Question

Construct a cyclic quadrilateral $PQRS$ in which $PQ$ $=5$ cm, $QR$ $=6$ cm, $PR$ $=3.5$ cm and $RS$ $=5$ cm.

Answer 1

Steps of Construction:

• Draw line segment $PR=3.5$ cm
• With P as center, cut an arc of radius $5$ cm. Similarly, with $R$ as center, cut an arc of radius $6$ cm, intersecting the previous arc at $Q$.
• Join $PQ$ and $QR$.
• Draw perpendicular bisectors of $PQ$ and $QR$. Let them intersect at $O$.
• Since $PQRS$ is cyclic, $PQ$ and $QR$ are chords of the circum-circle. Hence, perpendicular bisectors of the chords will meet at the center. Therefore, $O$ is the center of the circum-circle.
• With $O$ as center, draw a circle passing through $R,P$ and $Q$.
• With $R$ as center, draw an arc of radius $5$ cm, intersecting the circle at $S$.
• Join $RS$ and $QS$.

Thus $PQRS$ is a required quadrilateral.