1. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?

(a) 145° (b) 55° (c) 110° (d) None of the above

2. Two tangents are drawn to a circle from points A and B on its circumference, and they intersect at point P. If \(\angle\)APB = 68°, then what is the measure of \(\angle\)PAB?

3. AB and AC are two tangents drawn from point A to a circle with center O. The line OA intersects the chord BC (which joins the points of contact) at point M. If AM = 8 cm and BC = 12 cm, then what is the length of each tangent?

(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm

4. In a circle, AB and AC are two chords. Tangents are drawn at points B and C, and they intersect at point P. If \(\angle\)BAC = 54°, then what is the measure of \(\angle\)BPC?

5. In the adjacent figure, a circle is centered at point O. From an external point C, two tangents are drawn to the circle, touching it at points P and Q respectively. Another tangent is drawn at point R on the circle, which intersects the tangents CP and CQ at points A and B respectively. If CP = 11 cm and BC = 7 cm, then find the length of BR.

6. In a circle with center O, PQ and PR are two chords. Tangents drawn at points Q and P intersect at point S. If ∠QSR = 70°, then what is the measure of ∠QPR?

7. If two tangents are drawn from points P and Q on a circle and intersect at point A such that ∠PAQ = 60°, then what is the measure of ∠APQ?

8. From an external point A of a circle, two tangents AB and AC are drawn, touching the circle at points B and C respectively. A tangent is drawn at point X, which lies on the arc BC, and it intersects AB and AC at points D and E respectively. Prove that the perimeter of triangle ∆ADE = 2 × AB.

9. From an external point \(P\), two tangents \(PS\) and \(PT\) are drawn to a circle with center \(O\). \(QS\) is a chord of the circle that is parallel to \(PT\). If \(\angle SPT = 80^\circ\), then what is the measure of \(\angle QST\)?

10. AB is a chord of a circle with center O. From point O, a perpendicular OP is drawn to the chord AB. The extended line OP intersects the circle at point C. If AB = 6 cm and PC = 1 cm, then what is the radius of the circle?

11. From an external point P, two tangents PA and PB are drawn to a circle centered at O. If PA = 9 cm and ∠APB = 60°, then what are the measures of ∠PAB and the length of chord AB?

(a) 9 cm (b) 3 cm (c) 6 cm (d) 12 cm

12. AB is a diameter of a circle with center O. P is any point on the circle. Tangents are drawn at points A and B, which are intersected by the tangent drawn from point P at points Q and R respectively. If the radius of the circle is \(r\), prove that \(PQ \cdot PR = r^2\).

13. "Two tangents are drawn from points P and Q on a circle, which intersect at point A. If ∠PAQ = 60°, determine the value of ∠APQ."

14. I draw a circle with center O where two radii, OA and OB, are perpendicular to each other. Tangents are drawn at points A and B, which intersect each other at point T. Prove that AB = OT and that they bisect each other perpendicularly.

15. Rumela has drawn a circle centered at point O, in which QR is a chord. Tangents are drawn at points Q and R, and they intersect at point P. If QM is a diameter of the circle, prove that ∠QPR = 2∠RQM.

16. QR is a chord of a circle with center O. Tangents are drawn at points Q and R, and they intersect each other at point P. QM is the diameter of the circle. Prove that \(\angle\)QPR = 2\(\angle\)RQM.

17. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

18. The radius of a circle centered at point O is 5 cm. Point P is located 13 cm away from point O. PQ and PR are two tangents drawn from point P to the circle. What is the area of quadrilateral PQOR?

19. Two radii, OA and OB, of a circle centered at O form an angle of 130°. Tangents are drawn at points A and B, and they intersect at point T. Find the measures of \(\angle\)ATB and \(\angle\)ATO.

20. From an external point \(A\), two tangents are drawn to a circle centered at \(O\), touching the circle at points \(B\) and \(C\) respectively. Prove that the line \(AO\) is the perpendicular bisector of the line segment \(BC\).

21. In the figure, triangle ABC is inscribed in a circle and touches the circle at points P, Q, and R. If AP = 4 cm, BP = 6 cm, AC = 12 cm, and BC = x cm, then what is the value of x?

22. In a circle centered at O, PQ and PR are two chords. The tangents drawn at points Q and R intersect at point S. If \(\angle\)QSR = 70°, determine the value of \(\angle\)QPR.

23. The centers of two circles are P and Q; the circles intersect at points A and B. A line parallel to the line segment PQ is drawn through point A, and it intersects the two circles at points C and D. Prove that CD = 2PQ.

24. From a point outside a circle, two tangents can be drawn. The line segments joining the external point to the points of contact of the tangents are equal in length, and they subtend equal angles at the center of the circle.

25. From an external point \(A\), two tangents are drawn to a circle centered at \(O\), touching the circle at points \(B\) and \(C\) respectively. Prove that the line \(AO\) is the perpendicular bisector of the chord \(BC\).

26. From an external point \(A\), two tangents \(AP\) and \(AQ\) are drawn to a circle centered at \(O\), touching the circle at points \(P\) and \(Q\) respectively. If \(PR\) is a diameter of the circle, prove that \(OA \parallel RQ\).

27. We have drawn two circles with centers \(A\) and \(B\), which touch each other externally at point \(C\). A point \(O\) lies on the common tangent at point \(C\), and tangents \(OD\) and \(OE\) are drawn from point \(O\) to the circles centered at \(A\) and \(B\), touching them at points \(D\) and \(E\) respectively. It is given: - \(\angle COD = 56^\circ\) - \(\angle COE = 40^\circ\) - \(\angle ACD = x^\circ\) - \(\angle BCE = y^\circ\) We are to prove: - \(OD = OC = OE\) - \(x - y = 4^\circ\)

28. PQRS is a cyclic quadrilateral, and the sides PQ and SR are extended to meet at point T. The center of the circle is O; if ∠POQ = 110°, ∠QOR = 60°, and ∠ROS = 80°, then calculate and write the values of ∠RQS and ∠QTR.

29. In a circle with center \(O\), \(\bar{AB}\) is a diameter. On the opposite side of the circumference from the diameter \(\bar{AB}\), there are two points \(C\) and \(D\) such that \(\angle AOC = 130°\) and \(\angle BDC = x°\). Find the value of \(x\).

(a) 25° (b) 50° (c) 60° (d) 65°

30. The radius of a circle with center \(O\) is 5 cm. Point \(P\) is located at a distance of 13 cm from \(O\). From point \(P\), two tangents \(PQ\) and \(PR\) are drawn to the circle. Find the area of the quadrilateral \(PQOR\).

(a) \(60\) square cm (b) \(30\) square cm (c) \(120\) square cm (d) \(150\) square cm