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Draw and label on a circle: Centre Radius Diameter Circumference Chord Tangent Arc

Draw and label on a circle: Centre Radius Diameter Circumference Chord Tangent Arc Sector (major/minor) Segment (major/minor). 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. 180 - 2a. r. a.

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Draw and label on a circle: Centre Radius Diameter Circumference Chord Tangent Arc

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  1. Draw and label on a circle: Centre Radius Diameter Circumference Chord Tangent Arc Sector (major/minor) Segment (major/minor)

  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 180 - 2a r a

  3. Circle Fact 2. Tangent and Radius The tangent to a circle is perpendicular to the radius at the point of contact.

  4. Circle Fact 3. Two Tangents The triangle produced by two crossing tangents is isosceles.

  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.

  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.

  7. Circle Theorem 2: Semicircle The angle in a semicircle is a right angle.

  8. Circle Theorem 3: Segment Angles Angles in the same segment are equal.

  9. Circle Theorem 4: Cyclic Quadrilateral The sum of the opposite angles of a cyclic quadrilateral is 180o.

  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.

  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.

  12. b 180 - 2b 2a + 2b = 2(a + b) b 180 – 2a a a

  13. Circle Theorem 2: Semicircle Proof The angle in a semicircle is a right angle.

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

  15. Circle Theorem 3: Segment Angles Proof Angles in the same segment are equal.

  16. a a 2a

  17. Circle Theorem 4: Cyclic Quadrilateral Proof The sum of the opposite angles of a cyclic quadrilateral is 180o.

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

  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.

  20. 90 - a a 180 – 2(90 – a) 2a 180 – 180 + 2a 90 - a a

  21. Double Angle Semicircle 1 2 3 Segment Angles 4 5 Cyclic Quadrilateral Alternate Segment

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