1. If the chord of the larger circle of two concentric circles with radii 3 cm and 5 cm is a tangent to the smaller circle, what is its length?

(a) 4 cm (b) 6 cm (c) 8 cm (d) 12 cm

2. The radii of two concentric circles are 8 cm and 3 cm, respectively. If a chord of the larger circle is tangent to the smaller circle, determine the length of the chord.

3. The radii of two circles are 4 cm and 3 cm respectively, and the distance between their centers is 13 cm. Find the length of a direct common tangent to the two circles.

4. If two identical circles with a radius of 4 cm are externally tangent to each other, what is the length of their common chord?

5. AB and AC are two tangents drawn from point A to a circle with center O. The line OA intersects the chord BC (which joins the points of contact) at point M. If AM = 8 cm and BC = 12 cm, then what is the length of each tangent?

(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm

6. The radii of two circles are 4 cm and 3 cm respectively, and the distance between their centers is 13 cm. Find the length of a direct common tangent between the two circles.

7. 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.

8. Two chords of a circle have their lengths in the ratio 4:3. If their distances from the center are 9 cm and 12 cm respectively, what is the length of the chord closer to the center?

9. Two concentric circles have radii of 13 cm and 15 cm, respectively. A chord AB of the larger circle intersects the smaller circle at points P and Q. If PQ = 10 cm, then AB will be:

(a) 28 cm (b) 20 cm (c) 18 cm (d) 16 cm

10. 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?

11. In a circle with center O, AB and CD are two equal chords. If \(\angle\)AOB = 60° and CD = 10 cm, then what is the length of OD?

12. 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.

13. 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

14. The distance between the centers of two circles is 4 cm, and the circles are externally tangent to each other. If the radius of the first circle is 5 cm, determine the radius of the second circle.

(a) 13 cm (b) 6.5 cm (c) 3 cm (d) none of the above

15. If two circles with radii of 10 cm and 7 cm are internally tangent to each other, the distance between their centers will be—

(a) 4 cm (b) 6 cm (c) 17 cm (d) 3 cm

16. The radii of two circles are 8 cm and 3 cm, respectively, and the distance between their centers is 13 cm. Find the length of a common external tangent of the circles.

(a) 10 cm (b) 14 cm (c) 15 cm (d) 12 cm

17. The radii of two circles are 8 cm and 3 cm respectively, and the distance between their centers is 1.3 cm. Find the length of a direct common tangent between the two circles.

18. If the radii of two circles are 3 cm and 11 cm, and the distance between their centers is 17 cm, find the length of a direct common tangent to the circles.

19. The radii of two circles are 5 cm and 3 cm. If the two circles are externally tangent to each other, the distance between their centers will be -

(a) 2 cm (b) 2.5 cm (c) 1.5 cm (d) 8 cm

20. 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

21. Two circles with centers A and B have a common tangent PO of length 16 cm, and the length of PB is 10 cm; the area of triangle \(\triangle\)POB will be—

(a) 24 square cm (b) 48 square cm (c) 16 square cm (d) 12 square cm

22. Two solid spheres with radii of 8 cm and 10 cm are melted to form a right circular cone with a height of 42 cm. What is the radius of the cone?

(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm

23. 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.

24. 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.

25. If two circles with radii of 7 cm and 3 cm touch externally, then the distance between their centers will be 4 cm.

26. The radius of a circle centered at point O is 5 cm. Point P is located 13 cm away from point O. PQ and PR are two tangents drawn from point P to the circle. What is the area of quadrilateral PQOR?

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

28. If two identical circles with a radius of 4 cm intersect such that their centers lie on a straight line, what is the length of their common chord?

(a) \(4\sqrt3\) cm (b) \(5\sqrt3\) cm (c) \(6\sqrt3\) cm (d) \(7\sqrt3\)

29. From an external point P, two tangents PA and PB are drawn to a circle centered at O. If PA = 9 cm and ∠APB = 60°, then what are the measures of ∠PAB and the length of chord AB?

(a) 9 cm (b) 3 cm (c) 6 cm (d) 12 cm

30. 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.