Chapter 12 Angle Properties of a Circle

Recall O is the centre of circle OA = OB ( radius of Circle ) Major sector Major Arc AB Minor sector Minor Arc AB

Recall tangent chord diameter Centre Tangent is perpendicular to the diameter

Recall If AC = BC then  OCA =  OCB = 90  OA = OB  OAC =  OBC  OAB is an isosceles triangle If AB = CD, then OE = OF

Recall OA = OC (radius) AB and BC are the tangents  OAB =  OCB = 90  Hence AB = CB  OBA =  OBC  BOA =  BOC j B O C A

Introduction Major Segment Minor Segment The segment bounded by the arc APB and the chord AB is the minor segment of the circle. The segment bounded by the major arc AQB and the chord AB is the major segment of the circle. Chord AB

Angle at the circumference  ACB is subtended by the minor arc AB at the point C on the circumference.

Angle at the centre  AOB is the angle at the centre. Reflex  AOB is also the angle at the centre.

Time to work Skill Practice 12A Page 109 Q1a to e

Angles at the same segment  APB =  AQB Angles in the same segment are equal.

Relationship between Angle at centre and angle at circumference The angle subtended by an arc of a circle at the centre is twice that subtended by the same arc at any point on the remaining part of the circumference. If O is the centre,  AOB = 2  ACB  at centre = 2  at circumference 

 at centre = 2  at circumference  AOB = 2  ACB

 at centre = 2  at circumference reflex  AOB = 2  ACB  AOB = 2  ACB

Time to work Classwork SP 12A Pg 110 Q2 b, d, f Q3 a, c, e Q4 a, b Homework SP 12A Pg 110 Q2 a, c, e Q3 b, d, f

Angle in a semicircle The angle in a semicircle is a right angle. If AB is the diameter,  ACB = 90 .  in semicirle

Angles in opposite segments Angle in opposite segments are supplementary.  ABC +  ADC = 180   BCD +  BAD = 180 

Exterior angle of a cyclic quadrilateral The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.  ABC =  CDY  BCD =  BAX D O C B A X Y

Angle in the alternate segment ST is a tangent at A The angles formed by a tangent to a circle and a chord through its point of contact are equal to the angles in the alternate segments.  SAC =  ABC  BAT =  ACB A B C S T

Time to work Classwork SP12B Pg 113 Q1a, c Q2b, d Q3c,d Q4b,d Q5a, b Homework SP12B Pg 113 Q1b, d Q2a, c Q3a,b Q4a,c