Chapter 16- Year 9 Maths

Chapter 16Chapter 16

This chapter deals with applying reasoning to prove circle theorems and to solve problems.

After completing this chapter you should be able to:

Circle GeometryCircle Geometry

✓ identify and name parts of a circle

✓ use terminology associated with angles in circles

✓ identify the arc on which an angle at the centre or circumference stands

✓ demonstrate that at any point on a circle there is a unique tangent and that this tangent is perpendicular to the radius at the point of contact

✓ prove and apply theorems involving chord, angle, tangent and secant properties of circles.

Sy l labus re ference SGS5.3 .4 WM: S5 .3 .1–S5.3 .5

566 Circle Geometry (Chapter 16) Syllabus reference SGS5.3.4

Diagnostic testDiagnostic test

Which of the following statements is not correct?

A A is a sector of a circle

B B is a chord C C is a secant

D D is a tangent

The angle subtended at the circumference by the arc PQ is:

A ∠SQT

B ∠POQ

C ∠PSQ

D ∠QTS

The values of the pronumerals in the figure are:

A x = 50, y = 10

B x = 50, y = 6

C x = 65, y = 10

D x = 65, y = 6

The length of the chord AB is:

A 4 cm

B 6 cm

C 8 cm

D 10 cm

The line joining the centres of the circles intersect the common chord AB at X.OA = 10 cm, XB = 8 cm and ∠XBC = 43°. The values of x and y are:

A x = 43, y = 12 B x = 47, y = 6

C x = 43, y = 8 D x = 47, y = 8

The value of x is:

A 32

B 64

C 116

D 148

The value of x is:

A 35

B 50

C 65

D 100

AOB is a diameter. AB = 17 cm and BC = 15 cm. The length of AC is:

A 15 cm

B 17 cm

C 2 cm

D 8 cm

1

AB C

D

2

S

Q

P

T

O

3

O

50°

10 cm

10 cm

x°

y cm

6 cm

4

O 3 cm

A

B

2 cm

5

OC

A

B

10 cm

8 cmx° X43°

y cm

6

O x°

32°

7

x°100°

65°

8

O

A

C

B

17 cm

15 cm

567Circle Geometry (Chapter 16) Syllabus reference SGS5.3.4

The values of x and y are:

A x = 84, y = 116

B x = 42, y = 58

C x = 168, y = 232

D x = 96, y = 64

x =

A 35

B 55

C 70

D 110

PT is a tangent to the circle with centre O. The value of x is:

A 10

B 40

C 50

D 70

PQ and PR are the tangents drawn to the circle from P. O is the centre. The size of ∠PQR is:

A 40°

B 50°

C 65°

D 90°

TP is a tangent. The value of x is:

A 20

B 40

C 60

D 80

The two circles touch externally at Q.C and O are the centres of the circles.The size of ∠PQR is:

A 80°

B 90°

C 100°

D 190°

PQ and RS are chords that intersect at T. The length of the interval TQ is:

A 8 cm

B 7 cm

C 4.5 cm

D 2 cm

PT is a tangent. The value of x is:

A 3

B 4

C 6

D 20

The value of x is:

A 5

B 9

C 14

D 18

9

84°116°

x°y°

10

2x°

110°

11

x°40° O

P

T

12

50° P

Q

R

O

13

PT

60°40°

x°

14

60°40°

O

P

R

Q

C

15

6 cm 4 cm

3 cm

T

PS

Q

R

16

R

T

S P4 cm5 cm

x

17

R

T

S

QP

6 cm

4 cm

6 cm

x cm

If you have any difficulty with these questions, refer to the examples and questions in the sections listed in the table.

Question 1, 2 3–5 6–10 11–17

Section A C D E


A. CIRCLE TERMINOLOGY

Exercise 16A

a Draw diagrams and identify the following parts of a circle:• centre • radius • diameter • circumference• arc • minor arc • major arc • semicircle • sector • minor sector • major sector • chord• segment • minor segment • major segment • tangent• secant

b Share your diagrams with others in a group and make a summary in your exercise book.

Given interval AB and point P, join A to P and B to P to form ∠APB.

Then ∠APB is the angle subtended by the interval AB at the point P.

In a circle:

1 draw a chord AB and mark a point P on the circumference

2 join A to P and B to P to form ∠APB.

Then ∠APB is the angle subtended by the chord AB at the point P. Since P is also on the circumference we say that ∠APB is an angle subtended by the chord AB at the circumference.

Name two angles subtended by:a chord AB at the circumferenceb chord XY at the circumferencec chord AY at the circumferenced chord BX at the circumference

Copy the diagram into your exercise book. O is the centreof the circle. Draw and name the angle subtended at thecentre by the chord:a ABb CDc EF

Name the chord that subtends the angle:a EOD b CABc ABC d EFAe EDC

1

P

A

B

AB

P

2

A

B

ZY

X

3

E

FCD

B

A

O

4

A

B

C

DE

F

O


Name the angle subtended by the chord AB at thecircumference in:a the major segmentb the minor segment

Name two angles subtended by the chord XYat the circumference:a in the major segmentb in the minor segment

1 Draw a circle, mark an arc AB, the centre Oand a point P on the circumference.

2 Draw ∠APB and ∠AOB.

∠APB is the angle at P standing on the arc AB. (This is the same as the angle subtended by the chord AB at P.)

∠AOB is the angle at the centre O standing on the arc AB. (Again this is the same as the angle subtended bychord AB at O.)

a Name the angle at the centre standing on the arc PS.b Name an angle at the circumference standing on the arc PS.c Name two angles standing on the arc QR.

a Name two angles subtended at the circumference by the minor arc AB.

b Name two angles standing on the major arc AB.c Name two angles in the same segment

as ∠AQB, standing on the arc AB.d Name an angle in the same segment as

∠ATB, standing on the arc AB.

Identify the minor or major arc on which each of the following angles stands:a ∠BECb ∠EDC c ∠ABEd ∠ECBe ∠EDA

5

A

P

B

Q

O

6

P Q

Y

RS

X

O

AB

PO

7 Q

P

SR

O

8

O

AT S

B

RQ

P

9

O

A

B

CD

E


B. CHORD PROPERTIES OF A CIRCLE

Exercise 16B

a In a circle draw two chords, AB and CD, of the same length. Join the endpoints of each chord to the centre O, as shown in the figure. Measure the size of the angles AOB and COD.

b Using a set square, measure the distance of each chordfrom the centre.

c Compare your results with those of the rest of the class.

Complete the following proofs.

a Data: AB = CD, O is the centre of the circle.Aim: To prove ∠AOB = ∠COD.Proof: In ∆s AOB, COD:

AB = CD (data)AO = __ ( )BO = __ ( )

∴ ∆AOB ≡ ∆COD ( )Hence ∠AOB = __ ( )

Also ∠ABO = ∠ODC (corresponding angles of ≡ ∆s). We will need this inpart b.

b Data: AB = CD and OP, OQ are perpendicularsfrom O to AB, CD respectively.

Aim: To prove OP = OQ.Proof: In ∆s OPB, OQD

∠OPB = ∠OQD ( )∠OPB = __ (from part a above)

OB = __ ( )∴ ∆OBP ≡ ∆ODQ ( )

∴ OP = OQ ( )

From parts a and b we can conclude that:Chords of equal length in a circle subtend equal angles at the centre and are equidistant from the centre.

1

O

A

BD

C

O

2

A

B

C

D

O

O

A

B

C

QP

D


Draw any chord AB in a circle with centre O. Using a setsquare, draw the perpendicular AP from O to the chord AB.Measure the lengths of AP and PB. Compare your resultswith the class.

a O is the centre of the circle andOP ⊥ chord AB.i Prove that ∆OAP ≡ ∆OBP.ii Hence, deduce that PA = PB, i.e.

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

b Prove the converse of the theorem in part a,i.e. given M is the midpoint of chord AB, prove that OM ⊥ AB, i.e.the line from the centre of a circle to the midpoint of a chord is perpendicular to the chord.

c From parts a and b above it follows that O (the centre) is on the perpendicular bisectorof the chord AB, i.e.the perpendicular bisector of a chord passes through the centre of the circle.

a P is a point on the perpendicular bisector of AB.i Prove ∆APM ≡ ∆MPB.ii PA = PB.

b Complete the following.Data: A, B, C are three non-collinear points.

PQ and MN are the perpendicular bisectors of AB and BC respectively. X is the point of intersection of PQ and MN.

Aim: To prove that X is the centre of the circle which passes through A, B and C.

Proof: Since X is on the perpendicular bisector of AB and BC, then XA = and XB = i.e. XA = = .∴ A, B, C are equidistant from so X is the centre of the circle that passes through

, i.e. given any three non-collinear points, the point of intersection of the perpendicular bisectors of any two sides of the triangle formed by the three points is the centre of a circle through all three points.

O

A

P

B

3

4

O

A

P

B

O

AM

B

5

A M B

P

A

B

C

M

N

X

Q

P


a Copy the circle and mark any three points, A, B and C on the circumference.

b Construct the perpendicular bisectors of AB and BC.c Mark the position of O, the centre of the circle.

Copy the following circles and, by construction, find the centre of each one.

a b

Draw any two intersecting circles. Join the centresP and Q. Draw the common chord AB.a Measure the lengths of XA and XB.b Measure the angle AXQ.c Discuss your results with the class.

Data: P and Q are the centres of the circles.AB is the common chord.X is the point of intersection of PQ and AB.

a i Prove that ∆PAQ ≡ ∆PBQ.ii List all the equal angles in these two triangles.

b i Prove that ∆XAQ ≡ ∆XBQ.ii Hence prove that AX = XB and ∠AXQ = ∠BXQ = 90°, i.e.

when two circles intersect, the line joining their centres bisects their common chord at right angles.

6

A

C

B

7

8

X

P Q

A

B

9

X

P Q

A

B


C. USING CHORD PROPERTIES

Exercise 16C

Find the values of the pronumerals in these circles. O is the centre.a b c

d e f

g h i

Example 1

Find the value of the pronumerals in the figures below.a b c

a x = 54 (equal chords subtend equal angles at the centre)b x = 3 (equal chords are equidistant from the centre)c ∠AOC = x° (equal chords subtend equal angles at the centre)

∴ x° + x° + 130° = 360° (angles at a point)∴ x = 115

Ox°

54°

7 cm

7 cm10 cm

O

10 cm3 cm

x cmO x°130°

A

BC

15 cm 15 cm

1

O

6 cm

6 cm

50°

x° O

40 cm

40 cm70°

x° O10 m 10 m

110°x°

O20° 20°30 m

x

O

x

4 cm

5 cm

6 cm

z

y

O

x

5 cm

3 cm

3 cm

O15 m 15 m

12 m

x O

x°

O

x°

☞


i k l

m n

Find the values of the pronumerals.a P and Q are the centres of the circles.

b P and Q are the centres.

O

14 cm

x

O

x

13 cm12 cm x O

20 cm

O

x

3 cm8 cm

O

70°

x°

Example 2

P and Q are the centres.∠PAX = 60°, AX = 5 cm and XQ = 12 cm.Find the value of the pronumerals.

∠AXP = 90° (The line joining the centres ofintersecting circles bisects the common chordat right angles.) ∴ z = 30 (angle sum of ∆ = 180°)

XB = XA = 5 cm ∴ y2 = 52 + 122 (Pythagoras’)

= 169y = 13

QP

A

B

12 cm

y cm

z°x

60° 5 cm

2

y6 cm

QP8 cmx°

40°

xQ P

13 cm5 cm


AN = 18 cm and NB = 8 cm. Find the length of PQ.

The figure shows two concentric circles, centre O.OL = 13 cm, OK = 15 cm and LM = 10 cm. Find KN.

A hemispherical bowl has a diameter of 10 cm.Water is poured into the bowl until its surface is3 cm below the top of the bowl as shown. Find thediameter of the water surface.

OP is perpendicular to the chord AB. If AB = 14 cm andPQ = 1 cm, find the radius of the circle.

If a chord of length 100 mm is 120 mm from the centre of a circle, find the radius of the circle.

A chord 8 cm long is drawn in a circle of radius 6 cm. What is the distance of the chord from the centre?

D. ANGLE PROPERTIES OF A CIRCLE

Exercise 16D

O is the centre of the circle in all diagrams.

In each of the following diagrams:i state the size of the angle at the circumference standing on the arc AB (i.e. ∠ACB)ii find the size of the angle at the centre standing on the arc AB (i.e. ∠AOB)

3

NOA

P

B

Q

4 KL

M

N

O

5 3 cm

6

AQ

B

P

O

7

8

1

☞


a b c d

a Draw a circle and mark the position of the centre Oand any three points A, B and P on thecircumference, as shown.

b Measure the size of the angle subtended at thecentre by the arc AB.

c Measure the angle subtended at P by the arc AB.d Discuss your results with the class.

Complete the following general proofs.a Data: O is the centre of the circle.

Aim: To prove ∠AOB = 2 × ∠ACB.Proof: Let ∠ACO = x° and ∠OCB = y°.

In ∆AOC, ∠OAC = ∠OCA ( )= x°

Now, ∠POA = + (exterior angle of ∆ = sum of interior opposite angles)

= 2x°Similarly ∠POB = 2y°

∴ ∠AOB = 2x + 2y= 2( )= 2 × ∠ACB, i.e.

the angle at the centre of a circle is double the angle at the circumference, standing on the same arc.

b Data: O is the centre of the circle.Aim: To prove ∠AOB = 2 × ∠ACB.Proof: Let ∠OCA = x° and ∠OCB = y°.

Show that ∠POA = 2x° and ∠POB = 2y°.Now ∠AOB = ∠POB – ∠POA

= = 2(y – x)

But ∠ACB = ∴ ∠AOB = 2 × ∠ACB, i.e.

the angle at the centre of a circle is double the angle at the circumference, standing on the same arc.

35°

A

B

C

O30°

23°

AP

B

C

O

18°

26°A

B

C

P O65°

56°

P

A

B

C

O

2 P

AB

O

3

x°y°

AP

B

C

O

A

B

C

P Ox°

y°


Find x, using the results of question 3.a b c

d e f

g h

Find x in each of the following (give reasons).a b c d

Find x, using the results of question 3.

a b c

a x = 2 × 35 b 88 = 2 × x c 240 = 2 × x= 70 ∴ x = 44 ∴ x = 120

Ox°

35°O x°

88°O

240°

x°

Example 1

4

48°

x°O x°O

23°

35°

x°O

110°

O

x° O86°

x°

O

x°

100°

Ox°

72°O x°256°

5

Ox°

30°

O52°

x°

130°

x°

O

68°O

x°


a Draw a circle and mark the position of any fourpoints A, B, P and Q, as shown.i Measure the angle subtended at P by the arc AB.ii Measure the angle subtended at Q by the arc AB.iii Discuss your results with the class.

b Complete the following proof.Data: O is the centre of the circle.Aim: To prove ∠APB = ∠AQB.Proof: ∠APB = × ∠AOB ( )

∠AQB = × ( )∴ ∠APB = ∠AQB, i.e.

angles at the circumference of a circle, standing on the same arc, are equal.This result is often stated as: angles in the same segment of a circle are equal.

Find the values of the pronumerals.a b c

6P

AB

Q

12---

O

A B

QP

12---

Find the values of the pronumerals.

a b

a x = 70 (∠s in same segment are equal)y = 60 (∠s in same segment are equal)

b ∠ADB = 40° (∠s in same segment are equal)∠CPD = ∠PAD + ∠ADP (exterior angle of ∆ = sum of interior opposite angles)

110° = x° + 40°∴ x = 70

70°y° x°

60°

x°

40°110°

P

A

D

B

C

Example 2

7

x°

25°

y°

35° 140°

x°100°

60°

x°


d e

a In a circle draw a diameter AB. Join the endpointsof AB to any point P on the circumference.Measure the size of ∠APB.

b i Name the angle at the centre standing on thearc AB.

ii What is the size of this angle?iii Name the angle at the circumference standing

on the arc AB.iv Calculate the size of this angle.

c Complete the following proof.Data: AB is a diameter. O is the centre.Aim: To prove ∠ACB = 90°Proof: ∠AOB = 2 × ∠ACB ( )

But ∠AOB = ° ( )∴ 2 × ∠ACB = °

∴ ∠ACB = 90°, i.e. the angle in a semicircle is a right angle.

d Prove the above theorem using the properties of isosceles triangles.(Let ∠OAC = x° and ∠OBC = y°.)

w°z°

30°

40°

35°

x°y°

50°

10°x°50°

10°x°

8

A

B

P

O

OA

B

C

O

A

B

C

O

A

B

C


Find the values of the pronumerals.a b

a Draw a circle using a template.b Use a set square to find the end points of a

diameter and draw the diameter.c Repeat part b to find another diameter and

hence the centre of the circle.d Explain why the method is valid.

a Join any four points A, B, C and D, that lie onthe circumference of a circle to form aquadrilateral.i Measure the size of all the interior angles of the

quadrilateral.ii What do you notice about the sum of angles:

• BAD and BCD• ABC and ADC?

iii Discuss your results with the class.

Example 3


a b

a x + 45 + 90 = 180 b x2 + 122 = 132

x = 55 x2 + 144 = 169x2 = 25x = 5

O

x°

35°O

r = 6.5 cm

x cm12 cm

9

63°28°O

x°

y°

r = 8.5 cmx cm

15 cm

O

10

Set square

11

AB

C

D


b Giving reasons, find the size of:i ∠AOCii reflex ∠AOCiii ∠ABCiv ∠ADC + ∠ABCv ∠DAB + ∠DCB

Concyclic points are points through which a circle can be drawn. Any three non-collinear points are concyclic (see Exercise 16B, question 5).

If the four vertices of a quadrilateral lie on a circle then it is known as a cyclic quadrilateral.

c Complete the following proof.Data: ABCD is a cyclic quadrilateral.Aim: To prove i ∠ADC + ∠ABC = 180°

ii ∠DAB + ∠DCB = 180°Proof: i ∠AOC = 2 × ∠ADC ( )

reflex ∠AOC = 2 × ( )But ∠AOC + reflex ∠AOC = ( )

i.e. 2 × ∠ADC + 2 × ∠ABC = 360°i.e. 2 × (∠ADC + ∠ABC) = 360°

∴ ∠ADC + ∠ABC = ii ∠ABC +∠BCD + ∠CDA + ∠DAB = 360° ( )

∴ ∠BCD + ∠DAB + 180° = 360° (using the result of i)∴ ∠BCD + ∠DAB =

i.e. the opposite angles of a cyclic quadrilateral are supplementary.

55°B

A

C

DO

a cyclicquadrilateral

not a cyclicquadrilateral

A

B

C

DO

Example 4


x + 80 = 180∴ x = 100

y + 108 = 180∴ y = 72

108°80°

x° y°


Find the values of the pronumerals.a b

a Draw a cyclic quadrilateral ABCD and produce DC to E.i Measure angles DAB and BCE.ii Discuss your results with the class.

b Find x and y.

c Complete the following proof.Data: ABCD is a cyclic quadrilateral. CB is produced

to P.Aim: To prove ∠ABP = ∠ADC.Proof: Let ∠ABP = x°

∴ ∠ABC = (CBP a straight line)∴ ∠ADC = ( )∴ ∠ABP = ∠ADC, i.e.

an exterior angle at a vertex of a cyclic quadrilateral is equal to the interior opposite angle.

12

54°

105°

x°y°110°

38°

x°

O

13

A

DC E

B

125°

y°

x°

x°

A

D

CB

P

Example 5


x = 115y = 95(external ∠ of cyclic quadrilateral equal to interior opposite angle) x° 95°

115°y°


Find x.a b


d e f

g h i

j

14

x°

152°

2x°

94°

15

x°

65°

70°y°

O20°

30° x°

80°

55°

x° 58°

O

z°

w°

120°100°

O

50° 110°

28°

z°

82°

50°106°x°

130°

y°O

50°O

x°


a Complete the following proof.Data: O is the centre of the circumcircle of

∆ABC. Let OB = R. Using the usualtriangle notation, let the lengthsAB = c, BC = a and CA = b.

Construction: Draw the diameter BOP and join AP.Aim: To find the radius, R, of the circumcircle.Proof: In ∆ABP, ∠PAB = (angle in a semicircle)

∴ sin ∠APB =

=

∴ 2R × sin ∠APB = c

∴ R =

But ∠APB = ∠ACB ( )= ∠C

∴ R =

b Use ∆BCP to show that R = .

c Draw diameter AOQ and show that R = .

d Use parts a, b and c above to derive the sine rule for any triangle ABC.

Extension

The following is an example of what is known as an indirect proof. In an indirect proof we assume the conclusion to be false and show that this assumption leads to a contradiction, hence the original conclusion must be true.

a Data: ABCD is a quadrilateral in which the opposite angles are supplementary.Aim: To prove that A, B, C and D lie on a circle (i.e. ABCD is a cyclic

quadrilateral).Construction: Since a circle can be drawn through any three non-collinear points, we

draw the circle through A, B and C and assume that D does not lie on this circle. Two cases arise:

Read and discuss the following proof for Case 1 and when you understand all the concepts involved complete the proof for Case 2.

16P

A

B

O

C

PB-------

c2R-------

c2 APB∠sin--------------------------------

c2 Csin--------------------

a2 Asin--------------------

b2 Bsin--------------------

17

Case 1 D outside circle

A

B C

DP

Case 2 D inside circle

A

B C

PD


Case 1: Let P be the point where the circle cuts AD and let ∠ABC = x°.

Proof: ∠ADC = 180° – x° (data)∠APC = 180° – x° (opposite angles of a cyclic quadrilateral are

supplementary)∴ ∠ADC = ∠APC

But ∠APC = ∠ADC + ∠DCP (exterior angle of ∆PDC = sum of interior opposite angles)

∴ ∠ADC = ∠ADC + ∠DCPThis is impossible and thus the original assumption that D does not lie on the circle is false, i.e. A, B, C and D are concyclic points.

Case 2: Let P be the point where the circle cuts AD produced and let ∠ABC = x°.

Complete the proof. We have proved that:if the opposite angles of a quadrilateral are supplementary then the quadrilateral is cyclic.

b Determine whether or not the following quadrilaterals ABCD are cyclic. (Give reasons.)i ii iii iv

c Which of the special quadrilaterals can always have a circle drawn through their vertices?

Use the method of indirect proof to prove:

If an interval subtends equal angles at two points on the same side of the interval, then the two points andthe two endpoints of the interval are concyclic,i.e. if ∠ADB = ∠ACB then ABCD is a cyclic quadrilateral.

E. TANGENTS AND SECANTS

For example, AB is a secant.

A B

CD

135°

45°

A B

CD

55°

65°

A

D

C B130°

50°D C

A B

130° 130°

18

A B

C

D

A secant is a straight line that intersects a circle at two distinct points.

A

B


For example, ABC is a tangent.

The point of intersection (B ) is called the point of contact of the tangent with the circle. Through a given point on a circle only one tangent can be drawn.

Exercise 16E

a Draw a circle and a radius OB. At B draw thetangent ABC. Measure the size of ∠OBC.

b Draw any tangent PQR to a circle. Join the centreto the point of contact. Measure the size of ∠OQR.

From parts a and b we infer that:

A tangent to a circle is perpendicular to the radius drawn to the point of contact.

A tangent is a straight line that intersects (or touches) a circle at one point only.

B

A

C

1

B

A

C

O

Q

P

R

O

Example 1

Find the values of the pronumerals. PT is a tangent. O is the centre.

a b

a x + 68 + 90 = 180 b x2 + 122 = 152

∴ x = 22 x2 + 144 = 225x2 = 81

∴ x = 9

T

P

O

x°68°

O

T P12 cm

15 cmx cm


Find the values of the pronumerals. PT is a tangent to the circle centre O.a b c

d e f

g h i

j The concentric circles with centre O have radii 6 cmand 8 cm. AB is a chord of the larger circle and atangent to the smaller circle. Calculate the length of AB.

a Draw a circle of radius 30 mm. Mark the centre Oand any point T on the circle. Using a straight edgeand compasses only, construct the tangent at T.

b Draw a circle of radius 30 mm and mark the centre O. Using a set square, draw all thetangents to the circle from an external point P.

c i Draw a circle of radius 30 mm and mark the centre O.Mark any external point P. Bisect OP at M and drawthe circle centre M and radius = MO.Draw PT and explain why it is a tangent. Draw theother tangent from P.

2

T P

O

x°70°

8 cm6 cm

T

P

Oy cm

40°w°

T

P

O

T P

38°

z°

Ox° 36°T

O

P

z°y°

80°

T P

x° y°

O

O

TP

x°

50°

O

40°P T

x°

y°

O35°

TPx°

O

A B

3

T

O

P

O

T

PO M

☞


ii Draw a circle centre O. Use the method of part i to construct the tangents to this circle from any external point.

d Draw a circle using a template. Find the centre of the circle using tangents. Explain how your method works.

e Draw a circle of radius 25 mm. Construct the smallestsquare that contains this circle.

f Draw any line AB and mark a point O above the line.Construct a circle centre O such that AB is atangent to this circle.

Find the length of the tangent PT.a b

c Find the length of the tangent to a circle of radius 10 cm from a point P if P is 35 cm from the centre.

a Draw the tangents PA and PB to any circle fromthe point P. Measure the lengths of PA and PB.

b PT and PQ are the tangents from P to the circle,centre O.i Prove ∆PTO ≡ ∆PQO.ii Hence prove PT = PQ.So, the two tangents drawn to a circle from anexternal point are equal in length.

A

B

O

The length of a tangent from an external point P to a circle is defined to be the distance from P to the point of contact of the tangent.

4

T

P

O

15 cm

6 cm

PT

O

4 cm 24 cm

5

B

P

A

T

Q

PO


a Find the values of the pronumerals. PA and PB are tangents.i ii

iii

b The circle touches the sides of the quadrilateral ABCDat P, Q, R and S as shown. Find the angles of thequadrilateral PQRS.

c PQ, QR and RP are tangents. If PR = 9 cm, RW = 5 cmand RQ = 12 cm, find PQ.

Example 2

Find the values of the pronumerals. PA and PB are tangents. O is the centre.

x = 122y + 40 = 180

2y = 140∴ y = 70 y°

40°

A

P

B

x cm

12 cm

6

B

P

A

y°

30°

21 cm

x cm

15 cm

y cm

B

A

x°O P80°

y cm

54 cm

B

A

65°x°O P

100°

110°

60°

A B

CD

S

P

Q

R

P

U

QVR

W

☞


d I is the incentre of ∆PQR, RQ = 8 cm,PR = 9 cm and PQ = 7 cm. Find RB.

a Find the values of the pronumerals. PQ is a tangent.i ii iii

iv v vi

b Complete the following proof.Data: PQ is a tangent to the circle centre O.Aim: To prove ∠BCQ = ∠CAB.Proof: Join CO and BO. Let ∠BCQ = x°.

∠OCB = (tangent ⊥ radius)∴ ∠OBC = 90° – x° ( )∴ ∠COB = (angle sum of ∆ = 180°)

= 2x°∴ ∠BAC = (angle at centre = 2 × angle at circumference)∴ ∠BCQ = ∠CAB, i.e.

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

The chord BC divides the circle into two segments. ∠BCQ is in the minor segment while ∠BAC is in the alternate or other (major) segment.

P

B

A

QR

CI

7

O

P

Q

y°

z°

w°

x°38°

O

Q

y°

z°

x°56°P

O

P

Q

y°

x° 78°

Ox°

80°P Q

w° z°

x°

y°

v°150°

P Q

O

x°

126°

P

Q

O

PC

Q

AB



d e f

g h i

j

Example 3

Find the values of the pronumerals. PQ is a tangent.

a b

a x = 52° b x = 130°

52°

x°

P

Q

x°

130°

P Q

8

55°

x°136°

x°130°

x°

y°

70°

x°

100°

36°x°

49°75°

y°

x°

O

48°x°

70°

85°

x°

120°

75°

x°

O

48° 25°

x°

y°


Four cases arise:

1 AB and CD are known as direct common tangents.

2 PQ and RS are indirect common tangents.

3 PQ is a common tangent to the two circles that touch externally.

4 PQ is a common tangent to the two circles that touch internally.

Two circles of radii 2 cm and 7 cm have theircentres 13 cm apart. Find the length of thedirect common tangent PQ.

a Draw two circles, centres C and O, that touch externallyat the point P. Join CP and OP. Is CPO a straight line?

b Complete the following proof.Data: KPL is the common tangent to two circles with

centres O and C that touch externally at P.Aim: To prove C, P and O are collinear.Proof: ∠KPO = ° (tangent ⊥ radius at point

of contact)∠KPC = ° ( )

∴ ∠CPO = °∴ CPO is a straight line, i.e. C, P and O are collinear.

A common tangent to two circles is a straight line that touches both circles.

AB

CD

PS

RQ P

QP

Q

9

O

PQ

C13 cm

7 cm

x

2 cm

10

C OP

K

L

C OP


c KPL is a common tangent to the two circles withcentres C and O that touch internally at P. Provethat C, O and P are collinear.

From parts b and c we conclude:

When two circles touch, their centres and thepoint of contact are collinear.

a Draw two chords that intersect at P. Measure thelengths of the intervals AP, PB, CP and PD.Calculate: i AP × PB

ii CP × PD

b AB and CD are two chords of the circle that intersect at P.a Prove that ∆APD is similar to ∆CPB.b Write down all the equal ratios of sides in these two triangles.c Hence show that AP × PB = DP × PC

i.e. the products of the intercepts of two intersecting chordsof a circle are equal.

a Find x. b Find x.

c PQ = 24 cm d AB = 40 cmXQ = 8 cm PD = 25 cmXL = 10 cm CD bisects AB.Find KX. Find x.

K

L

CO

P

11

PD

BC

A

A

B

C

D

P

Example 4

Find x.

x × 8 = 7 × 48x = 28x = 3.5

8 cm

4 cm7 cm

x cm

126 cm

8 cmx cm

3 cm5 cm

8 cm

x cm

10 cm

P

Q

K

X

L

A BP

C

x

D

☞


e Find x.

a Draw the tangent PT and the secant PQR. Measure the lengths of the intervals PT, PR and PQ.Calculate: i PT2 ii PR × PQ

b Complete the following proof.Data: PT is a tangent to the circle.Aim: To prove PT2 = TA × TB.Proof: In ∆s PTB and ATP

∠TPB = (angle between chord and tangent= angle in alternate segment)∠PTB = ∠ATP ( )∴ ∆PTB is similar to ∆ATP. ( )

c Let us separate and redraw these triangles. Write down all the equal ratios of sides.

d Hence show that PT2 = TA × TB, i.e.the square of the length of a tangent from an external point equals the product of theintercepts of any secant from the point.

x cm

3 cm

4 cm8 cm

13

P

Q

T

R

A

B

TP

P T

B

A

P T

Example 5

PT = 7 cm and PB = 5 cm. Find AB.

72 = 5(5 + x)49 = 25 + 5x24 = 5xx = 4.8

A

B

PT


a TA = 8 cm b CA = 12 cmTB = 2 cm BC = 3 cmFind PT. Find DC.

c PQ = 10 cm d KL = 12 cmQR = 5 cm LN = 16 cmFind QS and RS. Find LM and MN.

e Find x. f Find x.

g Find x. h Find x.

A person of height h metres stands on the surfaceof the earth assumed to be a sphere of radius R.a Show that the distance, d, to the horizon is

given by d = .

b Calculate the distance across level ground tothe horizon that can be seen by a person ofheight 1.8 m. Use R = 6400 km.

c The formula in part a is often given asd ≈ . Use this formula to calculate the distance tothe horizon seen by a person of height 1.8 m.

d Explain why this second formula gives the same answer as the first, for many cases.

14P

B

A

T

D

B

A

C

P

S

R

Q

K

M

N

L

x cm

4 cm

5 cmx cm

12 cm

4 cm

4 cm

10 cm

x cm32 cm

x cm

15

R

dh

O

h2 2hR+

2hR


a Draw two secants, PAB and PCD, to a circlefrom an external point P. Measure the lengthsof the intervals PA, PB, PC and PD.Calculate:i PA × PB ii PC × PD

b PT = 12 cm, TB = 6 cm, TC = 18 cmFind TA and TD.Is TA × TB = TC × TD?

c QS = 10 cm, RS = 40 cm, SV = 16 cmFind SW.Is SQ × SR = SV × SW?

d Complete the following proof.Data: AB and CD are two secants that meet at T.Aim: To prove TB × TA = TD × TC.Construction: Draw the tangent TP.Proof: PT2 = TB × TA ( )

also PT2 = ( )∴ TB × TA = TD × TC, i.e.

the products of the intercepts of two intersecting secants to a circle from an external point are equal.

16

PA

C

D

B

P T

B

AC

D

12 cm

T

SQ

V

W

R

P

TB

D

C

A

Example 6

Find x.

8 × (8 + x) = 6(6 + 9)64 + 8x = 90

8x = 26x = 3.25

6 cm

8 cm

x cm

9 cm


Find x.a b

c Find the values of the pronumerals.

F. PROOFS USING CIRCLE THEOREMS

Exercise 16F

Prove that if two chords subtend equal angles at the centre of a circle then they are equal in length.

Prove that if two chords are equidistant from the centre of a circle then the chords are equal in length.

a Prove that ∆BTD is similar to ∆CTA.b Hence prove that TA × TB = TC × TD.

O is the centre of the two concentric circles.ABCD is a straight line. Prove that AB = CD.

PQ and RS are two equal chords of the circlecentre O. K and L are the midpoints of PQand RS respectively. Prove that ∠PKL = ∠KLR.

17

2 cm

5 cm

x cm

3 cm

6 cm

4 cm

x cm

8 cm

yz

xw4 cm

3 cm

6 cm8 cm9 cm

1

2

3 A B

D

C

T

4

A B C D

O

5

P

Q

R

S

O

K L


P and Q are the centres of the circles that intersect atK and L. AKC is parallel to PQ. Prove that AC = 2 × PQ.

AB and CD are parallel chords of the circle centre O.AP ⊥ CD and BQ ⊥ CD.

Prove: a CP = QD

b AC = BD

O is the centre of the circle and AB = AC.Prove that ∠COB = 4 × ∠OBA.

PQRS is a cyclic quadrilateral. PQ has been producedto T such that PTRS is a parallelogram. Prove that∆QRT is isosceles.

AC and BC are equal chords and CD is a tangent.Prove that AB || CD.

PQ is a tangent, QTS is a straight line and PQ || RS.Prove ∠QRS = ∠QTR.

6K

L

P Q

A C

7

O

A

P Q

B

CD

8

C

B

A

O

9 P Q T

RS

10A B

C D

11Q P

T

R S


AB and CD are chords that intersect at P.If AC || DB prove that ∠CPB = 2 × ∠BDC.

If AT = PT, prove that AP || QB.

If AP = AD, prove that CP = CB.

Prove that ∠KSM = ∠RLM.

If BE || AD, prove that CA = CD.

If QS bisects ∠PQR and ∠PSR, prove that∠QRS = ∠QPS = 90°.

The two circles intersect at B and D. AB is adiameter and ABC and ADE are straight lines.Prove ∠ACE = 90°.

12

A

B

C

D

P

13

A

PB

Q

T

14

C

AD

B

P

15

K LM

TS

R

16A

B

ED

PC

17 P

R

Q

S

18

A

O

B

C

ED


O is the centre of the larger circle and OX is adiameter of the inner circle that touches thelarger circle at X. XY is a chord that cuts theinner circle at P. Prove XP = PY.

O is the centre of the circle. If ∠OPQ = ∠PRQ,prove that ∠POQ = 90°.

ABC and ADE are chords of a circle centre O andare also tangents at B and D to a concentriccircle as shown. Prove AB = BC.

O is the centre of the circle. ∠AOC = 90° and CDis a tangent. Prove CD = CB.

∆PQR is inscribed in a circle. The tangents at Rand Q meet at T. Prove RT = QT.

PQ is a tangent to the circle centre O.If PB bisects ∠QBA, prove that ∠PQB = 90°.

The two circles touch externally at X. If P is anypoint on the common tangent, prove that thetangents PQ and PT are equal in length.

19

O

X

Y

P

20

Q

P

R

O

21

B D

A

O

CE

22

A

O C

D

B

23P

Q

R

T

24

Q

B

P

AO

25 T

P

Q

X


PQ = PR and VW || QR. Prove that VWRQ isa cyclic quadrilateral.

TP is a tangent. If PA = 3 × PT, prove that AB = 8 × PB.

Language in Mathematics

Name an angle:a subtended at the circumference by the chord PQb subtended by the chord QR at the centrec standing on the arc SRd subtended at the circumference by the chord QS

in the major segmente subtended at the circumference by the chord QS

in the minor segmentf in the same segment as ∠STR

Draw a circle and mark any four points A, B, C and D on the circumference. On your diagram draw an angle:a subtended at the circumference by the chord ABb subtended by the chord BC at the centrec standing on the arc ACd subtended at the circumference by the chord BC in the major segmente subtended at the circumference by the chord AC in the minor segmentf in the same segment as ∠ADB

Explain in your own words the meaning of:a collinear points b concyclic points c a cyclic quadrilateral

26

Q

V

P

W

R

27T

P

B

A

1

O

S

R

QP

T

2

3


✓Explain the difference in meaning between:a sector and segment b supplementary angles and complementary anglesc alternate and alternative

Draw diagrams to illustrate:a the point of contact of a tangent b the perpendicular bisector of an intervalc a point equidistant from two lines d an exterior angle of a cyclic quadrilateral

Three of the words in the following list are spelt incorrectly. Find these words and write the correct spelling:

vertex, diametre, tangent, interior, circumfrence, intersept

Glossaryalternate arc bisector centrechord circle circumference collinearcommon concyclic contact cyclicdiameter distinct equidistant exteriorintercept interior intersect majorminor non-collinear perpendicular quadrilateralradius secant sector segmentsemicircle subtend supplementary tangentvertex vertices

Which of the following statements is not correct?A A is a tangentB B is a sectorC C is a segmentD D is a chord

The angle at the circumference standing on the arc BC is:A ∠BDCB ∠BCDC ∠BECD ∠BCE

4

5

6

CHECK YOUR SKILLSCHECK YOUR SKILLS

1

A

C

B

D

2

D

CE

B

A


✓The values of the pronumerals are:A x = 35, y = 9B x = 35, y = 4C x = 55, y = 9D x = 55, y = 4

The length of the chord PQ is:A 5 cmB 6 cmC 8 cmD 10 cm

C and O are the centres of the circles. PR = 6 cm, RO = 8 cm and ∠PCR = 35°. The values of x and y are:A x = 55, y = 6B x = 35, y = 10C x = 55, y = 10D x = 35, y = 6

The value of x is:A 40B 50C 100D 160


AOB is a diameter and CD ⊥ AB. The value of x is:A 25B 40C 50D 90

3

O 4 cm

9 cm

9 cm y cmx°

55°

4

O4 cm

1 cmP Q

5

OCR

Q

P6 cm

y cm

x° 8 cm35°

6

O

100°

x°

7

x° 30°

50°

8

A BOD

C

50°

x°


✓The values of x and y are:A x = 110, y = 80B x = 55, y = 40C x = 70, y = 100D x = 90, y = 120

x =A 28B 56C 62D 124

PT is a tangent and O is the centre. The value of x is:A 25B 40C 50D 80

AB and AC are tangents and O is the centre. The size of ∠ACB is:A 20°B 55°C 70°D 90°

PQ is a tangent. The value of x is:A 10B 50C 80D 100

The two circles touch externally at B. C and O are the centres of the two circles. The size of ∠ABD is:A 80°B 90°C 100°D 140°

9

x°

y°80°

110°

10

2x

124°

11

40°

T

P

O

x°

12

70°O

B

C

A

13

80°

x°

P

Q

14

C B

AD

O50° 30°


AB and CD are chords that intersect at P. The value of x is:A 3B 8C 12D 24

PT is a tangent. The length of PT is:A 4 cmB 8 cmC 16 cmD 48 cm


If you have any difficulty with these questions, refer to the examples and questions in the sections listed in the table.

Copy the following diagrams and label as many parts as possible.a b

Question 1, 2 3–5 6–10 11–17

Section A C D E

15

x cm

2 cm

4 cm

6 cm

A

B

D

P

C

16

12 cm

4 cm

T

QP

R

17

8 cm 6 cm

4 cm

x cm

BD

A

E

C13---

REVIEW SET 16AREVIEW SET 16A

1


Name an angle:a subtended by the chord AB at the circumferenceb subtended at the centre by the chord ABc standing on the arc BCd subtended by the chord CE at the circumference

in the minor segment

Find the values of the pronumerals. O is the centre.a b c

Find the values of the pronumerals. P and Qare the centres of the circles.

Find the values of the pronumerals. O is the centre of each circle.a b c

d e f

Find the values of the pronumerals. O is the centre of each circle.a b c

2

O

D

E

C

A

B

3

x°40°

5 cm O

5 cm

15 cmOx

5 cmO

r

24 cm

4

QP

3 cm

x cm

4 cm50°y°

5

Oy°

x°30°

40° Ox°35° O

x°

20°

O

x°

40° Ox°

y°

120°80°

O

x°

100°

6

O

T

P

x°35°

O P

T

U

56°x°

T

P

75°

x°

PT is a tangent. PT and PU are tangents. PT is a tangent.


d e f

g

Draw diagrams and identify the following parts of a circle: centre, radius, minor sector, major sector, tangent, secant, chord, arc, diameter, semicircle, segment

Name an angle:a subtended by the chord BA at the circumferenceb subtended at the centre by the chord ABc standing on the arc EDd subtended by the chord DB at the circumference,

in the minor segment


Find the values of the pronumerals.P and Q are the centres of the circles.

C O44°x°38°

P

x cm

7 cm

5 cm4 cm 5 cm

4 cm

T

Px

C and O are the centres. P is the point of contact of the circles.

PT is a tangent.

4 cm6 cm

5 cmx cm

REVIEW SET 16BREVIEW SET 16B

1

2 D C

B

A

E O

3

O

100°

x°3 cm

3 cm

4 cm

O

x cm

O

x°

40°

4

QP6 cm

35°

y°x cm



d e f

Find the values of the pronumerals. O is the centre of each circle.a b

PT is a tangent. PQ and PR are tangents.

c d

PT is a tangent. C and O are the centres. P is the point of contactof the circles.

e f g

PT is a tangent.

5

O84°

x°

70° 100°

x°

y°

x cm

8 cm

5 cm

O

94°

2x°

Ox° 115° O 80°

x°y°

70°

6

116°Q

R

x° PO

6 cm

10 cmx cm

T

P

x°

70°P

T

C P O100° 110°

x°

x cm

3 cm

2 cm

2 cmx cm

6 cm

2 cm

T P

x cm

8 cm

9 cm

6 cm

Documents

Chapter 16- Year 9 Maths