1. O is the center of a circle. PQ is a diameter, and R is a point on the circumference. If \(\angle\)PQR = 40°, then what is the measure of \(\angle\)POR? (a) 80° (b) 40° (c) 20° (d) 100°

2. A chord of length 4 2 meters subtends a right angle at the center of a circle. What is the radius of the circle? (a) \(4\sqrt2\) meters (b) \(2\sqrt2\) meters (c) \(2\) meters (d) \(4\) meters

3. In a circle with center O, chords AB and CD intersect each other at point P. Given: PA = 2 cm, PB = 10 cm, and PC = 5 cm Find: The length of PD. (a) 6 cm (b) 8 cm (c) 3 cm (d) 4 cm

4. Given two parallel chords AB and CD, each of length 16 cm, and the radius of the circle is 10 cm, what is the distance between the two chords? (a) 12 cm (b) 16 cm (c) 20 cm (d) 5 cm

5. In the circle with center \( O \), if \( AOB \) is a diameter and point \( C \) is on the circle such that \( AC = 3 \) cm and \( BC = 4 \) cm, what is the length of \( AB \)? (a) 3 cm (b) 4 cm (c) 5 cm (d) 8 cm

6. In a circle with center \( O \), \( AB \) and \( CD \) are two equal-length chords. \( E \) is the midpoint of \( CD \), and \( \angle AOB = 70^\circ \). The value of angle \( \angle COE \) is: (a) 70° (b) 110° (c) 35° (d) 55°

7. In the center circle O, AB is a diameter. C is any point on the circumference of the circle where AC = 3 cm and BC = 4 cm. Find the length of AB. (a) 3 cm (b) 4 cm (c) 5 cm (d) 7 cm

8. In a circle with center O, a tangent PT is drawn from an external point P to the circle, with T being the point of tangency. If PT = 12 cm and OP = 13 cm, the diameter of the circle will be: (a) 5 cm (b) 8 cm (c) 6 cm (d) 10 cm

9. Two concentric circles have radii of 13 cm and 15 cm, respectively. A chord AB of the larger circle intersects the smaller circle at points P and Q. If PQ = 10 cm, then AB will be: (a) 28 cm (b) 20 cm (c) 18 cm (d) 16 cm

10. A tangent is drawn from an external point A to a circle centered at O, touching the circle at point B. Given: OB=5 cm, AO=13 cm, Find the length of AB. (a) 12 cm (b) 13 cm (c) 6.5 cm (d) 6 cm

11. "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

12. Two circles touch each other externally at point C. AB is a common tangent to both circles and touches the circles at points A and B, respectively. Find the measure of \(\angle\)ACB. (a) 60° (b) 45° (c) 30° (d) 90°

13. No of tangent can be drawn to a circle from an external point- (a) 1 (b) 3 (c) 4 (d) 2

14. 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°

15. How many non-collinear points are required at minimum to draw a unique circle? (a) two (b) three (c) one (d) None of the above

16. A circle with center O has a radius of 10 cm. PQ is a chord with a length of 16 cm. The length of the perpendicular drawn from O to PQ is — (a) 8 cm (b) 10 cm (c) 16 cm (d) 6 cm

17. Two circles intersect at points A and B, and the circumference of each passes through the center of the other. If a straight line drawn through point A intersects the two circles again at points C and D respectively, then what type of triangle is BCD? (a) right-angled isosceles (b) scalene (c) isosceles (d) equilateral

18. A circle with a radius of 10 cm has two parallel chords of lengths 4 cm and 6 cm. What is the distance between the two chords? (a) \(\sqrt7(\sqrt3-\sqrt{12})\) cm (b) \(\sqrt3(\sqrt7+\sqrt{12})\) cm (c) \(\sqrt{13}(\sqrt{12}-\sqrt{7})\) cm (d) \(\sqrt{7}(\sqrt{13}+\sqrt{12})\) cm

19. QR is a chord of the circle, and POR is a diameter of the circle. OD is perpendicular to the chord QR. If OD = 4 cm, then the length of PQ is – (a) 4 cm (b) 2 cm (c) 8 cm (d) None of these

20. A chord is drawn in a circle with a radius of 5 cm, at a distance of 3 cm from the center. What will be the length of that chord? (a) 8 cm (b) 2 cm (c) 5 cm (d) 3 cm

21. A chord of length \(4\sqrt{2}\) meters subtends a right angle at the center of a circle. What is the radius of the circle? (a) \(4\sqrt2\) \(4\sqrt2\) meters (b) \(8\) meters (c) \(4\) meters (d) \(8\sqrt2\) meters

22. Two chords, \(AB\) and \(CD\), of a circle with center \(O\), intersect at point \(P\). If \(\angle APC = 40°\), find the value of \(\angle AOC + \angle BOD\). (a) 60° (b) 80° (c) 120° (d) None of these

23. The radius of a circle is 13 cm and the length of a chord is 10 cm. What is the distance from the center of the circle to the chord? (a) 12.5 cm (b) 12 cm (c) \(\sqrt{69}\) cm (d) \(\sqrt{24}\) cm

24. In two concentric circles, a chord of the larger circle is tangent to the smaller circle. If the radii of the two circles are 10 cm and 4 cm respectively, then what is the length of the chord? (a) 8 cm (b) 9 cm (c) 11 cm (d) 12 cm

25. Two circles with centers A and B have a common tangent PO of length 16 cm, and the length of PB is 10 cm; the area of triangle \(\triangle\)POB will be— (a) 24 square cm (b) 48 square cm (c) 16 square cm (d) 12 square cm

26. A chord of length \(4\sqrt{2}\) meters subtends a right angle at the center of a circle. What is the radius of the circle? (a) \(4\sqrt2\) meters (b) \(8\) meters (c) \(4\) meters (d) \(8\sqrt2\)meters

27. A ও B কেন্দ্রবিশিষ্ট দুটি বৃত্তের PO সাধারণ জ্যার দৈর্ঘ্য 16 সেমি. এবং PB-এর দৈর্ঘ্য 10 সেমি.; \(\triangle\)POB-এর ক্ষেত্রফল হবে - translate in english (a) 48 square cm (b) 20square cm (c) 16 square cm (d)

28. If AB and CD are chords of a circle centered at O and their lengths are equal, then if \(\angle\)AOB = 60°, the measure of \(\angle\)COD will be—? (a) 30° (b) 60° (c) 120° (d) 180°

29. If a circle has a radius of 10 cm and two parallel chords of lengths 16 cm and 12 cm, what is the perpendicular distance between the two chords? (a) \(14\sqrt3\) cm (b) 8 cm (c) 2 cm (d) 10 cm

30. A chord is drawn at a distance of 5 cm from the center of a circle with a radius of 10 cm. What is the length of the chord? (a) \(5\sqrt3\) cm (b) 6 cm (c) 8 cm (d) 3 cm

31. Only one circle can be drawn through three non-collinear points. True / False

32. 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.

33. A circle is centered at point O with a radius of 10 cm. A perpendicular is drawn from O to a chord AB, and the length of this perpendicular is 6 cm. What is the length of the chord AB?

34. If two chords of a circle are equidistant from the center, they must be parallel.

35. In a circle with center O, AB is a diameter. From a point P on the circle, a perpendicular PN is drawn to AB. Prove geometrically that \[ PB^2 = AB \times BN \]

36. 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?

37. A straight line intersects one of two concentric circles at points A and B, and the other at points C and D. Prove that AC = BD.

38. If a circle is centered at point 'O' with a radius of 13 cm and chord AB has a length of 10 cm, what is the distance from point 'O' to chord AB?

39. Two equal circles, each with a radius of 10 cm, intersect, and the length of their common chord is 12 cm. Determine the distance between the centers of the circles.

40. Two circles intersect each other. The radius of each circle is 10 cm. The length of the common chord is 16 cm. Find the distance between the centers of the two circles.

41. Two parallel chords of a circle with a radius of 5 cm have lengths of 6 cm and 8 cm. What is the distance between the two chords?

42. AB and CD are two chords of a circle. When extended, BA and DC intersect at point P. Prove that \(\angle\)PCB = \(\angle\)PAD.

43. 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?

44. Two identical circles, each with a radius of 10 cm, intersect, and the length of their common chord is 12 cm. Determine the distance between the centers of the two circles.

45. Prove that if a chord (which is not a diameter) of a circle is bisected by a straight line drawn from the center of the circle, then that line is perpendicular to the chord.

46. Two chords AB and CD of a circle, centered at O, intersect each other internally at point P. Prove that \(\angle\)AOD + \(\angle\)BOC = 2\(\angle\)BPC.

47. Two identical circles, each with a radius of 10 cm, intersect each other, and the length of their common chord is 12 cm. Find the distance between the centers of the two circles.

48. Prove that if a perpendicular is drawn from the center of a circle to a chord that is not a diameter, then it bisects the chord.

49. Prove that two equal chords of a circle are equidistant from the center.

50. Two parallel chords AB and CD of lengths 10 cm and 24 cm respectively lie on opposite sides of the center O of a circle. If the distance between the chords AB and CD is 17 cm, find the radius of the circle.