1. 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.

2. 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?

3. 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.

4. Two equal circles, each with a radius of 10 cm, intersect each other. The length of their common chord is 12 cm. Find the distance between their centers.

5. Two circles intersect each other. The radius of each circle is 10 cm. The length of the common chord is 16 cm. Find the distance between the centers of the two circles.

6. 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

7. Given two parallel chords AB and CD, each of length 16 cm, and the radius of the circle is 10 cm, what is the distance between the two chords?

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

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

9. 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

10. 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

11. Two circles are touching each other externally, and the distance between their centers is 7 cm. If the radius of one circle is 4 cm, find the radius of the other circle.

12. 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

13. 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

14. 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

15. Two circles touch each other externally. The distance between their centers is 7 cm. If the radius of one circle is 4 cm, then the radius of the other circle is —

(a) 5 cm (b) 4 cm (c) 3 cm (d) 2 cm

16. If a circle has a radius of 10 cm and two parallel chords of lengths 16 cm and 12 cm, what is the perpendicular distance between the two chords?

(a) \(14\sqrt3\) cm (b) 8 cm (c) 2 cm (d) 10 cm

17. 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.

18. 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.

19. 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) \(7\sqrt3\)

20. 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.

21. 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.

22. 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.

23. If two circles with radii \(r_1\) and \(r_2\) touch each other externally, and the distance between their centers is \(d\), then which of the following is correct?

(a) \(r_1+d=r_2\) (b) \(r_2+d=r_1\) (c) \(r_1+r_2=d\) (d) \(r_1-r_2=d\)

24. 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) \(\sqrt3 r^2\)

25. 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.

26. Two parallel chords of a circle with a radius of 5 cm have lengths of 6 cm and 8 cm. What is the distance between the two chords?

27. A vertical hollow cylindrical pipe has an outer radius of 16 cm and an inner radius of 12 cm, with a height of 36 cm. If the pipe is melted and solid cylindrical rods are made from it, each with a diameter of 2 cm and a length of 6 cm, determine how many such solid rods can be produced.

28. 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.

29. 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

30. The radii of two circles are 5 cm and 3 cm, respectively. If the circles touch each other externally, then the distance between their centers will be –

(a) 2 cm (b) 3 cm (c) 8 cm (d) 1.5 cm