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**Draw and label on a circle:**

Centre Radius Diameter Circumference Chord Tangent Arc Sector (major/minor) Segment (major/minor)

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**Circle Fact 1. Isosceles Triangle**

Any triangle AOB with A & B on the circumference and O at the centre of a circle is isosceles. r a a r a

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**Circle Fact 2. Tangent and Radius**

The tangent to a circle is perpendicular to the radius at the point of contact.

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**Circle Fact 3. Two Tangents**

The triangle produced by two crossing tangents is isosceles.

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Circle Fact 4. Chords If a radius bisects a chord, it does so at right angles, and if a radius cuts a chord at right angles, it bisects it.

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**Circle Theorem 1: Double Angle**

The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.

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**Circle Theorem 2: Semicircle**

The angle in a semicircle is a right angle.

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**Circle Theorem 3: Segment Angles**

Angles in the same segment are equal.

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**Circle Theorem 4: Cyclic Quadrilateral**

The sum of the opposite angles of a cyclic quadrilateral is 180o.

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**Circle Theorem 5: Alternate Segment**

The angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.

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**Circle Theorem 1: Double Angle**

Proof The angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference.

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b b 2a + 2b = 2(a + b) b 180 – 2a a a

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**Circle Theorem 2: Semicircle**

Proof The angle in a semicircle is a right angle.

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b 360 – 2(a + b) = 180 180 = 2(a + b) 180 – 2b 90 = (a + b) 180 – 2a b a a

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**Circle Theorem 3: Segment Angles**

Proof Angles in the same segment are equal.

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a a 2a

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**Circle Theorem 4: Cyclic Quadrilateral**

Proof The sum of the opposite angles of a cyclic quadrilateral is 180o.

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a 2a + 2b = 360 2(a + b) = 360 2b a + b = 180 2a b

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**Circle Theorem 5: Alternate Segment**

Proof The angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.

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90 - a a 180 – 2(90 – a) 2a 180 – a 90 - a a

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**1 2 3 4 5 Double Angle Semicircle Segment Angles Cyclic Quadrilateral**

Alternate Segment

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