1. Draw and label on a circle:
Centre
Radius
Diameter
Circumference
Chord
Tangent
Arc
Sector (major/minor)
Segment (major/minor)
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2. 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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3. 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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4. Circle Fact 3. Two Tangents
The triangle produced by two crossing tangents is isosceles.
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5. 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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6. 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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7. Circle Theorem 2: Semicircle
The angle in a semicircle is a right angle.
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8. Circle Theorem 3: Segment Angles
Angles in the same segment are equal.
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9. Circle Theorem 4: Cyclic Quadrilateral
The sum of the opposite angles of a cyclic quadrilateral is 180o.
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10. 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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11. 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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12. b b
2a + 2b
= 2(a + b) b
180 – 2a a a
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13. Circle Theorem 2: Semicircle
Proof
The angle in a semicircle is a right angle.
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14. b 360 – 2(a + b) = 180 180 = 2(a + b) 180 – 2b 90 = (a + b) 180 – 2a b
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15. Circle Theorem 3: Segment Angles
Proof
Angles in the same segment are equal.
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16. a a 2a
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17. Circle Theorem 4: Cyclic Quadrilateral
Proof
The sum of the opposite angles of a cyclic quadrilateral is 180o.
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18. a
2a + 2b = 360
2(a + b) = 360 2b
a + b = 180 2a b
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19. 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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20. 90 - a a
180 – 2(90 – a) 2a
180 – a
90 - a a
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21. 1 2 3 4 5 Double Angle Semicircle Segment Angles Cyclic Quadrilateral
Alternate Segment
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