1. 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) \(r_1-r_2\lt d\)
2. 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
3. If two circles with radii of 7 cm and 3 cm touch externally, then the distance between their centers will be 4 cm.
4. 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
5. 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
6. 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
7. 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.
8. 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
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. 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
11. 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.
12. 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
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 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.
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.
16. 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
17. 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.
18. 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.
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. 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
22. 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
23. 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.
24. We have drawn two circles with centers \(A\) and \(B\), which touch each other externally at point \(C\). A point \(O\) lies on the common tangent at point \(C\), and tangents \(OD\) and \(OE\) are drawn from point \(O\) to the circles centered at \(A\) and \(B\), touching them at points \(D\) and \(E\) respectively. It is given: - \(\angle COD = 56^\circ\) - \(\angle COE = 40^\circ\) - \(\angle ACD = x^\circ\) - \(\angle BCE = y^\circ\) We are to prove: - \(OD = OC = OE\) - \(x - y = 4^\circ\)
25. Two fixed circles with centers A and B touch each other internally. Another circle touches the larger circle internally at point X and the smaller circle externally at point Y. If O is the center of this third circle, prove that AO + BO is constant.
26. I have drawn two circles with centers A and B that externally touch each other at point O. A straight line is drawn through point O, which intersects the two circles at points P and Q respectively. Prove that AP is parallel to BQ.
27. 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.
28. 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.
29. Match the items on the left with those on the right (any two):
