1. The radii of two circles are 3.5 cm and 2 cm. The circles touch each other internally. The distance between the centers of the two circles will be –

(a) 5.5 cm (b) 1 cm (c) 1.5 cm (d) None of the above

2. 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

3. The radii of two circles are 5 cm and 3 cm respectively. The two circles touch each other externally. Find the distance between the centers of the two circles.

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

4. The radii of two circles are 3.5 cm and 2 cm respectively. The two circles touch each other internally. Find the distance between the centers of the two circles.

(a) 5.5 cm (b) 1 cm (c) 1.5 cm (d) None of the above

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

6. 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.

7. The radii of two circles are 5.5 cm and 2 cm. The circles touch each other internally. The distance between the centers of the two circles is —

(a) 3.5 cm (b) 2.5 cm (c) 1.5 cm (d) 7.5 cm

8. 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) \(r_1-r_2=d\)

9. Two circles touch each other externally. The diameters of the circles are 7 cm and 6 cm respectively. What is the distance between their centers?

(a) 6 cm (b) 13 cm (c) 12.5 cm (d) 6.5 cm

10. 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

11. If two circles with diameters of 5 cm and 7 cm touch each other internally, then the distance between their centers is —

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

12. 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

13. 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?

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. 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

16. 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.

17. 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

18. 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.

19. 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.

20. 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.

21. If two circles with radii \(R\) and \(r\) are externally tangent to each other, then the distance between their centers is \(R + r\).

(a) \(R+r\) (b) \(R-r\) (c) 0 (d) none of the above

22. The radii of two circles are \(r_1\) and \(r_2\) units, respectively. The distance between their centers is \(d\) units. If the circles are externally tangent, which of the following will be correct?

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

23. 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.

24. 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.

25. 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.

26. 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.

27. If the ratio of the radii of two vertical solid circular cylinders is 2:3, and the ratio of their heights is 5:3, then the ratio of their curved surface areas will be –

(a) 2:5 (b) 8:7 (c) 10:9 (d) 16:9

28. If two arms are externally tangent to each other, the distance between their centers will be the sum of their radii.

29. Three circles with centers \(A\), \(B\), and \(C\) externally touch each other. If \(AB = 5\) cm, \(BC = 7\) cm, and \(CA = 6\) cm, then find the radius of the circle centered at \(C\).

30. In the adjacent figure, three circles with centers A, B, and C touch each other externally. If AB = 5 cm, BC = 7 cm, and CA = 6 cm, then find the radius of the circle centered at A.