To construct a quadrilateral PQRS, where PQ = PS = 4 cm, QR = QS = 5 cm, and SR = 6.2 cm, follow these steps:

Step 1: Draw a line segment PQ of length 4 cm.
- Start by drawing a straight line using a ruler and a pencil. Label this line as PQ, and mark a point P on the left end and a point Q on the right end.

Step 2: Construct a perpendicular bisector of line segment PQ.
- Use a compass to bisect the line segment PQ. Place the compass point on point P and draw an arc that intersects the line segment PQ. Without changing the compass width, place the compass point on point Q and draw another arc that intersects the previous arc. Label the points of intersection as A and B.
- With a ruler, draw a straight line passing through points A and B. Label the midpoint of line segment PQ as O.

Step 3: Construct a perpendicular line at point O.
- With point O as the center and any convenient radius, draw two arcs intersecting the line segment PQ. Label the points of intersection as C and D.
- With C and D as centers and a radius greater than half of line segment PQ, draw arcs intersecting each other. Label the points of intersection as E and F.
- Draw a straight line passing through point O and point E. Extend this line to the left side of PQ. Label the point of intersection as R.

Step 4: Construct a perpendicular line at point R.
- With point R as the center and any convenient radius, draw two arcs intersecting the line segment OR. Label the points of intersection as G and H.
- With G and H as centers and a radius greater than half of line segment OR, draw arcs intersecting each other. Label the points of intersection as I and J.
- Draw a straight line passing through point R and point I. Extend this line to the right side of OR. Label the point of intersection as S.

Step 5: Complete the quadrilateral.
- Connect the points P, Q, S, and R with straight lines to form the quadrilateral PQRS.

The resulting quadrilateral PQRS will have the given measurements: PQ = PS = 4 cm, QR = QS = 5 cm, and SR = 6.2 cm.