1. 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.

2. 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.

3. 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?

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. 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

6. 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

7. 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

8. 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

9. 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

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

11. 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.

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

13. 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

14. 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.

15. 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.

16. 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

17. 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

18. 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.

19. 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

20. 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.

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

22. 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 \(C\).

23. A right-angled triangle where the hypotenuse is 12 cm and one of the other sides is 5 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

24. A right-angled triangle in which the hypotenuse is 9 cm and one of the other sides is 5.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

25. 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

26. If two circles do not touch or intersect each other, the number of common tangents between them is—

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

27. If two circles do not touch or intersect each other, the number of common tangents between them is—?

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

28. 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.

29. 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.

30. 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.