1. 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.

2. 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

3. 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.

4. 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.

5. 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

6. 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?

7. 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.

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

9. 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.

10. 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.

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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.

17. 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

18. 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

19. In two concentric circles, a chord of the larger circle is tangent to the smaller circle. If the radii of the two circles are 10 cm and 4 cm respectively, then what is the length of the chord?

(a) 8 cm (b) 9 cm (c) 11 cm (d) 12 cm

20. 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

21. 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.

22. 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.

23. 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) \(r_1-r_2\lt d\)

24. 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.

25. 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?

26. 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

27. 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

28. 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

29. 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

30. 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.