A tangent PQ touches a circle with centre O at point A. If the radius OA is 5 cm, what is the magnitude of angle OA^Q?

Two tangents PA and PB are drawn to a circle from an external point P. If PA = 12 cm, what is the length of PB?

In the figure for question 2, if AP^B=70∘, find the magnitude of AO^B, where O is the centre.

A tangent XY touches a circle at point C. A chord CD is drawn such that YC^D=65∘. Find the magnitude of the angle subtended by the chord CD in the alternate segment.

A circle with centre O touches the sides of triangle ABC at points P, Q, R on sides AB, BC, CA respectively. If AP = 4 cm, BQ = 6 cm, and CR = 5 cm, find the perimeter of triangle ABC.

Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

In a circle with centre O, PT is a tangent from an external point P. If OT is the radius and OP = 25 cm, PT = 24 cm, find the radius of the circle.

Two circles touch each other externally at P. A common tangent AB touches the circles at A and B. Prove that the tangent at P bisects AB.

The tangent to a circle at point P is ST. QR is a chord parallel to ST. Prove that triangle PQR is an isosceles triangle.

From an external point P, two tangents PA and PB are drawn to a circle with centre O. Prove that OP is the perpendicular bisector of the chord AB.

A circle is inscribed in a quadrilateral ABCD. Prove that AB + CD = AD + BC.

Two circles intersect at A and B. A common tangent touches the circles at P and Q. Prove that the line AB, when produced, bisects PQ.

In triangle ABC, a circle is drawn passing through B and touching AC at its midpoint D, and intersecting AB at P. Prove that AB = 4AP.

State and prove the alternate segment theorem.