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Cambridge IGCSE0607

Circle theorems I

International Mathematics 0607 Chapter Notes

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Circle theorems I
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Circle theorems I notes

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In this chapter

  1. 1Angle in a semicircle
  2. 2Angle between tangent and radius
  3. 3Chaining theorems with reasons
  4. 4Angle at the centre
  5. 5Angles in the same segment
  6. 6Cyclic quadrilateral
  7. 7Alternate segment theorem

1. Angle in a semicircle

The angle in a semicircle is 90°. A straight line through the centre of a circle with both ends on the circle is a diameter.

When a diameter is a side of a triangle with its third corner on the circle, the angle at that corner is 90°. Write the reason as angle in a semicircle = 90°.

angle in a semicircle = 90^(°)

Examiner insight

“Occasionally a candidate gave the answer 90, incorrectly believing BE was the diameter and hence BFE was an angle in a semi-circle.” “Many gave the total of degrees in a triangle.”

Worked example 13 marks

AB is a diameter of a circle. C is a point on the circle. Angle ABC = 35°. Find angle BAC. Give a geometrical reason for each angle you find.

  1. 1

    angle ACB = 90^(°), angle in a semicircle [1]

  2. 2

    angle BAC = 180 - 90 - 35 [1]

  3. 3

    = 55^(°) [1]

2. Angle between tangent and radius

A tangent touches the circle at one point. The angle between the tangent and the radius at that point is 90°.

This gives a right-angled triangle whenever a radius ends at a tangent.

angle between tangent and radius = 90^(°)

Examiner insight

“Some candidates measured the angles OAB or BOA. This is not how to approach circle theorem questions as the question paper clearly says that the diagram is not to scale.” “Candidates were expected to realise that Angle BAO is 90° as the tangent AB is at right angles to the radius, AO.”

Worked example 13 marks

A tangent touches a circle, centre O, at P. Q is a point on the tangent. Angle POQ = 62°. Find angle OQP. Give a geometrical reason.

  1. 1

    angle OPQ = 90^(°), angle between tangent and radius [1]

  2. 2

    angle OQP = 180 - 90 - 62 [1]

  3. 3

    = 28^(°) [1]

3. Chaining theorems with reasons

Most questions need two or more theorems in a row. Write one line of working and its reason for each angle.

When a question asks for geometrical reasons, a calculation alone is not enough. Use the correct language and name the theorem.

Examiner insight

“This question proved to be tricky with many candidates knowing some circle theorems but not having a complete understanding, as there were very few candidates who scored full marks.” “Whilst some candidates gave correct geometrical reasons using the required language, many gave reasons using alternative wording or only providing calculations.”

Worked example 14 marks

A tangent touches a circle, centre O, at P. T is a point on the tangent. Q is a point on the circle on the same side of the line PO as T. Angle TPQ = 50°. Find angle POQ. Give a geometrical reason for each angle you find.

  1. 1

    angle OPQ = 90 - 50 = 40^(°), angle between tangent and radius is 90^(°) [1]

  2. 2

    OP = OQ, so angle OQP = 40^(°): base angles in an isosceles triangle are equal [1]

  3. 3

    angle POQ = 180 - 40 - 40 [1]

  4. 4

    = 100^(°) [1]

4. Angle at the centre

The angle at the centre is twice the angle at the circumference. Both angles stand on the same arc.

Two radii and a chord make an isosceles triangle. Base angles in an isosceles triangle are equal.

angle at centre = 2 × angle at circumference

Examiner insight

“For angle ACO candidates needed to state that angle at centre is twice angle at the circumference and that there were base angles in an isosceles triangle.”

Worked example 14 marks

A, B and C are points on a circle, centre O. Angle ACB = 38°, and C is on the major arc AB. Find angle OAB. Give a geometrical reason for each angle you find.

  1. 1

    angle AOB = 2 × 38 = 76^(°), angle at the centre is twice the angle at the circumference [1]

  2. 2

    OA = OB, so triangle OAB is isosceles: base angles are equal [1]

  3. 3

    angle OAB = (180 - 76) ÷ 2 [1]

  4. 4

    = 52^(°) [1]

5. Angles in the same segment

Angles in the same segment are equal.

Two such angles are drawn from the two ends of one chord to points on the same side of it.

angles in the same segment are equal

Examiner insight

“This circle theorem question proved challenging as many candidates were unable to recognise which circle theorems could be used, as they were not presented in the most usual positions.”

Worked example 12 marks

A, B, C and D are points on a circle. C and D are on the same side of the chord AB. Angle ACB = 47°. Find angle ADB. Give a geometrical reason.

  1. 1

    angle ADB = 47^(°) [1]

  2. 2

    angles in the same segment are equal [1]

6. Cyclic quadrilateral

A cyclic quadrilateral has its four corners on a circle. Opposite angles of a cyclic quadrilateral sum to 180° (they are supplementary).

Check the shape has four sides before using this. A pentagon on a circle does not follow the rule.

opposite angles of a cyclic quadrilateral sum to 180^(°)

Examiner insight

“For angle ADC candidates needed to state that opposite angles in a cyclic quadrilateral add up to 180.” “Candidates did not realise that ABCE was a cyclic quadrilateral.”

Worked example 13 marks

ABCD is a cyclic quadrilateral. Angle DAB = (2y + 30)^(°) and angle BCD = (y + 6)^(°). Find the value of y and angle BCD. Give a geometrical reason.

  1. 1

    (2y + 30) + (y + 6) = 180, opposite angles of a cyclic quadrilateral sum to 180^(°) [1]

  2. 2

    3y = 144, so y = 48 [1]

  3. 3

    angle BCD = 48 + 6 = 54^(°) [1]

7. Alternate segment theorem

The alternate segment theorem links a tangent to the angles inside the circle.

The angle between a tangent and a chord equals the angle in the alternate segment. The alternate segment is the one on the other side of the chord.

angle between tangent and chord = angle in alternate segment

Examiner insight

“The alternate segment theorem in part (b) proved to be the most troublesome.” “However, it was very common for candidates to make incorrect assumptions about the diagram.”

Worked example 13 marks

A tangent touches a circle at T. A and B are points on the circle. The angle between the tangent and the chord TA is 54°. B is on the other side of TA from this angle, and angle TAB = 71°. Find angle ATB. Give a geometrical reason.

  1. 1

    angle TBA = 54^(°), alternate segment theorem [1]

  2. 2

    angle ATB = 180 - 54 - 71 [1]

  3. 3

    = 55^(°) [1]

Recap

  • Angle in a semicircle = 90°.
  • Angle between tangent and radius = 90°.
  • Extended: angle at the centre is twice the angle at the circumference.
  • Extended: angles in the same segment are equal.
  • Extended: opposite angles of a cyclic quadrilateral sum to 180°.
  • Extended: alternate segment theorem.
  • Give a geometrical reason for every angle. Never measure: the diagram is not to scale.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. AB is a diameter of a circle and C is a point on the circle. Write down the size of angle ACB and give a geometrical reason.2 marks
  2. Do not use a calculator. AB is a diameter of a circle and C is a point on the circle. Angle CAB = 24°. Find angle ABC. Give a geometrical reason for each angle you find.3 marks
  3. A tangent touches a circle, centre O, at P. Q is a point on the tangent. Angle OQP = 33°. Find angle POQ. Give a geometrical reason for the right angle.3 marks
  4. Do not use a calculator. AB is a diameter of a circle. C is a point on the circle and AC = BC. Find angle CAB. Give a geometrical reason for each angle you find.3 marks
  5. Do not use a calculator. A, B and C are points on a circle, centre O. Angle AOB = 148° and C is on the major arc AB. Find angle ACB. Give a geometrical reason.2 marks
  6. A, B, C and D are points on a circle. C and D are on the same side of the chord AB. Angle ACB = (3x + 8)^(°) and angle ADB = (5x - 12)^(°). Find x and angle ADB.3 marks
  7. ABCD is a cyclic quadrilateral. Angle DAB = (4p + 10)^(°) and angle BCD = (2p + 20)^(°). Find p and angle DAB.3 marks
  8. A tangent touches a circle, centre O, at T. A is a point on the circle. The angle between the tangent and the chord TA is 62°. B is a point on the circle in the alternate segment. Find (a) angle TBA, (b) the obtuse angle TOA. Give a geometrical reason for each.4 marks
  9. Do not use a calculator. Four seats A, B, C and D are fixed in that order round the rim of a big wheel, centre O. Angle ABC = 116°. Find (a) the reflex angle AOC, (b) angle OAC. Give a geometrical reason for each angle you find.5 marks
  10. A circular lake has centre O. A straight path touches the lake at T, with points Q and R on the path either side of T. A is a point on the edge of the lake with angle QTA = 70°. B is a point on the edge of the lake on the opposite side of the chord TA from Q, and BT = BA. Find (a) angle BTA, (b) angle BTR, (c) angle OTB. Give a geometrical reason for each step.7 marks

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