Chapter 12 Angle Properties of a Circle

1
Chapter 12 Angle Properties of a Circle
2
Recall

• O is the centre of circle
• OA OB ( radius of Circle )

Major Arc AB
Major sector
Minor sector
Minor Arc AB
3
Recall
diameter
chord
Centre
tangent
Tangent is perpendicular to the diameter
4
Recall

• If AC BC
• then ?OCA ?OCB 90?
• OA OB
• ?OAC ?OBC
• ?OAB is an isosceles
• triangle

If AB CD, then OE OF
5
Recall

• OA OC (radius)
• AB and BC are the tangents
• ?OAB ?OCB 90?
• Hence AB CB
• ?OBA ?OBC
• ?BOA ?BOC

6
Introduction

• 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.

Major Segment
Minor Segment
Chord AB
7
Angle at the circumference
?ACB is subtended by the minor arc AB at the
point C on the circumference.
8
Angle at the centre

• ?AOB is the angle at the centre.
• Reflex ?AOB is also the angle at the centre.

9
Time to work ?

• Skill Practice 12A
• Page 109
• Q1a to e

10
Angles at the same segment

• ?APB ?AQB
• Angles in the same segment are equal.

11
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?

12
? at centre 2? at circumference
?AOB 2?ACB
?AOB 2?ACB
13
? at centre 2? at circumference
?AOB 2?ACB
reflex ?AOB 2?ACB
14
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

15
Angle in a semicircle

• The angle in a semicircle is a right angle.
• If AB is the diameter, ?ACB 90?.
• ? in semicirle

16
Angles in opposite segments

• Angle in opposite segments are supplementary.
• ?ABC ?ADC 180?
• ?BCD ?BAD 180?

17
Exterior angle of a cyclic quadrilateral

• The exterior angle of a cyclic quadrilateral is
  equal to the interior opposite angle.
• ?ABC ?CDY
• ?BCD ?BAX

18
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

19
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