1. 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

2. What is the length of a chord of a circle with a radius of 10 cm, if the chord subtends a right angle at the center?

3. In a circle with a radius of 10 cm, if a chord subtends a right angle at the center, then the length of the chord will be 5 cm.

4. A chord 18 cm long in a circle subtends a right angle (90°) at the center of the circle. What is the perpendicular distance from the center to the chord?

(a) \(6\sqrt{2}\) cm (b) 6 cm (c) 9 cm (d) \(9\sqrt{2}\) cm

5. 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?

6. Given: In a circle with center O, MN is any chord that is not a diameter, and AC is a diameter. To Prove: AC > MN, i.e., the diameter is the longest chord of a circle. Construction: From the center O, draw a perpendicular OD to the chord MN. Join O and M. Proof: OM > MD [∵ Triangle OMD is a right-angled triangle and OM is the hypotenuse] Or, OA > MD [∵ OA = OM, both are radii of the same circle] Or, ½AC > ½MN Or, AC > MN Therefore, the diameter is the longest chord of a circle.

7. 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

8. 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.

9. 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.

10. 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.

11. 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.

12. Niyamat has drawn a circle with a radius of 13 cm. I have drawn a chord AB of length 10 cm in this circle. Calculate and write the distance from the center of the circle to this chord AB.

13. In the adjacent figure, O is the center of the circle and AB is the diameter. The length of chord CD is equal to the radius of the circle. AC and BD are extended to meet at point P. Find the value of ∠APB.

14. A horse runs along a circular path with a radius of 21 meters. In 5 seconds, it covers an arc that subtends a 60° angle at the center of the circle. Find the speed of the horse.

15. 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°

16. From a point outside a circle, a tangent of length 8 cm is drawn to the circle. If the radius of the circle is 6 cm, what is the distance from the center of the circle to the external point?

(a) 10 cm (b) 8 cm (c) 6 cm (d) 14 cm

17. If the length of the median drawn from the right-angled vertex of a right-angled triangle is 5 cm, what is the area of the triangle’s circumcircle?

(a) 78\(\frac78\(\frac{2}{7}\) square cm{2}{7}\ (b) 78\(\frac{3}{7}\) square cm (c) 78\(\frac{5}{7}\) square cm (d) 78\(\frac{4}{7}\) suare cm

18. 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

19. 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

20. "If the length of the median drawn from the right angle vertex of a right-angled triangle is 10 cm, what is the area of the triangle's circumcircle?"

(a) \(100\pi\) (b) \(25\pi\) (c) \(50\pi\) (d) \(120\pi\)

21. 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

22. From point A, which is located 26 cm away from the center O of a circle, a tangent is drawn to the circle with a length of 10 cm. Find the radius of the circle.

23. 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.

24. 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.

25. The radius of a circle is 28 cm. What is the radian measure of the central angle subtended by an arc of length 5.5 cm on this circle?

26. 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?

27. A circle has center O and a radius of 5 cm. AB is a chord of the circle with a length of 8 cm. Calculate and write the distance from point O to the chord AB.

28. 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?

29. 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?

30. Two identical circles, each with a radius of 13 cm, intersect each other, and the length of their common chord is 10 cm. What is the distance between the centers of the two circles?