Construct a cyclic quadrilateral PQRS such that PQ = 5.5 cm, QR = 4.5 cm, ∠QPR = 45° and PS = 3 cm.

We need to construct the required quadrilateral, when three sides and one diagonal of a cyclic quadrilateral are given.

Steps of construction:

Step 1: Draw a rough diagram and mark the measurements.

Step 2: Draw a line segment PQ = 5.5 cm.

Step 3: Through P draw PX such that ∠ QPX = 45°

Step 4: With Q as centre and radius 4.5 cm, draw an arc intersecting QX at R and join QR.

Step 5: Draw the perpendicular bisectors of PQ and QR intersecting each other at O.

Step 6: With O as centre and OP (= OQ = OR )as radius, draw the circumcircle of DPQR.

Step 7: With P as centre and radius 3 cm, draw an arc intersecting the circle at S.

Step 8: Join PS and RS.

Thus, PQRS is the required cyclic quadrilateral.