angles of a cyclic quadrilateral are supplementry]
By A A AA postulate of similarity , △ T R Q △ T P S . ,\triangle TRQ~\triangle TPS.
By similarity properties, ( T R P T ) 2 = A r . △ T R Q A r . △ P T S ⇒ A r . △ T R Q = 3. {(\frac{TR}{PT})}^2=\frac{Ar.\triangle TRQ}{Ar.\triangle PTS}\Rightarrow Ar.\triangle TRQ=3.
Therefore,Area of quadrilateral P Q R S = A r . △ P T S − A r . △ T R Q = 27 − 3 = 24 . PQRS=Ar.\triangle PTS-Ar.\triangle TRQ=27-3=\boxed{24}.


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