The big picture
Most rules here start from one point: the centre.
| Idea | Remember | Example |
|---|---|---|
| The four names | Chord straight, arc curved, segment one chord, sector two radii. | A slice from the centre is a sector. |
| Chord and centre | Take a chord that is not a diameter. A 90° line from the centre meets it at its middle. | 30 cm stick, 17 cm radius: 8 cm out. |
| Equal chords | Same distance out, same length. Further out, shorter. | 7 cm out gives 48 cm; 24 cm out gives 14 cm. |
| Twice at the centre | The centre angle is double the edge angle on the same arc. | 100° at the centre, 50° at the rim. |
| Same segment | One arc, one fixed angle, everywhere on it. | P and Q on one arc both read 50°. |
| On a diameter | 180° at the centre gives exactly 90° at the rim. | AB a diameter: angle ACB is 90°. |
| Four on one circle | Opposite angles of a cyclic quadrilateral total 180°. | 85° and 95° are opposite: 85 + 95 = 180. |
| Concyclic test | Take the circle through three points, measure the fourth. Or add opposite angles. | (2, 1.5), radius 2.5: the fourth is 2.5 out. |
| Circumcentre | Two perpendicular bisectors cross at the centre. | A right angle puts it on the longest side. |
| Two points | A circle passes through A and B. Its centre lies on the perpendicular bisector of AB. | Centres (0, 0) and (0, 4): radii 3 and 5. |
| Turning | A circle looks the same after any turn about its centre. Every diameter is a mirror line. | A square fits after 90°, a hexagon after 60°. |
| Equal angles | In one circle, equal angles at the centre give equal chords. | 90° in a circle of radius 10 cm: a chord of 14.1 cm. |
| One side | AB makes equal angles at C and D. C and D are on the same side of line AB. Then A, B, C and D lie on one circle. | A = (0, 0), B = (8, 0): 53.1° at C and at D. |
This is Chapter 5. It is called I'm Up and Down, and Round and Round.
The centre matters in most of these ideas. Some measure a length from the centre. Some measure an angle at the centre.