1. Prove that the angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference.

2. Prove that the angle subtended at the center by an arc of a circle is twice the angle subtended by the same arc at any point on the circle.

3. Prove that the angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circumference.

4. Prove that the central angle subtended by an arc of a circle is twice any inscribed angle subtended by the same arc.

5. Prove that the central angle subtended by an arc of a circle is twice any inscribed angle subtended by the same arc.

6. Prove that the central angle subtended by an arc of a circle is twice any inscribed angle subtended by the same arc.

7. If the central angle of a circle centered at point O is 300°, then the angle subtended at the circumference by the same arc will be 60°.

8. Prove that the central angle subtended by an arc of a circle is twice the inscribed angle subtended by the same arc.

9. Prove that the central angle subtended by an arc of a circle is twice the inscribed angle subtended by the same arc.

10. If the circle drawn by Shubho has a radius of 7 cm, then calculate and write the radian measure of the central angle subtended by an arc of length 5.5 cm on that circle.

11. Let O be the center of a circle, and let AOB be a diameter. Let C be any point on the circle. Now, by joining A to C, B to C, and O to C, show that \[ \tan \angle ABC = \cot \angle ACO \]

12. Let O be the center of any circle, and let AOB be a diameter. Let C be any point on the circumference of the circle. Now, by joining A to C, B to C, and O to C, show that \[ \sin^2 \angle BCO + \sin^2 \angle ACO = 1 \]

13. Let O be the center of any circle, and let AOB be a diameter. Let C be any point on the circumference of the circle. Now, by joining A to C, B to C, and O to C, show that \[ \csc^2 \angle CAB - 1 = \tan^2 \angle ABC \]

14. AB is the diameter of a circle with center O. A straight line drawn from point A intersects the circle at point C, and the tangent drawn from point B intersects this line at point D. Prove that: For any such straight line, the area of the rectangle formed by AC and AD is always constant.

15. Prove that all angles subtended by the same arc of a circle are equal.

16. 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?

17. 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.

18. Prove that all angles subtended by the same arc at the circumference of a circle are equal.

19. An inscribed angle subtended by the same arc is half of the central angle subtended by that arc.

20. In a circle centered at O, AB is a diameter, and C is any point on the circle. Given that \(\angle\)OAC = 45°, determine the measure of \(\angle\)OCB.

(a) 90° (b) 45° (c) 30° (d) 60°

21. The radius of a circle is 28 cm. What is the circular measure of the central angle subtended by an arc of length 5.5 cm in that circle?

22. If O is the circumcenter of an equilateral triangle, then the central angle subtended by any one of its sides is _____________.

23. The central angle subtended by an arc of a circle is twice any inscribed angle subtended by the same arc.

24. If the radius of a circle is 6 cm, calculate and write the radian measure of the central angle subtended at the center by an arc of length 15 cm.

25. 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 \(PQ \cdot PR = r^2\).

26. The angle subtended at the center of a circle by equal arcs is ------

27. The angle formed between the two tangents drawn from an external point to a circle is bisected by the straight line segment connecting that point to the center of the circle.

28. ABC is an equilateral triangle inscribed in a circle. If P is any point on the arc BC opposite to vertex A, then prove that \(AP = BP + CP\).

29. ABC is an equilateral triangle inscribed in a circle. Point P is any point on the arc BC (excluding B and C). Prove that: PA = PB + PC.

30. If the radius of a circle is 7 cm, then what will be the radian measure of the central angle subtended by an arc of length 5.5 cm?