Question

Construct a cyclic quadrilateral $PQRS$ such that $PQ=QR=4.3cm$ and diagonals $PR=6.2cm,QS=5.8cm$. Then, radius of the circle circumscribing the quadrilateral $PQRS$ is:

• A. $4.2cm$
• B. $3.1cm$
• C. $2.4cm$
• D. $3.7cm$

Answer 1

The correct option is B $3.1cm$

• Draw a line segment $QR$ of length $4.3cm$.
  
• From $Q$ make an arc of length $4.3cm$.
  
• From $R$ make an arc of length $6.2cm$.
  
• Both these arcs cuts each other at point $P$.
  
• Join $PQ$ and $PR$.
  
• Draw perpendicular bisector of line $PQ$ and $QR$. The perpendiculars intersect each other at $O$.
  
• Draw a circle of radius $OQ$ from $O$ as center. This circle also touches $P,Q$ and $R$.
• From $Q$ mark an arc of $5.8cm$, this arc cuts the circle at point $S$
  
• Join $PQRS$, is the required quadrilateral.
  

$PQ=4.3QS=5.8cm\angle SPQ={111}^o$ (measurement)

Radius $=\frac{a}{2\sin A}$

In $△PSQ$

$R=\frac{5.8}{2\sin \left({111}^o\right)}=3.1cm$