CIRCLE THEOREMS. TANGENTS A straight line can intersect a circle in three possible ways. It can be: A DIAMETERA CHORD A TANGENT 2 points of intersection.

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Presentation transcript:

CIRCLE THEOREMS

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

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

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

O A B P 6 cm 8 cm AP is a tangent to the circle. a Calculate the length of OP. b Calculate the size of angle AOP. c Calculate the shaded area. c Shaded area = area of ΔOAP – area of sector OAB a b Example

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

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

SYMMETRY PROPERTIES OF CHORDS 2 O A B 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 If AB = CD then OP = OQ. C D ΙΙ P Q Ι Ι AB = CD

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

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

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

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

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

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

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

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

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

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

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

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