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CIRCLE THEOREMS

CIRCLE THEOREMS. TANGENTS. A straight line can intersect a circle in three possible ways. It can be:. A TANGENT. A DIAMETER. A CHORD. B. O. O. O. B. A. A. A. 2 points of intersection. 2 points of intersection. 1 point of intersection. TANGENT PROPERTY 1.

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CIRCLE THEOREMS

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  1. CIRCLE THEOREMS

  2. TANGENTS A straight line can intersect a circle in three possible ways. It can be: A TANGENT A DIAMETER A CHORD B O O O B A A A 2 points of intersection 2 points of intersection 1 point of intersection

  3. TANGENT PROPERTY 1 The angle between a tangent and a radius is a right angle. O A

  4. TANGENT PROPERTY 2 B ΙΙ O P ΙΙ The two tangents drawn from a point P outside a circle are equal in length. AP = BP A

  5. Example AP is a tangent to the circle. aCalculate the length of OP. bCalculate the size of angle AOP. cCalculate the shaded area. P B O 8 cm a b 6 cm A cShaded area = area of ΔOAP – area of sector OAB

  6. CHORDS AND SEGMENTS A straight line joining two points on the circumference of a circle is called a chord. A chord divides a circle into two segments. major segment minor segment

  7. SYMMETRY PROPERTIES OF CHORDS 1 The perpendicular line from the centre of a circle to a chord bisects the chord. O Note: Triangle AOB is isosceles. ΙΙ ΙΙ A B

  8. SYMMETRY PROPERTIES OF CHORDS 2 AB = CD D ΙΙ Q If two chords AB and CD are the same length then they will be the same perpendicular distance from the centre of the circle. ΙΙ Ι C O Ι If AB = CD then OP = OQ. ΙΙ ΙΙ A B P

  9. Example Find the value of x. O Triangle OAB is isosceles because OA = OB (radii of circle) So angle OBA = x. B A

  10. THEOREM 1 The angle at the centre is twice the angle at the circumference. O

  11. Example Find the value of x. O Angle at centre = 2 × angle at circumference

  12. Example Find the value of x. O Angle at centre = 2 × angle at circumference

  13. Example Find the value of x. O Angle at centre = 2 × angle at circumference

  14. Example Find the value of x. O Angle at centre = 2 × angle at circumference

  15. THEOREM 2 An angle in a semi-circle is always a right angle. O

  16. Example Find the value of x. O Angles in a semi-circle = 90o and angles in a triangle add up to 180o.

  17. THEOREM 3 Opposite angles of a cyclic quadrilateral add up to 180o.

  18. Example Find the values of x and y. Opposite angles in a cyclic quadrilateral add up to 180o.

  19. THEOREM 4 Angles from the same arc in the same segment are equal.

  20. Example Find the value of x. Angles from the same arc in the same segment are equal.

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