The shape of the lesson
By the end of the lesson, learners can
- The angle at the centre: In the angle-at-centre tab, drag C around the major arc and watch angle ACB stay at exactly half of angle AOB.
- Cyclic quadrilaterals: In the cyclic quadrilateral tab, drag any of the four corners around the circle.
- The alternate segment: In the alternate segment tab, drag the point C around the major arc.
- Chords and tangents: symmetry: In the chords tab, slide a chord across the circle.
- Centre involved? Angle at centre = 2 × angle at circumference on the same arc.
- Two angles on the same arc? Same segment: equal.
- Four points on the circle? Opposite angles add to \(180^\circ\).
- Tangent and chord? Alternate segment theorem; tangent and radius \(90^\circ\).
- Chord or two tangents? Perpendicular from centre bisects; tangents from a point are equal.
- Reasons in syllabus words for every step.
One picture every idea returns to
Angle at the centre \(= 2 \times\) angle at the circumference (same arc). Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral add to \(180^\circ\). The angle between a tangent and a chord equals the angle in the alternate segment. Tangents from one point are equal, and the perpendicular from the centre bisects a chord.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Twice at the centre.. Extended. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 11 min | Teach and check: The angle at the centre | In the angle-at-centre tab, drag C around the major arc and watch angle ACB stay at exactly half of angle AOB. Project tab 4 of the lab (Circle theorems lab, angle at the centre) and let a student drive it while the class predicts. Model the worked example Centre and circumference on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Cyclic quadrilaterals | In the cyclic quadrilateral tab, drag any of the four corners around the circle. Project tab 6 of the lab (Circle theorems lab, cyclic quadrilateral) and let a student drive it while the class predicts. Model the worked example Opposite angles on the board, one line at a time. Your turn: ABCD is a cyclic quadrilateral and side AB is extended beyond B to E. Angle ADC \(= 77^\circ\). Find angle CBE. | Pairs |
| 17 to 23 min | Teach and check: The alternate segment | In the alternate segment tab, drag the point C around the major arc. Project tab 7 of the lab (Circle theorems lab, alternate segment) and let a student drive it while the class predicts. Model the worked example Tangent and chord on the board, one line at a time. | Pairs |
| 23 to 30 min | Teach and check: Chords and tangents: symmetry | In the chords tab, slide a chord across the circle. Project tab 8 of the lab (Circle theorems lab, chords) and let a student drive it while the class predicts. Model the worked example Distance of a chord from the centre on the board, one line at a time. Your turn: In the two-tangents example, C is a point on the major arc AB. Find angle ACB in two different ways. | Round Table |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 22 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Chords and tangents: symmetry, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Round Table
One sheet travels round the team. Each student adds one line of working or one idea and passes it on, so the solution is built four hands at a time and every contribution is visible.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Halve the angle at the centre to get the circumference angle, do not double it. | And check which arc each angle stands on: if the angle at the circumference is obtuse, the matching centre angle is reflex. |
| Only OPPOSITE angles add to \(180^\circ\). | Adjacent angles of a cyclic quadrilateral have no special rule, and a quadrilateral whose vertices are not all on the circle has no such rule at all. |
| The equal angle is in the ALTERNATE segment, across the chord from the tangent angle. | Matching it with an angle in the same segment as the tangent angle is the usual error. |
| The right angles in the tangent kite are at A and B (where the tangents touch), not at P or O. | Angle AOB and angle APB are supplementary, not equal. |
Circle theorems II: centre, segments, cyclic quadrilaterals: worksheet
6 questions, 22 marks. Written in the style of the papers; no past paper question is reproduced.
Complete each statement.
(a) Angles in the same segment are ... (b) The tangents from an external point to a circle are ... (c) Equal chords are ... from the centre.
A, B, C and D lie on a circle with centre O. C and D are on the major arc AB, and angle ADB \(= 38^\circ\). Find (a) angle ACB (b) angle AOB. Give reasons.
ABCD is a cyclic quadrilateral. Angle ABC \(= 2x + 10^\circ\) and angle ADC \(= 3x - 5^\circ\). Find \(x\) and both angles.
A circle has centre O and radius 13 cm. Chord PQ has length 24 cm. Find the distance from O to PQ. A second chord RS is also 24 cm long: how far is it from O?
ABCD is a cyclic quadrilateral with centre O inside it. Let angle DAB \(= a\) and angle BCD \(= c\).
Prove that \(a + c = 180^\circ\).
A circle has centre O and radius 6 cm. P is a point 10 cm from O. PA and PB are tangents from P, touching the circle at A and B. C is a point on the major arc AB.
(a) Find PA. (b) Write down PB, giving a reason. (c) Find angle APB, to 1 decimal place. (d) Find angle ACB, to 1 decimal place.
Answer key and marking notes
- (a) equal
- (b) equal in length
- (c) equidistant (the same distance)
- (a) Angle ACB = angle ADB \(= 38^\circ\) (angles in the same segment are equal).
- (b) Angle AOB \(= 2 \times 38^\circ = 76^\circ\) (angle at the centre is twice the angle at the circumference).
- Opposite angles of a cyclic quadrilateral add to \(180^\circ\): \(5x + 5 = 180\).
- \(5x = 175\), so \(x = 35\).
- Angle ABC \(= 80^\circ\) and angle ADC \(= 100^\circ\).
- The perpendicular from O bisects PQ, so the half-chord is 12 cm.
- Distance \(= \sqrt{13^2 - 12^2} = \sqrt{25} = 5\) cm.
- RS is also 5 cm from O (equal chords are equidistant from the centre).
- Angle DAB stands on arc BCD, so the angle at the centre on that arc is \(2a\) (angle at centre is twice the angle at circumference).
- Angle BCD stands on arc DAB, so the angle at the centre on that arc is \(2c\).
- These two centre angles together make the full turn at O: \(2a + 2c = 360^\circ\) (angles at a point).
- Therefore \(a + c = 180^\circ\).
- (a) Angle OAP \(= 90^\circ\) (tangent and radius), so \(PA = \sqrt{10^2 - 6^2} = \sqrt{64} = 8\) cm.
- (b) PB = 8 cm: tangents from an external point are equal.
- (c) \(\sin(\text{APO}) = \dfrac{6}{10}\), so angle APO \(= 36.87^\circ\). By symmetry, angle APB \(= 2 \times 36.87 = 73.7^\circ\).
- (d) Angle AOB \(= 360 - 90 - 90 - 73.74 = 106.26^\circ\) (angles in quadrilateral OAPB).
- Angle ACB \(= 106.26 \div 2 = 53.1^\circ\) (angle at the centre is twice the angle at the circumference).
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
A, B and C lie on a circle with centre O, and C is on the major arc AB. Angle AOB \(= 136^\circ\). Find angle ACB, marked \(x\).
- A\(68^\circ\)
- B\(136^\circ\)
- C\(272^\circ\)
- D\(44^\circ\)
A and B lie on a circle with centre O, and angle AOB \(= 110^\circ\). Point E lies on the MINOR arc AB. Find angle AEB.
- A\(125^\circ\)
- B\(55^\circ\)
- C\(250^\circ\)
- D\(70^\circ\)
ABCD is a cyclic quadrilateral. Angle DAB \(= 2x + 15^\circ\) and angle BCD \(= 3x - 5^\circ\). Find angle BCD.
- A\(97^\circ\)
- B\(83^\circ\)
- C\(34^\circ\)
- D\(205^\circ\)
SAT is a tangent to the circle, centre O, at A. Angle TAB \(= 62^\circ\) and angle BAC \(= 50^\circ\). Find angle ABC, marked \(y\).
- A\(68^\circ\)
- B\(62^\circ\)
- C\(50^\circ\)
- D\(28^\circ\)
A circle has centre O and radius 17 cm. A chord PQ is 30 cm long. Find the distance from O to the chord PQ.
- A8 cm
- B22.7 cm
- C24.7 cm
- D2 cm
PA and PB are tangents from P to a circle with centre O, touching it at A and B. Angle APB \(= 52^\circ\). C is a point on the major arc AB. Find angle ACB.
- A\(64^\circ\)
- B\(128^\circ\)
- C\(26^\circ\)
- D\(52^\circ\)
Quiz key
The angle at the centre is twice the angle at the circumference, so \(x = 136^\circ \div 2 = 68^\circ\). The tempting \(272^\circ\) doubles instead of halving, \(136^\circ\) treats the two angles as equal, and \(44^\circ\) uses \(180^\circ - 136^\circ\) as if the angles were supplementary.
Angle AEB stands on the major arc AB, so use the reflex angle at the centre: \(360^\circ - 110^\circ = 250^\circ\). Then angle AEB \(= 250^\circ \div 2 = 125^\circ\). The tempting \(55^\circ\) halves \(110^\circ\), which is the angle at a point on the MAJOR arc. \(250^\circ\) forgets to halve, and \(70^\circ\) is \(180^\circ - 110^\circ\).
Opposite angles of a cyclic quadrilateral add to \(180^\circ\): \(2x + 15 + 3x - 5 = 180\), so \(5x = 170\) and \(x = 34\). Angle BCD \(= 3(34) - 5 = 97^\circ\) (check: angle DAB \(= 83^\circ\) and \(83 + 97 = 180\)). The tempting \(83^\circ\) is angle DAB, the other angle. \(34^\circ\) stops at \(x\), and \(205^\circ\) comes from setting the sum equal to \(360^\circ\).
By the alternate segment theorem, angle ACB = angle TAB \(= 62^\circ\). Angles in triangle ABC: \(y = 180^\circ - 62^\circ - 50^\circ = 68^\circ\). The tempting \(62^\circ\) is angle ACB, which is the angle equal to the tangent-chord angle, not angle ABC. \(50^\circ\) copies angle BAC, and \(28^\circ\) is \(90^\circ - 62^\circ\), wrongly bringing in the tangent-radius right angle. (Check with the other tangent-chord angle: angle SAC \(= 180 - 62 - 50 = 68^\circ = y\).)
The perpendicular from O meets PQ at its midpoint M, so PM = 15 cm and OP = 17 cm is the hypotenuse: \(OM = \sqrt{17^2 - 15^2} = \sqrt{64} = 8\) cm. The tempting 22.7 cm adds the squares (\(\sqrt{514}\)). 24.7 cm uses the whole chord, \(\sqrt{30^2 - 17^2}\), and 2 cm subtracts the lengths \(17 - 15\).
Angles OAP and OBP are \(90^\circ\) (tangent and radius), so angle AOB \(= 360^\circ - 90^\circ - 90^\circ - 52^\circ = 128^\circ\). C is on the major arc, so angle ACB \(= 128^\circ \div 2 = 64^\circ\) (angle at the centre is twice the angle at the circumference). The tempting \(128^\circ\) stops at the angle at the centre. \(26^\circ\) halves angle APB, and \(52^\circ\) assumes angle ACB equals angle APB.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.