Circle Geometry
Terminology
Diameter
Sector
Chord
Segment
Theorem 1
90°
1A The perpendicular drawn from the centre of a circle to a cord, bisects the cord.
Reason: Line from centre circle meets the midpoint of the chord
1B The perpendicular bisector of a chord passes through the centre of the circle.
Reason: ⊥ bisector
Converse: The line segment joining the centre of the circle to the midpoint of a cord, is
perpendicular to the chord.
Reason: Line from centre circle ⊥ to chord
Theorem 2
Ɵ
2Ɵ
Subtended by the
same arc or chord
, The angle which an arc or chord subtends at the centre of the circle is twice the angle it
subtends at the circumference of the circle.
Reason: Central angle theorem
Theorem 3
90°
The angle subtended at the circle by the diameter is a right angle.
Reason: Angles in semicircle
Converse: If the angle subtended by a chord at a point on the circle is 90°, then the chord is the
diameter.
Theorem 4
Ɵ
Ɵ
An arc or chord of a circle subtends equal angles at the circumference of the circle.
Reason: Angles in same segment
Converse: If a line segment joining two points subtends equal angles at two other points on the
same side of the line segment, then these 4 points lie on the circumference.
Corollaries of theorem:
1. Equal chords subtend equal angles at the circumference
2. Equal chords subtend equal angles at the centre
3. Equal chords of equal circles subtend equal angles at the circumference
Terminology
Diameter
Sector
Chord
Segment
Theorem 1
90°
1A The perpendicular drawn from the centre of a circle to a cord, bisects the cord.
Reason: Line from centre circle meets the midpoint of the chord
1B The perpendicular bisector of a chord passes through the centre of the circle.
Reason: ⊥ bisector
Converse: The line segment joining the centre of the circle to the midpoint of a cord, is
perpendicular to the chord.
Reason: Line from centre circle ⊥ to chord
Theorem 2
Ɵ
2Ɵ
Subtended by the
same arc or chord
, The angle which an arc or chord subtends at the centre of the circle is twice the angle it
subtends at the circumference of the circle.
Reason: Central angle theorem
Theorem 3
90°
The angle subtended at the circle by the diameter is a right angle.
Reason: Angles in semicircle
Converse: If the angle subtended by a chord at a point on the circle is 90°, then the chord is the
diameter.
Theorem 4
Ɵ
Ɵ
An arc or chord of a circle subtends equal angles at the circumference of the circle.
Reason: Angles in same segment
Converse: If a line segment joining two points subtends equal angles at two other points on the
same side of the line segment, then these 4 points lie on the circumference.
Corollaries of theorem:
1. Equal chords subtend equal angles at the circumference
2. Equal chords subtend equal angles at the centre
3. Equal chords of equal circles subtend equal angles at the circumference
