Circles

1. How many tangents can a circle have?
  a) 1
  b) 2
  c) Infinite
  d) None
  Answer: c) Infinite
2. What is the point where a tangent touches a circle called?
  a) Radius
  b) Chord
  c) Diameter
  d) Point of contact
  Answer: d) Point of contact
3. What is the line called that intersects a circle at exactly one point?
  a) Secant
  b) Tangent
  c) Diameter
  d) Radius
  Answer: b) Tangent
4. How many tangents can be drawn through a point inside a circle?
  a) 0
  b) 1
  c) 2
  d) Infinite
  Answer: a) 0
5. The tangent to a circle is perpendicular to which of the following?
  a) Radius at the point of contact
  b) Diameter
  c) Chord
  d) Secant
  Answer: a) Radius at the point of contact
6. If a tangent to a circle is drawn from a point outside the circle, how many tangents can be drawn?
  a) 1
  b) 2
  c) Infinite
  d) None
  Answer: b) 2
7. What is the distance between the center of a circle and a tangent to it?
  a) Equal to the radius
  b) Less than the radius
  c) Greater than the radius
  d) Equal to the diameter
  Answer: a) Equal to the radius
8. If a chord of a circle is bisected by a radius perpendicular to it, what can we conclude about the chord?
  a) It is a tangent
  b) It is the longest chord
  c) It is not related to the radius
  d) It is parallel to the tangent
  Answer: b) It is the longest chord
9. The length of the tangent from a point to a circle is 8 cm, and the distance of the point from the circle’s center is 10 cm. What is the radius of the circle?
  a) 4 cm
  b) 6 cm
  c) 8 cm
  d) 10 cm
  Answer: b) 6 cm
10. A circle has a radius of 5 cm. How many tangents can be drawn parallel to a given secant of the circle?
  a) 0
  b) 1
  c) 2
  d) Infinite
  Answer: c) 2
11. The lengths of tangents drawn from an external point to a circle are always:
  a) Unequal
  b) Equal
  c) Proportional to the radius
  d) Proportional to the diameter
  Answer: b) Equal
12. Prove that the tangents drawn at the ends of a diameter of a circle are parallel. Which theorem does this use?
  a) Tangent-radius theorem
  b) Tangent-length theorem
  c) Angle-bisector theorem
  d) Chord theorem
  Answer: a) Tangent-radius theorem
13. Two concentric circles have radii 5 cm and 3 cm. Find the length of a chord of the larger circle that touches the smaller circle.
  a) 4 cm
  b) 6 cm
  c) 8 cm
  d) 10 cm
  Answer: c) 8 cm
14. If two tangents are drawn from an external point to a circle, and the angle between them is 70°, what is the angle subtended at the center by the line joining the points of contact?
  a) 50°
  b) 60°
  c) 70°
  d) 80°
  Answer: a) 50°
15. A triangle circumscribes a circle of radius 4 cm. Two segments of the base are 6 cm and 8 cm. Find the remaining sides if the total perimeter is 36 cm.
  a) 10 cm, 12 cm
  b) 12 cm, 14 cm
  c) 14 cm, 16 cm
  d) 16 cm, 18 cm
  Answer: b) 12 cm, 14 cm
16. What is the geometric interpretation of a tangent as a special case of a secant?
  a) A secant becomes a tangent when its two points of intersection coincide.
  b) A tangent is the perpendicular bisector of a secant.
  c) A tangent divides the circle into two equal parts.
  d) A tangent is unrelated to a secant.
  Answer: a) A secant becomes a tangent when its two points of intersection coincide.
17. Prove that the perpendicular at the point of contact to the tangent of a circle passes through the center. Which property justifies this?
  a) The tangent-radius perpendicularity property
  b) Angle at the center property
  c) Equal tangents property
  d) Chord bisector property
  Answer: a) The tangent-radius perpendicularity property
18. If the distance from the center to a point outside the circle is r and the radius is $a$ , what is the formula for the length of the tangent?
  a)
  r2−a2\sqrt{r^2 – a^2}
  b)
  a2−r2\sqrt{a^2 – r^2}
  c)
  r+a\sqrt{r + a}
  d)
  r+ar + a
  Answer: a)
  r2−a2\sqrt{r^2 – a^2}
19. In a quadrilateral circumscribing a circle, what is true about the sum of opposite sides?
  a) Equal
  b) Unequal
  c) Proportional to the radius
  d) Proportional to the diameter
  Answer: a) Equal
20. If a tangent from a point is 13 cm long and the radius of the circle is 12 cm, what is the distance of the external point from the circle’s center?
  a) 10 cm
  b) 15 cm
  c) 25 cm
  d) 17 cm
  Answer: b) 15 cm
21. What is the length of the radius of a circle that has a diameter of 10 cm?
  a) 5 cm
  b) 10 cm
  c) 15 cm
  d) 20 cm
  Answer: a) 5 cm
22. If a circle has a radius of 7 cm, what is the length of its diameter?
  a) 7 cm
  b) 14 cm
  c) 21 cm
  d) 28 cm
  Answer: b) 14 cm
23. How many tangents can be drawn at a single point on a circle?
  a) 0
  b) 1
  c) 2
  d) Infinite
  Answer: b) 1
24. Which of the following is the longest chord of a circle?
  a) Tangent
  b) Radius
  c) Diameter
  d) Secant
  Answer: c) Diameter
25. What is the angle between the radius and a tangent at the point of contact?
  a) 0°
  b) 45°
  c) 90°
  d) 180°
  Answer: c) 90°
26. What is the length of the tangent from a point 10 cm away from the center of a circle with radius 6 cm?
  a) 5 cm
  b) 8 cm
  c)
  434\sqrt{3}
  cm
  d)
  525\sqrt{2}
  cm
  Answer: b) 8 cm
27. A tangent to a circle intersects a radius at the point of contact. What is the relationship between the radius and the tangent?
  a) They are parallel
  b) They are perpendicular
  c) They are equal in length
  d) They form an acute angle
  Answer: b) They are perpendicular
28. If a chord is bisected by a radius perpendicular to it, the chord is also the:
  a) Tangent
  b) Diameter
  c) Secant
  d) Radius
  Answer: b) Diameter
29. Two tangents are drawn from a point outside a circle. The distance between the points of contact is 12 cm, and the radius is 5 cm. Find the distance of the point from the center.
  a) 6 cm
  b) 8 cm
  c) 10 cm
  d) 13 cm
  Answer: c) 10 cm
30. A circle has a radius of 5 cm. A point lies 13 cm away from the center. Find the length of the tangent drawn from the point to the circle.
  a) 8 cm
  b) 10 cm
  c) 12 cm
  d) 15 cm
  Answer: b) 12 cm
31. Two tangents drawn from an external point to a circle form an angle of 120°. Find the angle subtended by the line joining the points of contact at the circle’s center.
  a) 30°
  b) 60°
  c) 90°
  d) 120°
  Answer: b) 60°
32. A quadrilateral circumscribes a circle. If three of its sides measure 6 cm, 7 cm, and 8 cm, find the length of the fourth side.
  a) 15 cm
  b) 13 cm
  c) 14 cm
  d) 16 cm
  Answer: c) 14 cm
33. The length of a chord of a circle is 8 cm, and its perpendicular distance from the center is 3 cm. Find the radius of the circle.
  a) 5 cm
  b) 6 cm
  c) 7 cm
  d) 8 cm
  Answer: a) 5 cm
34. The length of the tangents drawn from a point to a circle is 12 cm. The radius of the circle is 5 cm. Find the distance of the point from the center.
  a) 10 cm
  b) 13 cm
  c) 15 cm
  d) 17 cm
  Answer: b) 13 cm
35. Prove that the sum of the lengths of opposite sides of a quadrilateral circumscribing a circle is equal. Which property supports this statement?
  a) Equal tangents theorem
  b) Tangent-radius property
  c) Circumscribing property
  d) Parallel tangents property
  Answer: c) Circumscribing property
36. A chord of a circle is 10 cm long and is at a distance of 6 cm from the center. Find the radius of the circle.
  a) 6 cm
  b) 8 cm
  c) 10 cm
  d) 12 cm
  Answer: b) 8 cm
37. Two tangents are drawn from an external point to a circle. Prove that the angle between the two tangents is supplementary to the angle subtended by the line joining the points of contact at the center. What is the supplement of 50°?
  a) 90°
  b) 130°
  c) 150°
  d) 180°
  Answer: b) 130°
38. A tangent drawn to a circle intersects a secant at an external point. The square of the tangent is equal to the product of the secant and its external segment. Which theorem supports this?
  a) Tangent-secant theorem
  b) Equal tangents theorem
  c) Angle at the center theorem
  d) Radius tangent theorem
  Answer: a) Tangent-secant theorem
39. If the tangents drawn from an external point are 15 cm each and the distance between their points of contact is 24 cm, find the radius of the circle.
  a) 9 cm
  b) 10 cm
  c) 12 cm
  d) 13 cm
  Answer: b) 10 cm
40. A circle has two parallel tangents, 10 cm apart, and the radius of the circle is 5 cm. What is the distance between the center and each tangent?
  a) 5 cm
  b) 7.5 cm
  c) 10 cm
  d) 2.5 cm
  Answer: a) 5 cm
41. If two circles are tangent externally, how many tangents can be drawn to both circles?
  a) 1
  b) 2
  c) 3
  d) 4
  Answer: d) 4
42. A circle is inscribed in a triangle with sides 8 cm, 15 cm, and 17 cm. Find the radius of the circle.
  a) 2 cm
  b) 4 cm
  c) 6 cm
  d) 8 cm
  Answer: a) 2 cm
43. The length of the tangent from a point to a circle is 8 cm, and the radius is 6 cm. Find the distance of the external point from the center.
  a) 10 cm
  b)
  474\sqrt{7}
  cm
  c)
  525\sqrt{2}
  cm
  d) 12 cm
  Answer: d) 10 cm
44. What is the angle subtended at the center by the diameter of the circle?
  a) 30°
  b) 60°
  c) 90°
  d) 180°
  Answer: d) 180°
45. In a rhombus circumscribing a circle, what can be said about its diagonals?
  a) They are equal
  b) They bisect at 90°
  c) They are perpendicular to tangents
  d) They are parallel
  Answer: b) They bisect at 90°
46. The perpendicular distance from the center to a chord is 4 cm, and the chord is 12 cm. Find the radius of the circle.
  a) 5 cm
  b) 7 cm
  c) 10 cm
  d) 13 cm
  Answer: b) 7 cm
47. What is the area of the largest circle that can fit inside a square of side 10 cm?
  a)
  25π sq cm
  b)
  50π50\pi
  sq cm
  c)
  75π75\pi
  sq cm
  d)
  100π100\pi
  sq cm
  Answer: b)
  50π50\pi
  sq cm
48. A tangent and a radius meet at the point of contact. The radius is 6 cm, and the tangent is 8 cm. What is the distance from the center to the external point of the tangent?
  a) 8 cm
  b) 10 cm
  c) 12 cm
  d) 14 cm
  Answer: b) 10 cm
49. The length of a tangent to a circle is twice the radius. If the radius is 5 cm, find the distance from the external point to the center.
  a) 5 cm
  b) 10 cm
  c) 15 cm
  d) 25 cm
  Answer: c) 15 cm
50. In two concentric circles, the radius of the larger circle is 10 cm, and the smaller one is 6 cm. Find the length of a chord of the larger circle that touches the smaller circle.
  a) 6 cm
  b) 8 cm
  c) 12 cm
  d) 16 cm
  Answer: c) 12 cm