1. 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?

2. In a circle with center O, AB and CD are two equal chords. If \(\angle\)AOB = 60° and CD = 10 cm, then what is the length of OD?

3. In a circle centered at \(O\), two chords \(AB\) and \(CD\) have equal lengths. Given that \(\angle AOB = 60°\) and the radius of the circle is \(6\) cm, find the area of \(\triangle COD\).

(a) \(9\sqrt3\) square c.m (b) \(6\sqrt3\) square c.m (c) \(2\sqrt3\) square c.m (d) \(3\sqrt3\) square c.m

4. 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?

5. AB and CD are two chords of a circle with center O, and both have equal lengths. If \(\angle\)AOB = 60°, then what is the measure of \(\angle\)COD?

6. 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

7. In a circle centered at O, the chords AB and CD are of equal length. If \(\angle\)AOB = 60° and CD = 6 cm, determine the radius of the circle.

8. In a circle centered at \(O\), two equal chords \(AB\) and \(CD\) are given. If \(\angle AOB = 60^\circ\) and the radius of the circle is 6 cm, determine the area of \( \triangle COD \).

(a) \(6\sqrt3\) square cm (b) \(2\sqrt3\) square cm (c) \(2\sqrt3\) square cm (d) \(9\sqrt3\) square cm

9. In a circle centered at O, the chords AB and AC are perpendicular to each other. If AB = 6 cm and AC = 8 cm, then the radius of the circle will be -.

(a) 7 cm (b) 10 cm (c) 5 cm (d) 4 cm

10. In a circle centered at O, the chords AB and CD are of equal length. If \(\angle\)AOB = 60° and CD = 6 cm, find the radius of the circle.

11. In a circle centered at O, the chords AB and CD have equal lengths. If \(\angle\)AOB = 60°, then the value of \(\angle\)COD is -

(a) 60° (b) 30° (c) 120° (d) 90°

12. 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

13. 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

14. In triangle △ABC, if ∠ABC = 90°, AB = 5 cm, and BC = 12 cm, then what is the radius of its circumcircle?

15. 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?

16. In a circle with a radius of 5 cm, AB and AC are two equal chords. The center of the circle is located outside the triangle ABC. If AB = AC = 6 cm, determine the length of the chord BC.

17. In a circle centered at O, the chords AB and CD are of equal length. If \(\angle\)AOB = 60°, then determine the value of \(\angle\)COD.

(a) 40° (b) 30° (c) 60° (d) 90°

18. In a circle centered at \( O \), the chords \( AB \) and \( AC \) are perpendicular to each other. If \( AB = 8 \) cm and \( AC = 6 \) cm, determine the radius of the circle.

(a) 7 cm (b) 10 cm (c) 5 cm (d) 9 cm

19. In a circle centered at O, the chords AB and CD are equal in length. If \( \angle \)AOB = 60°, then find the value of \( \angle \)COD.

(a) 30° (b) 40° (c) 120° (d) 60°

20. A circle has center ‘O’, and a point P lies 26 cm away from it. If the length of the tangent drawn from point P to the circle is 10 cm, then what is the radius of the circle?

21. In a circle centered at O, there are two parallel chords AB and CD with lengths 10 cm and 24 cm, positioned on opposite sides of the center. If the distance between the chords AB and CD is 17 cm, then calculate and write the radius of the circle.

22. In a circle centered at O, the chords AB and CD are equal in length. If \(\angle\)AOB = 60°, then find the measure of \(\angle\)COD.

(a) 40° (b) 30° (c) 60° (d) 90°

23. 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.

24. 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?

25. Point P is any point inside a circle centered at O. If the radius of the circle is 5 cm and OP = 3 cm, then find the minimum length of the chord passing through point P.

26. 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?

27. 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?

28. 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?

29. In a circle, AB and AC are two mutually perpendicular chords. If AB = 4 cm and AC = 3 cm, then the radius of the circle is _____ cm.

30. From an external point \(P\), two tangents \(PA\) and \(PB\) are drawn to a circle centered at \(O\). Given \(PA = 9\) cm and \(\angle APB = 60°\), find the length of \(AB\).