chapter 3 circles theorem notes f4

8/12/2019 Chapter 3 Circles Theorem Notes F4

1/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+7

Chapter 3 Circles Theorem

Section 3.1Circle Properties

! ()& !"#!$%&'#'(!'", 8)& 9",812$&1#*429 8)& &96& *: 1 $"#$%&0

! ; !)*#+ ", 1 ,8#1"6)8 %"2& ,&6+&28/*"2"26 8 *2& =*"280! ;2 ,#!*: 1 $"#$%& ", 12> =1#8 *: 8)& $"#$%&?, $"#$4+:&2$&0! ; .'!-*#", 1 #&6"*2 @*429&9 @> 8"26 @&8 1 $)*#9 129 12 1#$ %>"26 @&8


2/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

A

Section 3.2 Theorem 1 The angle subtended by the arc at the centre of the

circle is twice the angle subtended at the circumference

Subtended Angles

B: ;C D 129 E 1#& 8)#&& =*"28, *2 8)& $"#$4+:&2$& *: 1 $"#$%& 8)18

:B;= ;=4 @>A ,6 @1< ;=4 A=B>< ,6 89:;41< ;=4 @1234 ,56 @; ;=4

A7>A9CD4>41A4

()& 1#$ 129 $)*#9 1%,* ,4@8&29 8)& 126%& ;FD 18 8)& !"#$%"0

H*8& 8)18 1 $)*#9 $12 ,4@8&29 12 *@84,& 126%& 18

8)& $"#$4+:&2$&0

B2 ,4$) 1 $1,& 8)& 126%& ", "2 8)& +"2*# ,&6+&280

The angle subtended by an arc at the centre of a circleis twice the angle subtended at the circumference

Angle AOB = 2 !ACB


3/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

I

-=4 @1234 89:;414 BD @ A7>A34 :? @1 @>A 78 ;E7A4 @1? @1234 89:;41A9CD4>41A4 :? ;=4 8@C4 @>A

&'()*+" -

= J A K .L

= J MN O18 $&28#& J A K 18 $"#$P

&'()*+" .

= J A K 77Q

= J AIR O

18 $&28#& J A K

18 $"#$P

&'()*+" /

= J A K Q7

= J 7RA O18 $&28#& J A K 18 $"#$P


4/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

.

&'()*+" 0 &71< ;=4 F@394 BD @1234 G1

&'()*+" 2

&71< ;=4 F@394 BD @1234 HI

.9GGB>; 'H4>A784E6 .L7 SK$",& AM! H* 7


5/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

Q

Section 3.3Theorem 2: Angles in the same segment are equal

,12348 89:;41A @>4 4J9@3I

&'()*+" -

&71< ;=4 C788712 @12348 27F712 >4@8B18 DB> ?B9> @18E4>I

;26%& ;!D J T. OU, ,4@8&29&9 @> 8)& ,1+& $)*#9P

&'()*+" .

&71< ;=4 C788712 @12348 @1< 27F4 >4@8B18 DB> ?B9> @18E4>I

1 J .7 K A J LA O18 $&28#& J A K 18 $"#$P

@ J .7 OU, ,4@8&29&9 @> 8)& ,1+& $)*#9P

;26%& ;ED J ;26%& ;VD


6/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

N

&'()*+" /

&71< ;=4 C788712 @12348 @1< 27F4 >4@8B18 DB> ?B9> @18E4>I

1 J 7AR W A J NR O18 $&28#& J A K 18 $"#$P

@ J NR OU, ,4@ @> 8)& ,1+& $)*#9P

$ J IA OU, ,4@ @> 8)& ,1+& $)*#9P

.9GGB>; 'H4>A784E6 .L7 SK$",& AM! H* ACNCL

X129*48

Section 3.4 Theorem 3: The angle subtended by the diameter is a right angle

()", $12 @& 9"G&9 :#*+ 8)& =#&G"*4, 8)&*#&+0 Y"2$& 8)& 126%& 18 8)& $&28#& O7LRP ", 8


7/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

T

&'()*+" -

&'()*+" .


8/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

L

&'()*+" /

&71< ;=4 91O1BE1 @12348

&'()*+" 0

"1 ;=4 @C PQ 78 @ BD ;=4 A7>A34 @1< ,RP 78 (I

641 8@?8 ;=@; ;=4 F@394 BD(78 SLI/7F4 >4@8B18 ;B 4HG3@71 E=? =4 78 E>B12I

.9GGB>; 'H4>A784E6 .L7 SK$",& AM! H* IC T


9/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

M

Section 3.5 Theorem 4: The sum of the opposite angles of a cyclic

quadrilateral is 180

; Z419#"%18% 4@8B1 DB> ?B9> @18E4>I

;26%& YEV [ ;26%& Y\V J TUL]

;29

;26%& EY\ [ ;26%& EV\ J TUL]


10/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

7R

&'()*+" .

&71< ;=4 C788712 @1234 27F712 @ >4@8B1 DB> ?B9> @18E4>I

&'()*+" /

&71< ;=4 C788712 @1234 27F712 @ >4@8B1 DB> ?B9> @18E4>I


11/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

77

&'()*+" 0

&71< ;=4 @12348 C@>O4< 71 34;;4>8I

.9GGB>; 'H4>A784E6 .L7 SK$",& AM! H* .C M

X129*48

Section 3.6 Theorem 5: The angle between a radius and tangent form a right

angle

A tangentto a circle is a line which just touches the circle.

Remember:

A tangent is always at right angles to the

radius where it touches the circle.


12/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

7A

&'()*+" -

&'()*+" .

&'()*+" /


13/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

7I

&'()*+" 0

.9GGB>; 'H4>A784E6 .TQ SK$",& AMD H*, 7C AC I

X129*48

Section 3.7 Theorem 6: Tangents from the same external points to a circle

are equal in length

B: 8


14/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

7.

&'()*+" -

&'()*+" .


15/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

7Q

&'()*+" /

&'()*+" 0


16/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

7N

&'()*+" 2

.9GGB>; 'H4>A784E6 .TQ SK$",& AMD H*, QC N

X129*48

Section 3.8 Theorem 7: Alternate Segment Theorem

The angle between the tangent and chord at the point of contact is equal to the angle in the

alternate segment.

This is the circle property that is the most difficult to

spot. Look out for a triangle with one of its vertices

(corners) resting on the point of contact of the tangent.


17/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

7T

The angle between a tangent and

chord is equal to the angle made

by that chord in the alternate

segment.

In this diagram we can use the rule

to see that the yellow angles are

equal, and the blue angles are

equal.

Example 1

Find the angles marked with letters.

K J NR] ^;%8& Y&6+&28 ()&*#&+_

> J 7LR` NR` .RJ LR^;26%&, "2 1 8#"126%&_

&'()*+" .


18/19

!"#$%& ()&*#&+, -*#+ .

/0$1+&234%",+$56+1"%0$*+

7L

&'()*+" /

&'()*+" 0


19/19

chapter 3 circles theorem notes f4

Documents