CHAPTER : CIRCLES

CIRCLE:A circle is a collection of all points in a plane which are at a constant distance from a fixed point.
 The fixed point is called the centre and the constant distance is known as the radius.

O is the centre and OA is the radius of the circle

SECANT: A line which intersects a circle in two distinct  points is called a secant of a circle.

Line PQ is a secant of a circle

TANGENT:  A tangent to a circle is a line that  intersects the circle in exactly one point.

 line PQ intersects a circle at a single point O, so it is tangent of a
              circle

THEORM 1: A tangent at any point of a circle is perpendicular to the radius through the point of contact.

Given:-  A circle (O,r) and a tangent AB at point P.

To Prove:- OP┴ AB

Construction:- Take any point Q, other than P, on the tangent
                        AB. Join OQ. Suppose OQ meets the circle at R.

Proof:-  Clearly,  OP = OR
                           OQ = OR + RQ
                           OQ > OR
                           OQ > OP
                           OP < OQ

Thus, OP is the shorter distance than any other segment joining O to any point of AB.

Hence, OP ┴ AB 

Length of tangent: The length of the segment of the tangent  from the external point and the point of contact with the circle is called the  length of the tangent.

THEORM 2: The lengths of tangents drawn from an external point to a circle are equal.

Given:-  PQ and PR are two tangents from point P
to a circle (O,r)

To Prove:- AP = AQ

Construction:- Join OP, OQ and OA.

Proof:- By  joining OP, OQ and OR,
            angle OQP and ORP are right angles.

            Now, in triangle OQP and ORP,
            OQ = OR            (radii of same circle)
            OP = OP             (Common)
            OQP= ORP         (RHS)
            Therefore, PQ = PR  ( by CPCT)