Chapter 8 Circles III

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CHAPTER 8 : CIRCLES III8.1 TANGENTS TO A CIRCLE

8.2 ANGLES BETWEEN TANGENTS

AND CHORDS

8.3 COMMON TANGENTS

Created By: Mohd Said B Tegoh







Tangent

A B

O

A tangent to a circle is a straight line which

touches the circle at a point. The point is

called the point of tangency or the contact

point.

8.1 (a) Identifying Tangents to A Circle



Tangent

A B

O

The tangent to a circle is a straight line

perpendicular to the radius that passes

through the contact point.

8.1(a) Identifying Tangents to A Circle



A

P

Q

O

The lengths of the tangents from a givenpoint A to each contact points are equal,

AP = AQ

8.1(b) Properties of Tangents to A Circle



A

P

Q

O

The line that joins the given point outside

the circle, point A, to the centre of the circlebisects;

The angle between the two tangents, that is,

< PAO = < QAO



A

P

Q

O

OPA and OQA are congruent

The angle between the two radii that

passes through the contact points, that is

< POA = < QOA



(a) A cord is a line that connects two distinctpoints on a circle. The cord CD divides the

circle into two portions

(b) The smaller portion is called the minor

segment whereas the larger portion is called

the major segment

C DMinor segment

Major segment

8.2 Angles Between Tangents and Chords



A

B

H

The segment FHG (the major segment) is called the

alternate segment for < AGF. The angle in the

alternate segment which is subtended by the chord

FG is <FHG

G

F

8.2 (a) Identifying the angle in the alternate

segment



D E

A

a

b

p

q a = b

P = q

C

B

The angle between a tangent and a chord

through the point of contact is equal to

the angle subtended by the chord

alternate segment

8.2 (b) Calculations involving the angle in

the alternate segment



a

b

c

da = b

c = d

A B

C

8.2 (b) Calculations involving the angle in

the alternate segment



A Common Tangent to two circles is a

straight line that touches each of the

circles at only one point.

Common Tangent

8.3 COMMON TANGENTS



P

Q

RS

O C

T Two Common

Tangents

PQ = RS

TOC is a straight line

No. of common tangents

= 2.

(a) Two Circles which intersect

at Two Points

8.3 (b) Properties of Common Tangents



P Q

R S

O C

Two parallel common

tangents

PQ = RS

PQ, OC, and RS are parallels



P Q

R S

O C

Three Common

Tangents

PQ = RSXP = XQ = XY = YZ = ZR = ZS

(b) Two Circles which intersect at one point

X

Z

Y

OYC is a straight lineNo. of common tangents = 3, that are PQ, RS,

and XZ



P

Q

R

S

OC

T Three Common Tangents

PQ = RS

XP=XQ=XY=YZ=ZR=ZSNo. of Common Tangents = 3, that are PQ,RS, and XZ

TOYC is a straight line

X

Y

Z



(c) Two Circles which Do Not Intersect

P

Q

R

S

OC

T Four common

tangents

PQ = RSKL = MN

No. of common tangents = 4, that are PQ,RS, KL and MN

OXC is a straight line

K

N

ML

X



P Q

R S

O C

PQ = RSKL = MNOXC is a straight line

No. of common tangents = 4, that are PQ, RS, KL and MN

K

L

M

N

X



Q

R

T

S

x 0

DIAGRAM 3

In Diagram 3, P QR is a tangent to the circle at point Q.

Given that P QT = 56º. The length of the arc TQ is

equal to the length of the arc SQ.

P

C l o n e d S

P M



Q

R

T

S

x 0

DIAGRAM 3

P

560

560

56

0

x = 180 ² 56 - 56

= 68

Solution



DIAGRAM 4

In Diagram 4, LJK is a tangent to the circle JNM at J

and KMN is a straight line. Arc MN = Arc MJ

The value of x isA 15º

B 25º

C 35º

D 65º

K

M

N

x

55º

J

L

C l o n e d

S P M



K

M

N

x

55º

J

550

700

550

1100

x = 180 ² 110 - 55

= 15

700

Solution

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Chapter 8 Circles III

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