1. 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?

2. Two tangents are drawn to a circle from points P and Q, and they intersect at point A. If ∠PAQ = 80°, then what is the value of ∠APQ?

3. 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

4. Masum has drawn a circle centered at point O, in which AB is a chord. A tangent is drawn at point B, which intersects the extended line AO at point T. If ∠BAT = 21°, then calculate the measure of ∠BTA.

5. 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 \(\angle QST\)?

6. "AB is a diameter of a circle with center O. P is any point on the circumference. If ∠ POA = 120°, then what is the measure of ∠ PBO?"

(a) 90\(^o\) (b) 60\(^o\) (c) 75\(^o\) (d) None of the above

7. 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

8. 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

9. AB is a diameter of a circle with center O. CD is a chord whose length is equal to the radius of the circle. When AC and BD are extended, they intersect at point P. What is the measure of ∠APB?

10. In a circle centered at O, chords AB and CD intersect at point P. The line segment OP is the bisector of ∠APC. Prove that AB = CD.

11. 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?

12. In a circle centered at O, chords AB and CD are equidistant from the center. If ∠AOB = 60° and CD = 6 cm, then what is the radius of the circle?

13. 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?

14. 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?

15. In the adjacent figure, O is the center of the circle, and BOA is the diameter. A tangent is drawn at point P on the circle, which intersects the extension of BA at point T. If ∠PBO = 30°, then find the measure of ∠PTA.

16. 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.

17. In triangle ABC, the incenter is I. When the internal bisector of ∠A (i.e., AI) is extended, it intersects the circumcircle at point P. If PB = 15 cm, then what is the length of PI?

(a) 5 cm (b) 15 cm (c) 10 cm (d) 20 cm

18. AB is a chord of a circle with center O. A tangent is drawn at point B, which intersects the extended line AO at point T. If ∠BAT = 21°, then find the value of ∠BTA.

19. 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?

20. In the adjacent figure, O is the center of the circle and BOA is the diameter. A tangent is drawn at point P on the circle, which intersects the extended line BA at point T. If \(\angle PBO = 30^\circ\), then what is the measure of \(\angle PTA\)?

21. AB is the diameter of a circle centered at O. C is any point on the circle. Given ∠BAC = 50° and CD is perpendicular to AB, Find the value of ∠BCD.

22. In the cyclic quadrilateral ABCD, side AB is extended up to point P. If ∠CAB = 35° and ∠CBP = 82°, then what is the measure of ∠ADB?

(a) 47° (b) 55° (c) 35° (d) 60°

23. In the adjacent figure, AB is the diameter of a circle centered at O. Point C is any point on the circle. Given that ∠BAC = 50° and CD is perpendicular to AB, find the value of ∠BCD.

24. If I draw a circle centered at point O, and from a point P located 26 cm away from the center a tangent is drawn to the circle which measures 10 cm in length, then calculate and write the length of the radius of the circle.

25. Two circles touch each other externally at point C. AB is a common tangent to both circles, touching them at points A and B respectively. Find the measure of ∠ACB.

(a) 60° (b) 45° (c) 30° (d) 90°

26. 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.

27. 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.

28. In the given figure, O is the center of the circle and AB is the diameter. ABCD is a cyclic quadrilateral. If ∠ADC = 120°, then find the measure of ∠BAC.

(a) 50° (b) 60° (c) 30° (d) 40°

29. AB is a chord of a circle centered at O. The tangent drawn at point B intersects the extended line AO at point T. If \(\angle\)BAT = 25°, then what is the value of \(\angle\)BTA?

30. A circle has a radius of 2.5 cm. Two chords AB = 4 cm and AC = 3 cm are drawn. What is the measure of angle ∠BAC?