Circle Geo

Maths Extension 1 – Circle Geometry

Circle Geometry
Properties of a Circle Circle Theorems: ! Angles and chords ! Angles ! Chords ! Tangents ! Cyclic Quadrilaterals

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Maths Extension 1 – Circle Geometry

Properties of a Circle

Radius Major Segment

Diameter

Chord Minor Segment

Tangent

Concyclic points form a Cyclic Quadrilateral

Sector

Arc

Tangents Externally and Internally Concentric circles

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Maths Extension 1 – Circle Geometry

Circle Theorems
l

! Equal arcs subtend equal angels at the centre of the circle. ! If two arcs subtend equal angles at the centre of the circle, then the arcs are equal.
O

θ θ

l = rθ

l

! Equal chords subtend equal angles at the centre of the circle.
||

! Equal angles subtended at the centre of the circle cut off equal chords.

θ
O

θ

||

A

||

B

θ
O

S OB = OC (radius of circle) A ∠BOA = ∠COD (vert. opp. Angles) S OA = OD (radius of circle) ∴ ∆BOA ≡ ∆COD (SAS) AB = DC (corresponding sides in ≡ ∆ ' s )

θ

D

|| C

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Maths Extension 1 – Circle Geometry

! A perpendicular line from the centre of a circle to a chord bisects the chord. ! A line from the centre of a circle that bisects a chord is perpendicular to the chord.
O

|

|

R ∠OMB = ∠OMA (straight line) H OB = OA (radius of circle) S OM = MO (common) ∴ ∆AOM ≡ ∆BOM (RHS) AM = BM (corresponding sides in ≡ ∆ ' s )
O

A

|

M

|

B

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Maths Extension 1 – Circle Geometry

! Equal chords are equidistant from the centre of the circle. ! Chords that are equidistant from the centre are equal.

O

||

A

N

R ∠ANO = ∠BMO = 90° (A line from the centre of a circle that bisects a chord is perpendicular to the chord) H AO = BO (Radius of Circle) S NO = MO (given) ∴ ∆ANO ≡ ∆BMO (RHS)
|| M

O

B... [continues]

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