Exemplar Exercise 9.2 (Circles)

If a chord AB subtends an angle of 60° at the centre of a circle, then angle between the tangents at A and B is also 60°.

The length of tangent from an external point on a circle is always greater than the radius of the circle.

The length of tangent from an external point P on a circle with centre O is always less than OP.

The angle between two tangents to a circle may be 0°.

If angle between two tangents drawn from a point P to a circle of radius a and centre O is 90°, then OP = a√2
 .

If angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then OP = a√3
 .

The tangent to the circumcircle of an isosceles triangle ABC at A, in which AB = AC, is parallel to BC.

If a number of circles touch a given line segment PQ at a point A, then their centres lie on the perpendicular bisector of PQ.

If a number of circles pass through the end points P and Q of a line segment PQ, then their centres lie on the perpendicular bisector of PQ.

AB is a diameter of a circle and AC is its chord such that ∠BAC = 30°. If the tangent at C intersects AB extended at D, then BC = BD.