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Maths · Circle Theorems

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Circle Theorem Proofs and Problems

Proving circle theorems, solving multi-step problems, and finding the equation of a tangent.

  • Higher
  • 9 key terms
  • All boards

Learning Objectives

  1. 1Prove the main circle theorems using isosceles triangles.
  2. 2Choose and combine circle theorems in multi-step problems.
  3. 3Find the equation of a tangent to a circle at a point.
  4. 4Write clear, complete reasons.

Proving and combining theorems

Higher tier questions can ask you to prove a circle theorem, or to combine several of them in one diagram. Proofs follow the same pattern each time: draw radii, which make isosceles triangles, then use the fact that the angles of a triangle add up to \(180^\circ\) and that an exterior angle equals the sum of the two opposite interior angles. In a multi-step problem, work out one angle at a time and write its reason, because each reason earns a mark.

Proof: the angle at the centre

The angle at the centre is twice the angle at the circumference. Radii give isosceles triangles.

  • Set up

    \(A\), \(B\) and \(C\) are on a circle with centre \(O\). Join \(CO\) and extend it to \(D\) on the circle.

  • Isosceles

    \(OA = OC\), so triangle \(OAC\) is isosceles, and angle \(OCA = a\) means angle \(OAC = a\).

  • Exterior angle

    Angle \(AOD = a + a = 2a\), because an exterior angle equals the sum of the two opposite interior angles.

  • The other side

    In the same way, if angle \(OCB = b\), angle \(BOD = 2b\).

  • Conclusion

    Angle \(AOB = 2a + 2b = 2(a + b) = 2 \times\) angle \(ACB\).

Proofs for the other theorems

The other circle theorems follow from the angle at the centre.

  • Semicircle

    If \(AB\) is a diameter, the angle at the centre is \(180^\circ\), so the angle at the circumference is \(90^\circ\).

  • Same segment

    Two angles at the circumference on the same chord are each half the same angle at the centre, so they are equal.

  • Cyclic quadrilateral

    The angles at the centre are \(2x\) and \(2y\) with \(2x + 2y = 360^\circ\), so \(x + y = 180^\circ\).

  • Alternate segment

    The tangent-radius right angle and an isosceles triangle give the angle in the alternate segment.

A multi-step problem

\(A\), \(B\), \(C\) and \(D\) are points on a circle with centre \(O\). Angle \(AOC = 140^\circ\), where \(B\) is on the major arc \(AC\). Work out angle \(ABC\) and angle \(ADC\), where \(D\) is on the minor arc \(AC\). Give reasons.

Show the solutionHide the solution
  1. 1 Centre and circumference Angle \(ABC = 140 \div 2 = 70^\circ\), because the angle at the centre is twice the angle at the circumference.
  2. 2 Cyclic quadrilateral \(ABCD\) has all four points on the circle.
  3. 3 Opposite angles Angle \(ADC = 180 - 70 = 110^\circ\).
  4. 4 Reason "Opposite angles of a cyclic quadrilateral add up to \(180^\circ\)."

AnswerAngle \(ABC = 70^\circ\) and angle \(ADC = 110^\circ\)

The tangent to a circle on a grid

A tangent is perpendicular to the radius, so the gradient of the tangent is linked to the gradient of the radius.

  • Gradient of the radius

    Work out the gradient from the centre to the point of contact.

  • Perpendicular gradient

    Turn the fraction upside down and change the sign: if \(m = \frac{3}{4}\), the tangent has gradient \(-\frac{4}{3}\).

  • Equation

    Use \(y - y_1 = m(x - x_1)\) with the point of contact.

  • Circle centred at the origin

    For \(x^2 + y^2 = r^2\), the radius to \((a, b)\) has gradient \(\frac{b}{a}\).

The equation of a tangent

The point \(P(3, 4)\) is on the circle \(x^2 + y^2 = 25\). Find the equation of the tangent to the circle at \(P\).

Show the solutionHide the solution
  1. 1 Gradient of the radius The centre is \((0, 0)\), so the radius \(OP\) has gradient \(\frac{4}{3}\).
  2. 2 Perpendicular The tangent is perpendicular to the radius, so its gradient is \(-\frac{3}{4}\).
  3. 3 Equation \(y - 4 = -\frac{3}{4}(x - 3)\).
  4. 4 Rearrange \(y = -\frac{3}{4}x + \frac{9}{4} + 4 = -\frac{3}{4}x + \frac{25}{4}\).

Answer\(y = -\frac{3}{4}x + \frac{25}{4}\)

Test yourself

  1. 1

    Why is a triangle with two radii isosceles?

    Show answerHide answer

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

  2. 2

    What is the angle at the centre in terms of the angle at the circumference?

    Show answerHide answer

    Twice as big.

  3. 3

    What is the gradient of a tangent in terms of the gradient of the radius?

    Show answerHide answer

    The negative reciprocal.

  4. 4

    What does an exterior angle of a triangle equal?

    Show answerHide answer

    The sum of the two opposite interior angles.

  5. 5

    What is the first step in a proof?

    Show answerHide answer

    Draw in the radii and mark the equal angles.

Exam technique: proof questions

Each line of a proof needs a reason, and the reasons are the marks.

  • Use letters

    Call unknown angles \(a\) and \(b\) to prove a general result.

  • Say why

    Write the reason after each step.

  • Finish

    State what you have proved, linking back to the question.

  • Do not use numbers

    A proof must work for any angle, so do not measure or use values from the diagram.

Summary and exam focus

  • Proofs use radii to make isosceles triangles.
  • The angle at the centre is twice the angle at the circumference.
  • Combine theorems one step at a time, with a reason for each angle.
  • The tangent at a point has the negative reciprocal gradient of the radius.

Exam focus

The point \(P(3, 4)\) is on the circle \(x^2 + y^2 = 25\). Find the equation of the tangent at \(P\). (3 marks) (3 marks)

The radius has gradient \(\frac{4}{3}\), so the tangent has gradient \(-\frac{3}{4}\). Then use \(y - 4 = -\frac{3}{4}(x - 3)\).

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Proof
A series of steps, with reasons, that shows a result is always true.
Theorem
A result that has been proved.
Exterior angle
An angle between a side extended and the next side.
Radius
The distance from the centre to the circle.
Isosceles triangle
A triangle with two equal sides.
Negative reciprocal
The number found by turning a fraction upside down and changing its sign.
Gradient
A measure of the steepness of a line.
Tangent
A straight line that touches a circle at one point.
Major arc
The longer arc between two points on a circle.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Prove 4 marks Core

Not drawn accurately. \(A\), \(B\) and \(C\) are points on the circumference of a circle, centre \(O\). Prove that angle \(AOB\) is twice angle \(ACB\). You may add lines to the diagram. [4 marks]

A circle diagram showing the angle at the centre and the angle at the circumference, for a proof.

Mark scheme — 4 marks available

  • Draws CO extended to D and states OA = OB = OC as radii — B1
  • Isosceles triangles, so OCA = OAC and OCB = OBC — M1
  • Exterior angles: AOD = 2 x OCA and BOD = 2 x OCB — M1
  • Concludes AOB = 2 x ACB — Q1

Model answer

Draw the line \(CO\) and extend it to meet the circle at \(D\). \(OA = OC = OB\), because they are radii. Triangles \(OAC\) and \(OBC\) are isosceles, so \(OAC = OCA\) and \(OBC = OCB\). The exterior angle \(AOD = 2 \times OCA\) and \(BOD = 2 \times OCB\). So \(AOB = 2(OCA + OCB) = 2 \times ACB\).

2. Exam question Work out 4 marks Core

Not drawn accurately. \(A\), \(B\), \(C\) and \(D\) are points on the circumference of a circle, centre \(O\). Angle \(AOC = 136^\circ\). (a) Work out the size of angle \(ABC\). Give a reason for your answer. [2 marks] (b) Work out the size of angle \(ADC\). Give a reason for your answer. [2 marks]

A circle diagram showing a cyclic quadrilateral and its centre.

Mark scheme — 4 marks available

  • (a) \(68\) — B1
  • (a) The angle at the centre is twice the angle at the circumference — Q1
  • (b) \(112\) — B1
  • (b) Opposite angles of a cyclic quadrilateral add up to 180 degrees — Q1

Model answer

(a) \(ABC = 136 \div 2 = 68^\circ\), because the angle at the centre is twice the angle at the circumference. (b) \(ADC = 180 - 68 = 112^\circ\), because opposite angles of a cyclic quadrilateral add up to \(180^\circ\).

3. Exam question Work out 4 marks Core

The point \(P(-3, 4)\) is on the circle \(x^2 + y^2 = 25\). Work out an equation of the tangent to the circle at \(P\). [4 marks]

Mark scheme — 4 marks available

  • Gradient of \(OP = -\dfrac{4}{3}\) — B1
  • Gradient of the tangent \(= \dfrac{3}{4}\) — B1
  • \(y - 4 = \dfrac{3}{4}(x + 3)\) or equivalent — M1
  • \(y = \dfrac{3}{4}x + \dfrac{25}{4}\) or \(3x - 4y = -25\) — A1

Model answer

The radius \(OP\) has gradient \(\dfrac{4}{-3} = -\dfrac{4}{3}\), so the tangent has gradient \(\dfrac{3}{4}\). Then \(y - 4 = \dfrac{3}{4}(x + 3)\), which gives \(y = \dfrac{3}{4}x + \dfrac{25}{4}\).

4. Exam question Show that 3 marks Core

Show that the line \(3x + 4y = 25\) is a tangent to the circle \(x^2 + y^2 = 25\) at the point \((3, 4)\). [3 marks]

Mark scheme — 3 marks available

  • Checks that \((3, 4)\) is on both the circle and the line — B1
  • Gradient of line \(= -\dfrac{3}{4}\) and gradient of radius \(= \dfrac{4}{3}\) — B1
  • Product of the gradients is -1, so the line is perpendicular to the radius, with a conclusion — Q1

Model answer

\(3^2 + 4^2 = 25\), and \(3 \times 3 + 4 \times 4 = 25\), so \((3, 4)\) is on both. The line is \(y = -\dfrac{3}{4}x + \dfrac{25}{4}\), with gradient \(-\dfrac{3}{4}\). The radius to \((3, 4)\) has gradient \(\dfrac{4}{3}\), and \(\dfrac{4}{3} \times -\dfrac{3}{4} = -1\), so the line is perpendicular to the radius, and is a tangent.

5. Exam question Work out 5 marks Stretch

Not drawn accurately. \(TAS\) is a tangent to the circle at \(A\). \(A\), \(B\), \(C\) and \(D\) are points on the circumference. Angle \(TAD = 56^\circ\) and angle \(CAD = 44^\circ\). (a) Write down the size of angle \(ACD\). Give a reason for your answer. [2 marks] (b) Work out the size of angle \(ADC\). [2 marks] (c) Work out the size of angle \(ABC\). Give a reason for your answer. [1 mark]

A circle diagram showing a tangent and a cyclic quadrilateral.

Mark scheme — 5 marks available

  • (a) \(56\) — B1
  • (a) Alternate segment theorem stated — Q1
  • (b) \(180 - 56 - 44\) — M1
  • (b) \(80\) — A1
  • (c) \(100\) with the cyclic quadrilateral reason — A1

Model answer

(a) \(ACD = 56^\circ\), by the alternate segment theorem. (b) \(ADC = 180 - 56 - 44 = 80^\circ\). (c) \(ABC = 180 - 80 = 100^\circ\), because opposite angles of a cyclic quadrilateral add up to \(180^\circ\).

6. Exam question Prove 3 marks Stretch

\(A\), \(B\), \(C\) and \(D\) are points on the circumference of a circle, centre \(O\). \(C\) and \(D\) are on the same side of the chord \(AB\). Prove that angle \(ACB\) equals angle \(ADB\). [3 marks]

Mark scheme — 3 marks available

  • States that the angle at the centre AOB is twice the angle at the circumference — B1
  • \(ACB = \dfrac{1}{2}AOB\) and \(ADB = \dfrac{1}{2}AOB\) — M1
  • Concludes that ACB = ADB — Q1

Model answer

The angle at the centre, \(AOB\), is twice the angle at the circumference. So \(ACB = \dfrac{1}{2}AOB\) and \(ADB = \dfrac{1}{2}AOB\). Therefore \(ACB = ADB\).

7. Multiple choice 1 mark Easier

Why is a triangle made by two radii and a chord isosceles?

  1. A The chord is a diameter
  2. B Two of its sides are radii, so they are equal Correct
  3. C All three sides are radii
  4. D The angles are all \(60^\circ\)

Why: Two radii are always equal.

8. Multiple choice 1 mark Easier

In a proof, what should be written next to every step?

  1. A A reason Correct
  2. B A number
  3. C A diagram
  4. D A question

Why: Each step needs a reason.

9. Multiple choice 1 mark Easier

What does the exterior angle of a triangle equal?

  1. A The third interior angle
  2. B \(90^\circ\)
  3. C \(360^\circ\) minus the interior angle
  4. D The sum of the two opposite interior angles Correct

Why: This is the exterior angle theorem.

10. Multiple choice 1 mark Core

The radius to a point on a circle has gradient \(\dfrac{2}{3}\). What is the gradient of the tangent at that point?

  1. A \(\dfrac{3}{2}\)
  2. B \(-\dfrac{2}{3}\)
  3. C \(-\dfrac{3}{2}\) Correct
  4. D \(\dfrac{2}{3}\)

Why: A tangent is perpendicular to the radius, so its gradient is the negative reciprocal.

11. Multiple choice 1 mark Core

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

  1. A \(\dfrac{3}{4}\)
  2. B \(\dfrac{4}{3}\) Correct
  3. C \(-\dfrac{4}{3}\)
  4. D \(7\)

Why: \(\dfrac{4 - 0}{3 - 0} = \dfrac{4}{3}\).

12. Multiple choice 1 mark Core

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

  1. A \(-\dfrac{3}{4}\) Correct
  2. B \(\dfrac{4}{3}\)
  3. C \(\dfrac{3}{4}\)
  4. D \(-\dfrac{4}{3}\)

Why: The radius has gradient \(\dfrac{4}{3}\), so the tangent has gradient \(-\dfrac{3}{4}\).

13. Multiple choice 1 mark Stretch

\(AOC\) is \(150^\circ\) at the centre, and \(D\) is on the minor arc \(AC\). What is angle \(ADC\)?

  1. A \(75^\circ\)
  2. B \(150^\circ\)
  3. C \(210^\circ\)
  4. D \(105^\circ\) Correct

Why: \(B\) on the major arc gives \(ABC = 75^\circ\), and \(ADC = 180 - 75 = 105^\circ\).

14. Multiple choice 1 mark Stretch

Which line is the tangent to \(x^2 + y^2 = 25\) at \((3, 4)\)?

  1. A \(4x + 3y = 25\)
  2. B \(3x - 4y = 25\)
  3. C \(3x + 4y = 25\) Correct
  4. D \(x + y = 7\)

Why: Gradient \(-\dfrac{3}{4}\) through \((3, 4)\) gives \(y - 4 = -\dfrac{3}{4}(x - 3)\), which rearranges to \(3x + 4y = 25\).

15. Multiple choice 1 mark Stretch

Why must a proof not rely on measuring angles in a diagram?

  1. A Diagrams are always wrong
  2. B The proof must work for every case, not just the one drawn Correct
  3. C Angles cannot be measured
  4. D It takes too long

Why: A proof uses letters and reasons so that it works for any angle.