Theory:

Working rule to construct a tangent to a circle (Using Alternate Segment Theorem):
Given the radius of a circle, let us learn how to construct a tangent to the circle using its alternate segment theorem.
Example:
Draw a circle of radius \(4\) \(cm\). At a point \(L\) on it, draw a tangent to the circle using the alternate segment.
 
Rough Sketch:
 
Const_R_2.png
 
Construction:
 
Step \(1\): With \(O\) as the centre, draw a circle of radius \(4\) \(cm\).
 
Step \(2\): Take a point \(L\) on the circle. Through \(L\), draw any chord \(LM\).
 
Step \(3\): Take two points \(M\) and \(N\) distinct from \(L\) on the circle, so that \(L\), \(M\) and \(N\) are in anti-clockwise direction. Join \(LN\) and \(NM\).
 
Step \(4\): Through \(L\) draw a tangent \(TT'\) such that \(\angle TLM\) \(=\) \(\angle MNL\).
 
Step \(5\): \(TT'\) is the required tangent.
 
Cons_2_GIF.gif