1. The radius of a circle is 13 cm, and the length of a chord in the circle is 10 cm. Determine the perpendicular distance from the center to the chord.

(a) 12 cm (b) \(\sqrt{69}\) cm (c) 24 cm (d) 12.5 cm

2. The radius of a circle is 13 cm and the length of a chord is 10 cm. Find the distance from the center of the circle to the chord.

(a) 12.5 cm (b) 12 cm (c) √69 cm (d) 24 cm

3. 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.

4. The radius of a circle is 13 cm. If the length of a chord AB in that circle is 10 cm, find the perpendicular distance from the center of the circle to the chord.

5. 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?

6. 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

7. 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.

8. 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?

9. 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?

10. Point P is an external point to a circle with center O. The distance from point P to the center of the circle is 26 cm, and the length of the tangent drawn from point P to the circle is 10 cm. The radius of the circle is ____ cm.

11. The length of a chord of a circle is 14 cm, and the perpendicular distance from the center to the chord is 24 cm. Find the radius of the circle.

12. The radius of a circle centered at point O is 5 cm, and AB is a chord of length 8 cm. Calculate and write the distance from point O to the chord AB.

13. The radius of a circle with center \(O\) is 5 cm. Point \(P\) is located at a distance of 13 cm from \(O\). From point \(P\), two tangents \(PQ\) and \(PR\) are drawn to the circle. Find the area of the quadrilateral \(PQOR\).

(a) \(60\) square cm (b) \(30\) square cm (c) \(120\) square cm (d) \(150\) square cm

14. The distance from the center of a circle to an external point is 13 cm. The length of the tangent from that point to the circle is 12 cm. The radius of the circle is—

(a) 25 cm (b) 1 cm (c) 5 cm (d) \(\cfrac{13}{12}\) cm

15. 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

16. 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.

17. AB is a chord (not a diameter) of a circle with center O. If the length of chord AB is 8 cm and the radius of the circle is 5 cm, find the perpendicular distance from the center O to the chord AB.

18. If the radius of a circle is \(r\sqrt{2}\) and the perpendicular distance from the center to a chord is \(r\), then what is the length of the chord?

19. 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

20. If the perpendicular distance from the center of a circle with a radius of 17 cm to a chord is 4 cm, what is the length of the chord?

21. For a circle centered at \(O\) with a radius of 5 cm and a chord length of 4 cm, the perpendicular distance from \(O\) to the chord is –?

(a) 3 cm (b) 1 cm (c) 4 cm (d) 2 cm

22. 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.

23. 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.

24. The diameter of a circle centered at O is 26 cm. The distance from point O to the chord PQ is 5 cm. Calculate and write the length of the chord PQ.

25. The length of chord PQ in a circle centered at O is 4 cm, and the distance from point O to PQ is 2.1 cm. Calculate and write the diameter of the circle.

26. 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.

27. I have drawn a circle with center O and radius 6 cm. From a point P located 10 cm away from the center O, a tangent PT is drawn to the circle. Calculate and write the length of the tangent PT.

28. The radius of a circle centered at point O is 5 cm. Point P lies at a distance of 13 cm from point O. From point P, two tangents PQ and PR are drawn to the circle. Find the area of quadrilateral PQOR.

(a) 60 square cm (b) 30 square cm (c) 120 square cm (d) 150 square cm

29. 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

30. 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