Draw a circle of radius 3.6 cm. Take any point M on the circle. Draw a tangent to the circle at point M without using the centre of the circle.

Answer: Line TM drawn through M with ∠NMT = ∠MKN is the required tangent.

Steps of construction:

1. Draw a circle of radius 3.6 cm and take a point M on it.
2. Draw any chord MN through M.
3. Take a point K on the major arc MN, other than M and N. Join KM and KN to form ∠MKN.
4. With K as centre and a convenient radius, draw an arc cutting seg KM at P and seg KN at Q.
5. With M as centre and the same radius, draw an arc cutting chord MN at R and extending to the side of chord MN opposite to point K.
6. With R as centre and radius equal to PQ, draw an arc cutting the previous arc at T, on the side of chord MN opposite to point K. Then ∠NMT = ∠MKN.
7. Draw line MT.
8. Line TM is the required tangent to the circle at M.

Reason: By the converse of the tangent-secant angle theorem, if the angle between a line and a chord at a point of the circle equals the inscribed angle on the other side of the chord, the line is a tangent at that point.