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

2. Prove with reasoning that two equal-length chords of a circle are equidistant from the center.

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

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

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

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

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

8. Let us prove logically that two equal chords of a circle are equidistant from the center.

9. Two equal chords of a circle are equidistant from the center.

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

11. Chords of a circle that are equidistant from the center have equal lengths.

12. Prove that from an external point to a circle, the two tangents drawn are equal in length, and the line segments connecting the external point to the points of contact form equal angles at the center.

13. "Chords PQ and RS of a circle intersect each other at point X inside the circle. By joining P to S and R to Q, prove that triangles ∆PXS and ∆RSQ are similar. From this, prove that: PX × XQ = RX × XS Or, when two chords of a circle intersect internally, the product of the two segments of one chord is equal to the product of the two segments of the other chord.

14. Prove that chords of a circle which are equidistant from the center are equal in length.

15. Prove that the two tangents drawn from an external point to a circle are equal in length from the point to the points of contact on the circle.

16. 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?

17. Prove that the two tangents drawn to a circle from an external point are equal in length, and the line segments joining the points of contact to the external point subtend equal angles at the center of the circle.

18. AB and CD are two chords of a circle with center O. OM and ON are perpendiculars drawn from the center to the chords such that OM = ON. Prove that AB = CD.

19. 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

20. O is the center of a circle where AB and CD are two chords of equal length. If the perpendicular distance from O to the chord AB is 4 cm, then the perpendicular distance from O to the chord CD will also be 4 cm.

(a) 2cm (b) 6cm (c) 8cm (d) 4cm

21. AB and AC are two equal chords of a circle. Prove that the bisector of \(\angle\)BAC passes through the center of the circle.

22. If the angle formed between two intersecting chords of a circle is bisected by a line that passes through the center, then prove that the two chords are equal.

23. Mohit has drawn two straight lines from an external point of a circle, which intersect the circle at points A, B and C, D respectively. Using reasoning, prove that two angles of ∆XAC and ∆XBD are equal.

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

25. 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°

26. In a circle with center \(O\), \(\bar{AB}\) is a diameter. On the opposite side of the circumference from the diameter \(\bar{AB}\), there are two points \(C\) and \(D\) such that \(\angle AOC = 130°\) and \(\angle BDC = x°\). Find the value of \(x\).

(a) 25° (b) 50° (c) 60° (d) 65°

27. 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

28. 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?

29. If AB and AC are chords of the larger of two concentric circles, and they touch the smaller circle at points P and Q respectively, prove that: \[ PQ = \frac{1}{2}BC \]

30. If two chords of a circle intersect each other, prove that the product of the segments of one chord is equal to the product of the segments of the other chord.