OpenRevise

Exam questions · Maths · Circle Theorems

Tangents and Chords

  • 6 exam questions
  • 20 marks
  • 9 quick checks
  1. 1 Calculate [3 marks]

    The diagram is not drawn to scale. \(TA\) and \(TB\) are tangents to a circle, centre \(O\). Angle \(ATB = 72^\circ\). Calculate angle \(TAB\), giving a reason for your answer. [3 marks]

    A circle diagram showing two tangents from a point T.
    Show answerHide answer

    Model answer

    \(TA = TB\), because tangents from a point to a circle are equal, so triangle \(TAB\) is isosceles. Angle \(TAB = (180 - 72) \div 2 = 54^\circ\).

    Mark scheme

    • Tangents from a point are equal, so triangle TAB is isosceles — B1
    • \((180 - 72) \div 2\) — M1
    • \(54\) — A1
  2. 2 Calculate [3 marks]

    \(PT\) is a tangent to a circle, centre \(O\), touching the circle at \(T\). The radius of the circle is 7 cm and \(PT = 24\) cm. Calculate the length of \(OP\). [3 marks]

    Show answerHide answer

    Model answer

    The tangent is perpendicular to the radius, so angle \(OTP = 90^\circ\). \(OP^2 = 7^2 + 24^2 = 49 + 576 = 625\), so \(OP = 25\) cm.

    Mark scheme

    • Angle OTP is a right angle, as a tangent is perpendicular to the radius — B1
    • \(7^2 + 24^2\) or \(49 + 576\) — M1
    • \(25\) — A1
  3. 3 Calculate [3 marks]

    The diagram is not drawn to scale. \(AB\) is a chord of a circle, centre \(O\), with radius 10 cm. \(AB = 16\) cm. \(M\) is the point on \(AB\) where \(OM\) is perpendicular to \(AB\). Calculate the length of \(OM\). [3 marks]

    A circle diagram showing a chord and the perpendicular from the centre.
    Show answerHide answer

    Model answer

    The perpendicular from the centre bisects the chord, so \(AM = 8\) cm. \(OM^2 = 10^2 - 8^2 = 100 - 64 = 36\), so \(OM = 6\) cm.

    Mark scheme

    • \(AM = 8\) — M1
    • \(10^2 - 8^2\) — M1
    • \(6\) — A1
  4. 4 Calculate [4 marks]

    The diagram is not drawn to scale. \(PA\) and \(PB\) are tangents to a circle, centre \(O\). Angle \(OAB = 40^\circ\). Calculate angle \(APB\), giving reasons for your answer. [4 marks]

    A circle diagram showing two tangents and a chord.
    Show answerHide answer

    Model answer

    Angle \(OAP = 90^\circ\), because a tangent is perpendicular to the radius, so \(PAB = 90 - 40 = 50^\circ\). \(PA = PB\), so \(PBA = 50^\circ\) and \(APB = 180 - 50 - 50 = 80^\circ\).

    Mark scheme

    • Angle \(PAB = 90 - 40 = 50\) — M1
    • \(180 - 50 - 50\) — M1
    • \(80\) — A1
    • Tangent perpendicular to the radius, and tangents from a point are equal — B1
  5. 5 Calculate [3 marks]

    A circle has centre \(O\) and radius 13 cm. A chord is 12 cm from \(O\). Calculate the length of the chord. [3 marks]

    Show answerHide answer

    Model answer

    Half the chord is \(\sqrt{13^2 - 12^2} = \sqrt{25} = 5\) cm, so the chord is 10 cm.

    Mark scheme

    • \(13^2 - 12^2\) or \(169 - 144\) — M1
    • \(5\) found as half the chord — M1
    • \(10\) — A1
  6. 6 Prove [4 marks]

    \(TA\) and \(TB\) are tangents to a circle, centre \(O\), touching the circle at \(A\) and \(B\). Prove that triangles \(OAT\) and \(OBT\) are congruent, and so that \(TA = TB\). [4 marks]

    Show answerHide answer

    Model answer

    \(OA = OB\), because they are radii. Angles \(OAT\) and \(OBT\) are both \(90^\circ\), because a tangent is perpendicular to the radius. \(OT\) is common to both triangles. So the triangles are congruent (RHS), and \(TA = TB\).

    Mark scheme

    • OA = OB because they are radii — B1
    • Angles \(OAT = OBT = 90^\circ\), because a tangent is perpendicular to the radius — B1
    • OT is common, so the triangles are congruent (right angle, hypotenuse, side) — B1
    • Concludes TA = TB because corresponding sides of congruent triangles are equal — B1

Quick check

  1. 1

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

    1. A\(45^\circ\)
    2. B\(180^\circ\)
    3. C\(60^\circ\)
    4. D\(90^\circ\)
    Show answerHide answer

    D: \(90^\circ\)

    A tangent is perpendicular to the radius.

  2. 2

    Two tangents from the point \(P\) touch a circle at \(A\) and \(B\). \(PA = 9\) cm. What is \(PB\)?

    1. A18 cm
    2. B4.5 cm
    3. C9 cm
    4. DIt cannot be found
    Show answerHide answer

    C: 9 cm

    Tangents from the same point are equal in length.

  3. 3

    A perpendicular from the centre of a circle meets a chord. What does it do to the chord?

    1. AIt doubles the chord
    2. BIt bisects the chord
    3. CIt is the same length as the chord
    4. DIt makes the chord a diameter
    Show answerHide answer

    B: It bisects the chord

    The perpendicular from the centre bisects the chord.

  4. 4

    \(PT\) is a tangent, \(O\) is the centre, the radius is 3 cm and \(OP = 5\) cm. How long is \(PT\)?

    1. A4 cm
    2. B2 cm
    3. C\(\sqrt{34}\) cm
    4. D8 cm
    Show answerHide answer

    A: 4 cm

    \(OTP\) is right-angled at \(T\), so \(PT^2 = 5^2 - 3^2 = 16\).

  5. 5

    Tangents \(PA\) and \(PB\) touch a circle with centre \(O\). Angle \(AOB = 100^\circ\). What is angle \(APB\)?

    1. A\(100^\circ\)
    2. B\(50^\circ\)
    3. C\(260^\circ\)
    4. D\(80^\circ\)
    Show answerHide answer

    D: \(80^\circ\)

    \(OAPB\) is a quadrilateral with two right angles, so \(APB = 360 - 90 - 90 - 100 = 80^\circ\).

  6. 6

    A circle has radius 5 cm and a chord is 8 cm long. How far is the chord from the centre?

    1. A4 cm
    2. B\(\sqrt{41}\) cm
    3. C3 cm
    4. D1 cm
    Show answerHide answer

    C: 3 cm

    Half the chord is 4 cm, so the distance is \(\sqrt{5^2 - 4^2} = 3\).

  7. 7

    Tangents \(PA\) and \(PB\) touch a circle at \(A\) and \(B\). Angle \(APB = 50^\circ\). What is angle \(PAB\)?

    1. A\(50^\circ\)
    2. B\(65^\circ\)
    3. C\(130^\circ\)
    4. D\(40^\circ\)
    Show answerHide answer

    B: \(65^\circ\)

    \(PA = PB\), so triangle \(PAB\) is isosceles and \(PAB = (180 - 50) \div 2 = 65^\circ\).

  8. 8

    The distance from the centre of a circle of radius 13 cm to a chord is 5 cm. How long is the chord?

    1. A24 cm
    2. B12 cm
    3. C\(2\sqrt{194}\) cm
    4. D10 cm
    Show answerHide answer

    A: 24 cm

    Half the chord is \(\sqrt{13^2 - 5^2} = 12\), so the chord is 24 cm.

  9. 9

    A line from the centre to a point \(P\) outside a circle of radius \(r\) has length \(d\). Which expression gives the tangent length from \(P\)?

    1. A\(\sqrt{d^2 + r^2}\)
    2. B\(d - r\)
    3. C\(d + r\)
    4. D\(\sqrt{d^2 - r^2}\)
    Show answerHide answer

    D: \(\sqrt{d^2 - r^2}\)

    The radius and tangent make a right angle, with \(d\) as the hypotenuse.