CHAPTER - Maths · PDF file474 CHAPTER 29 Circle geometry A line drawn from the centre of a circle perpendicular to a chord bisects the chord. AM BM The converse (opposite) of this

472

29.1 Parts of a circle The diagrams show the mathematical names for some parts of a circle.

● The circumference is the distance around the edge of a circle.

● A chord is a straight line segment joining two points on a circle.

● A diameter is a chord that passes through the centre of a circle.

● A radius is the distance from the centre of a circle to a point on the circle.

● A tangent is a line that touches the circle at only one point.

29.2 Isosceles trianglesTriangles formed by two radii and a chord are isosceles because they have two sides of equal length(the two sides that are radii). In an isosceles triangle, the angles opposite the equal sides are also equal.

A and B are points on the circumference of a circle, centre O.Angle OAB � 40°.

Calculate the size of angle AOB.Give reasons for your answer.

Solution 1

OA � OB

Angle OBA � 40°

In an isosceles triangle, the angles opposite the equal sides are equal.

Angle AOB � 180° � (40° � 40°)

� 100°

The angle sum of a triangle is 180°.

40°40°

100°

A

O

B

Example 1

Tangent

Chord

C ir

cu

mfe

r

en

c

e

Diameter

Radius

29C H A P T E R

Circle geometry

40°

A

O

B

OA and OB are radii.

Triangle OAB is isosceles.

Give the reason.

Add the equal angles and subtract the sum from 180°.

Give the reason.

At each step, mark the new information onthe diagram.

Exercise 29A

In questions 1–9 each diagram shows a circle, centre O.Calculate the size of each of the angles marked with a letter.The diagrams are NOT accurately drawn.

1 2 3

4 5 6

In Questions 7–9, give reasons for your answers.

7 8 9

29.3 Tangents and chordsHere are four geometric facts which involve tangents or chords.

● A tangent is perpendicular to the radius at the point of contact.

Angle OTP � 90°

Angle OTQ � 90°

● Tangents from an external point to a circle are equal in length.

PA � PB

42°

O

k

136°O

i

j

38°

O

h

70°

Of

g42°

30°e

O

270°

d

O

15°

cO

110°

b

O25°

aO

473

29.3 Tangents and chords CHAPTER 29

O

T

P

Q

O

A

B

P

474

CHAPTER 29 Circle geometry

● A line drawn from the centre of a circle perpendicular to a chord bisects the chord.

AM � BMThe converse (opposite) of this statement is also true.

● A line drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord.

PT is a tangent at T to a circle, centre O.TU is a chord of the circle.Angle PTU � 54°.

Find the size of angle TOU.Give reasons for your answer.

Solution 2

PA and PB are tangents to a circle.Angle APB � 68°.

Calculate the size of angle PAB.Give reasons for your answer.

Solution 3

PA � PB

Tangents from an external point to a circleare equal in length.

Angle PAB ��180°

2

� 68°�

� 56°

The angle sum of a triangle is 180° and inan isosceles triangle the angles oppositethe equal sides are equal.

68°

56°

56°

A

B

P

Example 3

Angle OTU � 90° � 54°

� 36°

Tangent is perpendicular to the radius.

Angle OUT � 36°

In an isosceles triangle, the angles opposite the equal sides are equal.

Angle TOU � 180° � (36° � 36°)

� 108°

Angle sum of a triangle is 180°.

O

U

T

P54°

108°

36°

36°

Example 2

O

A

M

B

O

U

T

P

54°

Subtract 54° from 90°.

Give the reason.

OT � OU.

Give the reasons.

Add the equal angles andsubtract the sum from 180°.

Give the reason.

68°

A

B

P

Give the reason.

Subtract 68° from 180°and divide the result by 2.Triangle PAB is isosceles.

Give the reasons.

Exercise 29B

The diagrams are NOT accurately drawn.

1 PT is a tangent at T to a circle, centre O.Angle POT � 37°.Find the size of angle a.Give reasons for your answer.

2 PA is a tangent at A to a circle, centre O.B is a point on the circumference of the circle.POB is a straight line.Find the size of each of the angles marked with letters.

a b

3 PA is a tangent at A to a circle, centre O.AB is a chord of the circle.Calculate the size of angles x and y.

a b

4 AB is a chord of a circle, centre O.M is the midpoint of AB.Angle BAO � 64°.Find the size of angle AOM.Give reasons for your answer.

5 PA and PB are tangents.Angle ABP � 61°.Calculate the size of angle APB.Give reasons for your answer.

34°A

B

Py

O

118°

A

B

Px

O

20°

A

B

y

P

O

40°

A

B

x

P

O

475

29.3 Tangents and chords CHAPTER 29

37°

aPT

O

64°

AB

M

O

61°

A

B

P

476


6 PA and PB are tangents to a circle, centre O.Find the size of angles x and y.

a b

7 PA is a tangent to the circle at A.AB is a diameter of the circle.D is a point on PB such that angle BAD � 72°.AP � AB.Calculate the size of angle PDA.

8 PA is a tangent to the circle, centre O.AB is a chord of the circle.Angle AOB � 152°.Angle APB � 71°.Find the size of angle PBA.

29.4 Circle theoremsTheorem 1 – the angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumferenceAngle AOB � 2 � angle ACB

ProofDraw the line CO and produce it to D.OA � OB � OC (radii).

Triangle OAC is isosceles so angle OAC � angle OCA � x (say).Triangle OBC is isosceles so angle OBC � angle OCB � y (say).Angle AOD � angle OAC � angle OCA (exterior angle of triangle), i.e. angle AOD � 2x.Similarly, angle BOD � 2y.

Angle AOB � 2x � 2y � 2(x � y) � 2 � angle ACB,Angle AOB � 2 � angle ACB.

A

B

C

D

xy

x y

2x 2y

O

78°

A

y

B

PO64°

A

x

B

OP

72°

A

BPD

152°

71°A

B

P

O

A

C

B

O

477

29.4 Circle theorems CHAPTER 29

P, Q and R are points on a circle, centre O.Angle PRQ � 41°.

Work out the size of angle POQ.Give a reason for your answer.

Solution 4Angle POQ � 2 � 41°

� 82°

The angle at the centre of a circle is twice the angle at the circumference.

Theorem 2 – the angle in a semicircle is a right angleAngle ACB � 90°

ProofThe angle subtended at O, the centre of the circle, by the arc AB is 180°, that is, angle AOB � 180°.

Angle AOB � 2 � angle ACB (angle at the centre of a circle is twice the angle at the circumference).

Angle ACB � �12� angle AOB

� �12� � 180°

� 90°

A, B and C are points on a circle.AB is a diameter of the circle.Angle BAC � 58°.

Work out the size of angle ABC.Give a reason for each step in your working.

Solution 5Angle ACB � 90°

The angle in a semicircle is a right angle.

Angle ABC � 180° � (90° � 58°)

� 180° � 148°

� 32°

The angle sum of a triangle � 180°.

Example 5

A

B

C

O

Example 4 R

P Q

41°

O

Double angle PRQ.

The reason may be shortened to this.

A

B

C

O

58°

A

BC

State the size of angle ACB.

Give the reason.

Add 90° and 58°.

Subtract the sum from 180°.

Give the reason.

478


Theorem 3 – angles in the same segment are equalAngle APB � angle AQB

ProofAngle APB � �

12� � angle AOB (angle at the centre of a circle

is twice the angle at the circumference).

Similarly, angle AQB � �12� angle AOB.

So angle APB � angle AQB.

A, B, C and D are points on a circle.Angle ADB � 63°.

Find the size of angle ACB.Give a reason for your answer.

Solution 6Angle ACB � 63°

The angles in the same segment are equal.

Cyclic quadrilateralsA quadrilateral whose vertices (corners) all lie on the circumference of a circle is called a cyclic quadrilateral.

The diagram below shows a cyclic quadrilateral PQRS.

Theorem 4 – the sum of the opposite angles of a cyclic quadrilateral is 180°Angle SPQ � angle SRQ � 180°and angle PSR � angle PQR � 180°.

ProofPQRS is a cyclic quadrilateral whose vertices lie on a circle, centre O.

Let angle SPQ � a, angle SRQ � b, angle SOQ � x and reflex angle SOQ � y.

Then x � 2a (angle at the centre of a circle is twice the angle at thecircumference).Similarly, y � 2b.

x � y � 360° (sum of angles at a point � 360°) so 2a � 2b � 360°.Dividing both sides by 2, a � b � 180°.

That is, angle SPQ � angle SRQ � 180°.

Also, angle PSR � angle PQR � 180° (the sum of the angles of a quadrilateral is 360°).

Example 6

A

BO

P

Q

State the size of angle ACB, whichis equal in size to angle ADB.

Give the reason.

P

Q

S

R

A

B

D

C63°

P

Q

S

Rb

ax

O

y

479


ABC is a straight line.B, C, D and E are points on a circle.Angle ABE � 81°.

Work out the size of angle CDE.Give a reason for each step in your working.

Solution 7Angle CBE � 180° � 81°

� 99°

The sum of angles on a straight line � 180°.

BCDE is a cyclic quadrilateral

so angle CDE � 180° � 99°

� 81°

The sum of opposite angles of a cyclic quadrilateral � 180°.

Notice that angle ABE � angle CDE.Angle ABE is an exterior angle of the cyclic quadrilateral and it is the same size as the oppositeinterior angle.

Theorem 5 – the angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segmentAngle PTB � angle BAT

ProofPTQ is a tangent to the circle at T.TB is a chord of the circle.Angle BAT is any angle in the alternate (opposite) segment to angle PTB.

Let angle PTB � x and angle BAT � y.

Draw the diameter TC.Angle CTB � 90° � x (tangent is perpendicular to a radius).Angle CBT � 90° (angle in a semicircle is a right angle).

In triangle CBT, 90° � 90° � x � angle BCT � 180° (angle sum of triangle).So angle BCT � x.

Angle BCT � angle BAT (angles in the same segment).That is x � y and angle PTB � angle BAT.

This theorem is known as the alternate segment theorem.

A, B and T are points on a circle.PTQ is a tangent to the circle.Angle PTB � 37°.Angle ATB � 68°.

Work out the size of angle ABT.Give a reason for each step in your working.

Example 8

Example 7

A B

C

D

E

81°


Give the reason.


Give the reason.

A

B

PT

Q

AC

B

PT

Qx

y

A

B

PT

Q37°

68°

480


Solution 8Method 1Angle PTB � angle BAT

� 37°

Alternate segment theorem.

Angle ABT � 180° � (37° � 68°)

� 75°

The angle sum of triangle � 180°.

Method 2Angle ATQ � 180° � (37° � 68°)

� 75°

The sum of angles on a straight line � 180°.

Angle ATQ � angle ABT� 75°

Alternate segment theorem.

ABCD is a cyclic quadrilateral.Angle ADB � 36°. Angle BDC � 47°.

a Find the size of i angle BAC ii angle ABC.Give reasons for your answers.

b Is AC a diameter? Explain your answer.

Solution 9a i Angle BAC � 47°

Angles in the same segment.

ii Angle ADC � 36° � 47°

� 83°

Angle ABC � 180° � 83°

� 97°

The sum of opposite angles of a cyclic quadrilateral � 180°.

b AC is not a diameter.If it were, angle ADC would be 90°(the angle in a semicircle) but it is 83°.

Example 9


Add 37° and 68°.


Give the reason.

Add 37° and 68°.


Give the reason.


AB

D

C

47°

36°


Add 36° and 47° to find the size of angle ADC.

Subtract 83° (the size of angle ADC) from 180°.

Give the reason.

The full answer consists of a statementand an explanation.

Exercise 29C

The diagrams are NOT accurately drawn.Dots show the centres of some of the circles.

In Questions 1–9, find the size of the angles marked with letters.Give a reason for each answer.

1 2 3

4 5 6

7 8 9

10 A, B and T are points on the circle.PT is a tangent to the circle at T.Angle PTB � 38°.AB � AT.

Work out the size of angle ABT.Give a reason for each step in your working.

11 A, B and C are points on a circle.Angle ABC � 28°.Angle BAC � 62°.

Is AB a diameter? Explain your answer.

mn

117°

j

k

l

64°

h

i

51°

g

f36°e

47°

d

77°

cb

48°a

74°

481


A

B

T

P

38°

62°

28°

A

B

C

12 A, B, C and D are points on a circle.Angle ABC � 76°.Angle ADB � 31°.

Work out the size of i angle BDC ii angle CAB.Give a reason for each step in your working.

13 a Is a rectangle a cyclic quadrilateral? Explain your answer.

b Is this quadrilateral cyclic? Explain your answer.

14 A, B, C and T are points on the circle.PTQ is a tangent to the circle.Angle PTC � 51°.Angle BAC � 23°.

Work out the size of angle BCT.Give a reason for each step in your working.

15 A, Q and R are points on the circle.PQ and PR are tangents to the circle.Angle QPR � 48°.

Work out the size of angle QAR.Give a reason for each step in your working.

Chapter summary

482


31°

76°

A

B

CD

98°

104°

75°

23°

51°

A

Q

T

P

C

B

48°

Q

R

P

A

You should know the meaning of:

● circumference

● chord

● diameter

● radius

● tangent

c i

rc

u

mfe

r

en

c

e

diameter

radius

tangent

chord

483

Chapter summary CHAPTER 29

You should now know these geometric facts and be able to use them:

a tangent is perpendicular to the radius at the point of contact

tangents from an external point to a circle are equal in length

a line drawn from the centre of a circle perpendicular to a chord bisects the chord

You should now know these geometric facts and be able to prove them:

the angle subtended by an arc at the centre of a circle is twice the angle subtended at the circumference

b � 2a

the angle in a semicircle is a right angle.

angles in the same segment are equal

a quadrilateral whose vertices (corners) all lie on the circumference of a circle is called a cyclic quadrilateral.The sum of the opposite angles of a cyclic quadrilateral is 180°.

a � c � 180° and b � d � 180°

�

�

�

�

�

�

�

a

b

a

b

cd

484


Chapter 29 review questionsThe diagrams are NOT accurately drawn.

1 P and Q are points on a circle, centre O.Angle POQ � 116°.

Work out the size of angle OPQ.Give reasons for your answer.

3 PT is a tangent at T to a circle, centre O.Angle OPT � 39°.

Work out the size of angle POT.Give reasons for your answer.

5 PQ is a chord of a circle, centre O.M is the midpoint of PQ.Angle POM � 57°.

Work out the size of angle OPM.Give reasons for your answer.

the angle between a chord and the tangent at the point of contact is equal to the angle in the alternate segment.

�

116°

Q

PO

47°

O

B

S

T

A

39°

OP

T

2 A and B are points on a circle, centre O.SBO and TBA are straight lines.Angle SBT � 47°.

Work out the size of angle AOB.Give reasons for your answer.

4 A and B are points on a circle.PA and PB are tangents to the circle.Angle APB � 54°.

Work out the size of angle PAB.Give reasons for your answer.

6 A, B, C and D are points on a circle centre O.Angle ADB � 38°.

a Give a reason why angle ACB � 38°.b i Find the size of angle AOB.

ii Give a reason for your answer.

54° P

A

B

PM

Q

O57°

A B

C

D

O

38°

485

Chapter 29 review questions CHAPTER 29

7 The diagram shows a circle with its centre at O.A, B, and C are points on the circumference of the circle.At C, a tangent to the circle has been drawn.D is a point on this tangent.Angle OCB � 24°.

a Find the size of angle BCD.Give a reason for your answer.

b Find the size of angle CAB.Give a reason for your answer. (1384 June 1995)

8 A, B, C and D are four points on the circumference of a circle.TA is a tangent to the circle at A.Angle DAT � 30°.Angle ADC � 132°.

a i Calculate the size of angle ABC.

ii Explain your method.

b i Calculate the size of angle CBD.

ii Explain your method.

c Explain why AC cannot be a diameter of the circle. (1385 June 2000)

9 A, B, C and D are points on a circle.AP and BP are tangents to the circle.Angle BAD � 80°.Angle BAP � 70°.

a Find the size of angle BCD,marked x° in the diagram.

b Find the size of angle APB.Give reasons for your answer.

c Find the size of angle DCA.Give reasons for your answer.

10 A, B, C and T are points on the circumference of a circle.Angle BAC � 25°.The line PTS is the tangent at T to the circle.AT � AP.AB is parallel to TC.

a Calculate the size of angle APT.Give reasons for your answer.

b Calculate the size of angle BTS.Give reasons for your answer. (1384 June 1997)

11 A, B, C and D are points on the circumference of a circle centre O.AC is a diameter of the circle.Angle BDO � x°.Angle BCA � 2x°.Express, in terms of x, the size of i angle BDA ii angle AOD iii angle ABD. (1385 November 1998)

O

C

D

BA

24°

Diagram NOTaccurately drawn

C

D

A T

B


30°

132°

C

DA

P

B


80°70°

x°

C

A

PST

B

Diagraaccura25°


C

B

D A

ODiagram NOTaccurately drawn

2x°

x°

486


12 The diagram shows a triangle ABC and a circle, centre O.A, B and C are points on the circumference of the circle.AB is a diameter of the circle.AC � 16 cm and BC � 12 cm.

a Angle ACB � 90°. Give a reason why.

b Work out the diameter AB of the circle.

c Work out the area of the circle.Give your answer correct to three significant figures. (1387 June 2005)

13 a Explain why angle OTP � 90°.

b Calculate the length of OT.Give your answer correct to three significant figures.

c Angle QOT � 36°.Calculate the length of OQ.Give your answer correct to three significant figures.

(4400 November 2004)

14 A, B and C are three points on the circumference of a circle.Angle ABC � Angle ACB.PB and PC are tangents to the circle from the point P.

a Prove that triangle APB and triangle APCare congruent.

Angle BPA � 10°.

b Find the size of angle ABC.

(1387 June 2004)

15 The diagram shows a circle with centre O and a triangle OPT.P is a point on the circumference of the circle and TP is a tangent to the circle.

a Angle OPT � 90°. Give a reason why.

The radius of the circle is 50 cm. TP � 92 cm.

b Calculate the length of OT.Give your answer correct to three significant figures.

c Calculate the size of the angle marked x°.Give your answer correct to three significant figures.

The region that is inside the triangle but outside the circle is shown shaded in the diagram.

d Calculate the area of the shaded region.Give your answer correct to two significant figures.

PT

Q

O Diagram NOTaccurately drawn

40°

36°6 cm

C B

A

O16 cm

12 cm


P

A

CB


P Ox°

T

50 cm

92 cm

P Ox°

T

50 cm

92 cm

Documents

CHAPTER - Maths · PDF file474 CHAPTER 29 Circle geometry A line drawn from the centre of a circle perpendicular to a chord bisects the chord. AM BM The converse (opposite) of this