1. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB? (a) 145° (b) 55° (c) 110° (d) None of the above

2. If the chord of the larger circle of two concentric circles with radii 3 cm and 5 cm is a tangent to the smaller circle, what is its length? (a) 4 cm (b) 6 cm (c) 8 cm (d) 12 cm

3. 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

4. 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

5. AB is a chord of a circle centered at O, and PT is a tangent to the circle at point A. If ∠ AOB = 120°, then what is the measure of ∠ BAT? (a) 60° (b) 30° (c) 90° (d) 45°

6. If PQ is a tangent to a circle centered at O and touches the circle at point C, then what is the value of \(\angle\)BCP? (a) 90\(^o\) (b) 80\(^o\) (c) 70\(^o\) (d) None of the above

7. From a point outside a circle, a tangent of length 8 cm is drawn to the circle. If the radius of the circle is 6 cm, what is the distance from the center of the circle to the external point? (a) 10 cm (b) 8 cm (c) 6 cm (d) 14 cm

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 at point \(C\). \(AB\) is a common tangent to the two circles, touching them at points \(A\) and \(B\), respectively. The measurement of \(\angle ACB\) is – (a) 60° (b) 45° (c) 30° (d) 90°

11. 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

12. If two circles touch each other internally, the number of common tangents will be – (a) 1 (b) 2 (c) 3 (d) 4

13. The distance from the center of a circle to an external point is 13 cm. The length of the tangent from that point to the circle is 12 cm. The radius of the circle is— (a) 25 cm (b) 1 cm (c) 5 cm (d) \(\cfrac{13}{12}\) 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. 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

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. 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

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

19. 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

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

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

22. Two concentric circles will have only one common tangent. True / False

23. Only one tangent can be drawn to a circle from any external point. True / False

24. In a circle with center O, PQ and PR are two chords. Tangents drawn at points Q and P intersect at point S. If ∠QSR = 70°, then what is the measure of ∠QPR?

25. Prove that the two tangents drawn to a circle from an external point are equal in length, and the line segments joining the points of contact to the external point subtend equal angles at the center of the circle.

26. Prove that the two tangents drawn from an external point to a circle are equal in length from the point to the points of contact on the circle.

27. If two circles lying in the same plane have 3 common tangents, then the circles will _____.

28. From an external point \(P\), two tangents \(PS\) and \(PT\) are drawn to a circle with center \(O\). \(QS\) is a chord of the circle that is parallel to \(PT\). If \(\angle SPT = 80^\circ\), then what is the measure of \(\angle QST\)?

29. The radius of a circle centered at point O is 5 cm. Point P is located 13 cm away from point O. PQ and PR are two tangents drawn from point P to the circle. What is the area of quadrilateral PQOR?

30. If two circles touch each other externally, prove that the point of contact lies on the straight line joining their centers.

31. Prove that the tangent to a circle and the radius drawn to the point of contact are perpendicular to each other.

32. Prove that the tangent to a circle and the radius drawn to the point of contact are perpendicular to each other.

33. What is the length of the tangent drawn from an external point located 17 cm away from the center of a circle with a radius of 4 cm?

34. A circle has center ‘O’, and a point P lies 26 cm away from it. If the length of the tangent drawn from point P to the circle is 10 cm, then what is the radius of the circle?

35. Prove that if two circles touch each other externally, then the point of contact lies on the straight line joining their centers.

36. If ABCD is a cyclic quadrilateral inscribed in a circle with center O, then prove that: \[ AB + CD = BC + DA \]

37. Two circles touch each other externally at point C. A common tangent AB touches the circles at points A and B respectively. Find the value of \(\angle ACB\).

38. Prove that two tangents can be drawn from any external point to a given circle.

39. AB is a chord of a circle with center O. A tangent is drawn at point B, which intersects the extended line AO at point T. If ∠BAT = 21°, then find the value of ∠BTA.

40. AB and AC are tangents to a circle with center O. Prove that AO bisects the angle between the tangents at the point of contact, forming a right angle.

41. From point A, which is located 26 cm away from the center O of a circle, a tangent is drawn to the circle with a length of 10 cm. Find the radius of the circle.

42. From an external point located 17 cm away from the center of a circle with a diameter of 16 cm, what is the length of the tangent drawn to the circle?

43. Prove that the centers of three equal circles, each touching the other, form the vertices of an equilateral triangle.

44. If AB and AC are chords of the larger of two concentric circles, and they touch the smaller circle at points P and Q respectively, prove that: \[ PQ = \frac{1}{2}BC \]

45. Prove that from an external point to a circle, the two tangents drawn are equal in length, and the line segments connecting the external point to the points of contact form equal angles at the center.

46. If two circles touch each other, then the point of contact lies on the straight line joining their centers — prove it.

47. 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.

48. Two tangents are drawn to a circle from points P and Q, and they intersect at point A. If ∠PAQ = 80°, then what is the value of ∠APQ?