Construction of a Tangent to a Circle at a given point when its centre is not known : 
Steps of construction : 

   Draw a circle of  radius r cm.

Mark a point P on it. Draw any chord PQ.

Take a point R on the major arc QP. Join RP & RQ.  

 Produce TP to T" such that T' PT is the required tangent at P.


Construction of Tangent to a Circle from a Point Outside it when the Centre of the Circle is

known : 
Steps of Construction : 

Draw a circle with O as centre and radius r cm. Mark a point P outside the circle such that OP = x cm.

Join OP and draw its perpendicular bisector, which cut OP at M.

Steps of Construction : 
 Draw a circle with O as centre and radius r cm and any diameter AOB of this circle.

Construct the given angle at O such that radius OC meets the circle at C. (Suppose given angle is 90º)

Draw perpendicular at A and C intersect each other at P.

Hence PA and PC are the required tangents to the given circle, inclined at a given angle.