Draw a circle of radius 2.8 cm. Take any point P on it and draw a tangent at P without using the centre of the circle.

Answer: Line PT drawn so that \(\angle TPQ \cong \angle PRQ\) is the tangent (by the converse of the tangent secant angle theorem).

Steps of construction:

1. Draw a circle of radius 2.8 cm and take point P on it.
2. Draw any chord PQ. Take a point R on the major arc and draw seg RP and seg RQ, forming \(\angle PRQ\).
3. With R as centre and any convenient radius, draw an arc cutting RP and RQ at L and M. With the same radius and P as centre, draw an arc cutting chord PQ at N.
4. Measure LM with the compass. With N as centre and radius LM, cut the arc drawn from P at T, on the side of PQ away from R.
5. Draw line PT. It is the required tangent.

Reason: \(\angle QPT \cong \angle PRQ\) by construction. If the angle between a chord and a line at its end equals the angle inscribed in the opposite arc, the line is a tangent (converse of the tangent secant angle theorem).

[Figure: circle with chord PQ, point R on the major arc; equal arcs copy \(\angle PRQ\) at P to give line PT touching the circle at P.]