Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments

The figure shows a circle of radius 9 cm with 0 as the centre. The diameter AB produced meets the tangent PQ at P. If PA = 24 cm, find the length of tangent PQ:

TA and TB are tangents to a circle with centre O from an external point T. OT intersects the circle at point P. Prove that AP bisects the angle TAB.

In the following figure; If AB = AC then prove that BQ = CQ.

In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If ∠ACO = 30°, find:

1. ∠BCO 
2. ∠AOB
3. ∠APB

In a triangle ABC, the incircle (centre O) touches BC, CA and AB at points P, Q and R respectively. Calculate:

i)`∠`QPR .

In the following figure, PQ and PR are tangents to the circle, with centre O. If `∠`QPR = 60°, calculate:

1. ∠QOR,
2. ∠OQR,
3. ∠QSR.

In the given figure, AC = AE. Show that:

1. CP = EP
2. BP = DP

In figure , ABC is an isosceles triangle inscribed in a circle with centre O such that AB = AC = 13 cm and BC = 10 cm .Find the radius of the circle.