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

Answer: Use the converse of the tangent-secant angle theorem (alternate segment angle).

Analysis: By the converse of the tangent-secant angle theorem, if a line through a point on the circle makes with a chord an angle equal to the angle inscribed in the opposite arc, the line is a tangent.

Steps of construction:

1. Draw a circle of radius 3.6 cm and take a point P on it.
2. Draw a chord PQ through P. Take any point R on the circle, different from P and Q (on the major arc). Join RP and RQ.
3. With R as centre draw an arc of any radius cutting RP and RQ at M and N.
4. With the same radius and centre P, draw an arc cutting chord PQ at S (on the side away from R).
5. With S as centre and radius equal to MN, cut the arc of step 4 at T. Draw line PT.
6. Line PT is the required tangent, since \(\angle QPT = \angle QRP\).