CIRCLE THEOREMS

TANGENTSA straight line can intersect a circle in three possible ways.It can be:A DIAMETERA CHORDA TANGENT2 points of intersection2 points of intersection1 point of intersectionABOOOABA

TANGENT PROPERTY 1OA

TANGENT PROPERTY 2OAPB

OABP6 cm8 cmAP 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 OAP area of sector OABabExample

CHORDS AND SEGMENTSmajor segmentminor segmentA 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 1OAB

SYMMETRY PROPERTIES OF CHORDS 2OABCDPQAB = CD

OFind the value of x.Triangle OAB is isosceles because OA = OB (radii of circle)ExampleABSo angle OBA = x.

THEOREM 1O

OFind the value of x.Angle at centre = 2 angle at circumferenceExample

OFind the value of x.Angle at centre = 2 angle at circumferenceExample

OFind the value of x.Angle at centre = 2 angle at circumferenceExample

OFind the value of x.Angle at centre = 2 angle at circumferenceExample

THEOREM 2O

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

THEOREM 3

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

THEOREM 4

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