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Flashcards · Maths

Circle Theorems

80 cards from 5 lessons

  1. What is a chord?

    A straight line joining two points on a circle.

  2. What is a tangent?

    A straight line that touches a circle at one point.

  3. What is an arc?

    A part of the circumference of a circle.

  4. What is the diameter?

    A chord that passes through the centre.

  5. What does the angle at the centre equal?

    Twice the angle at the circumference.

  6. If the angle at the circumference is \(35^\circ\), what is the angle at the centre?

    \(70^\circ\).

  7. If the angle at the centre is \(100^\circ\), what is the angle at the circumference?

    \(50^\circ\).

  8. What is the angle in a semicircle?

    \(90^\circ\).

  9. What must the line be for the semicircle theorem?

    A diameter, passing through the centre.

  10. What is the reason for a \(90^\circ\) angle at the circumference on a diameter?

    The angle in a semicircle is \(90^\circ\).

  11. What kind of triangle do two radii make?

    An isosceles triangle.

  12. What do the other two angles of a right-angled triangle add up to?

    \(90^\circ\).

  13. What are the two marks for in a circle theorem question?

    The angle and the reason.

  14. Where is the angle at the centre drawn from?

    The centre of the circle.

  15. Do both angles have to come from the same chord?

    Yes, from the same two points on the circle.

  16. How many degrees are in a straight line through the centre?

    \(180^\circ\).

  17. What is a segment of a circle?

    The part of a circle cut off by a chord.

  18. What is true of angles in the same segment?

    They are equal.

  19. What must be true of two angles for the same-segment theorem?

    They are made by the same chord, on the same side.

  20. What is a cyclic quadrilateral?

    A quadrilateral with all four corners on a circle.

  21. What do opposite angles of a cyclic quadrilateral add up to?

    \(180^\circ\).

  22. If one angle of a cyclic quadrilateral is \(74^\circ\), what is the opposite angle?

    \(106^\circ\).

  23. What is the reason for opposite angles of a cyclic quadrilateral?

    Opposite angles of a cyclic quadrilateral add up to \(180^\circ\).

  24. What is an exterior angle of a cyclic quadrilateral equal to?

    The interior opposite angle.

  25. What do the four angles of a quadrilateral add up to?

    \(360^\circ\).

  26. What do you do when angles are given as expressions?

    Form an equation, using the theorem, and solve it.

  27. Which pairs of angles are opposite in \(ABCD\)?

    \(A\) and \(C\), and \(B\) and \(D\).

  28. Can two angles on opposite sides of a chord be equal?

    Not by the same-segment theorem; they add up to \(180^\circ\) instead.

  29. What is the first step in a circle theorem problem?

    Decide which theorem each angle belongs to.

  30. What should you write on the diagram?

    Every angle you find.

  31. Is a rectangle a cyclic quadrilateral?

    Yes, its opposite angles add up to \(180^\circ\).

  32. Is a parallelogram that is not a rectangle cyclic?

    No.

  33. What is a tangent?

    A straight line that touches a circle at one point.

  34. What is a chord?

    A straight line joining two points on a circle.

  35. What is the angle between a tangent and the radius at the point of contact?

    \(90^\circ\).

  36. What is true of tangents from the same point to a circle?

    They are equal in length.

  37. What shape do two tangents and the two radii to the points of contact make?

    A kite.

  38. What do the angles of a quadrilateral add up to?

    \(360^\circ\).

  39. What does a perpendicular from the centre do to a chord?

    It bisects it.

  40. What is the reason for a right angle at a point of contact?

    A tangent is perpendicular to the radius.

  41. What is the shortest distance from the centre to a chord?

    The perpendicular distance.

  42. What triangle is formed by the centre and the ends of a chord?

    An isosceles triangle, because two sides are radii.

  43. What do you use to find the length of a tangent?

    Pythagoras in the triangle with the radius.

  44. A radius is 5 cm and the distance from the centre to the point is 13 cm. How long is the tangent?

    12 cm.

  45. Which side is the hypotenuse in a tangent triangle?

    The line from the centre to the outside point.

  46. A chord is 16 cm long and the radius is 10 cm. How far is the chord from the centre?

    6 cm.

  47. What should you draw first in a tangent question?

    The radius to the point of contact.

  48. Can a line touch a circle at two points and be a tangent?

    No, a tangent touches at exactly one point.

  49. What is the alternate segment theorem?

    The angle between a tangent and a chord equals the angle in the alternate segment.

  50. Where must the chord start for the theorem to apply?

    At the point where the tangent touches the circle.

  51. What does alternate mean here?

    On the other side of the chord.

  52. Which two lines make the angle between a tangent and a chord?

    The tangent and the chord.

  53. If the tangent-chord angle is \(62^\circ\), what is the angle in the alternate segment?

    \(62^\circ\).

  54. What is a tangent perpendicular to?

    The radius at the point of contact.

  55. What do the angles in a triangle add up to?

    \(180^\circ\).

  56. What are the base angles of an isosceles triangle?

    Equal.

  57. Why is a triangle made by two radii isosceles?

    Two of its sides are radii, so they are equal.

  58. What do you write for the alternate segment theorem?

    Angle between tangent and chord equals angle in the alternate segment.

  59. What should you do first when you see a tangent?

    Look for the radius and the chord from the point of contact.

  60. Can you use the theorem if the chord does not start at the point of contact?

    No.

  61. What do two tangents from one point have in common?

    They are equal in length.

  62. What angle is between a tangent and a diameter at the point of contact?

    \(90^\circ\).

  63. What is the best way to avoid losing marks?

    Write the full reason next to every angle.

  64. How do you check an answer?

    Check the angles of each triangle add up to \(180^\circ\).

  65. What does the angle at the centre equal?

    Twice the angle at the circumference.

  66. What makes a triangle isosceles in circle proofs?

    Two sides are radii.

  67. What does the exterior angle of a triangle equal?

    The sum of the two opposite interior angles.

  68. What do the angles of a triangle add up to?

    \(180^\circ\).

  69. What should each line of a proof include?

    A reason.

  70. Why must you not use numbers in a proof?

    The proof must work for any angle.

  71. How do you prove the angle in a semicircle is \(90^\circ\)?

    The angle at the centre is \(180^\circ\), so the angle at the circumference is half of it.

  72. How do you prove opposite angles of a cyclic quadrilateral add up to \(180^\circ\)?

    The angles at the centre add up to \(360^\circ\), and each is double.

  73. What is the gradient of a tangent to a circle at \((a, b)\), centre origin?

    \(-\frac{a}{b}\).

  74. What is the gradient of the radius to \((3, 4)\) from the origin?

    \(\frac{4}{3}\).

  75. What is the gradient of the tangent at \((3, 4)\) on \(x^2 + y^2 = 25\)?

    \(-\frac{3}{4}\).

  76. What formula gives a line through a point?

    \(y - y_1 = m(x - x_1)\).

  77. What do you do first in a multi-step circle problem?

    Work out one angle and write its reason.

  78. Which theorems often combine with the alternate segment theorem?

    Triangle angles and isosceles triangles.

  79. What should you write on the diagram?

    Every angle you find.

  80. Which type of triangle do two radii and a chord make?

    Isosceles.