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

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

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

4. 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

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

9. Prove that 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.

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

12. 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.

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

14. From a point outside a circle, two tangents can be drawn. The line segments joining the external point to the points of contact of the tangents are equal in length, and they subtend equal angles at the center of the circle.

15. Two identical circles, each with radius \(r\), intersect in such a way that each circle passes through the center of the other. The centers of the circles are labeled A and B, and they intersect at points P and Q. The area of triangle \(\triangle APB\) will be:

(a) \(\cfrac{\sqrt3}{4}r^2\) (b) \(\cfrac{\sqrt3}{2}r^2\) (c) \(\cfrac{\sqrt3}{3}r^2\) (d) \(\sqrt3 r^2\)

16. 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

17. 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?

18. 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?

19. 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: \[ PQ = \frac{1}{2}BC \]

20. 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.

21. 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?

22. 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?

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

24. If the lengths of two equal chords are _____ cm and the distance between them is 4 cm, then the diameter of the circle will be 10 cm.

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

26. Construct a right-angled triangle in which the two arms adjacent to the right angle are 7 cm and 9 cm. Draw an incircle (inscribed circle) of that triangle and measure its radius. (Each construction step must be marked.)

27. AB and CD are two chords of a circle with center O. When extended, they intersect at point P. Prove that \(\angle AOC - \angle BOD = 2\angle BPC\).

28. In a circle centered at O, there are two chords of lengths 6 cm and 8 cm. If the distance from the center to the shorter chord is 4 cm, calculate and write the distance from the center to the other chord.

29. 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.

30. AB and CD are two parallel chords, each of length 16 cm. If the radius of the circle is 10 cm, find the distance between the two chords.

(a) 12 cm (b) 16 cm (c) 20 cm (d) 5 cm