CHAPTER - Maths · PDF file474 CHAPTER 29 Circle geometry A line drawn from the centre of a...
Transcript of CHAPTER - Maths · PDF file474 CHAPTER 29 Circle geometry A line drawn from the centre of a...
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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.
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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
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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.
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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
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CHAPTER 29 Circle geometry
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
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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.
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CHAPTER 29 Circle geometry
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
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29.4 Circle theorems CHAPTER 29
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°
Subtract 81° from 180°.
Give the reason.
Subtract 99° from 180°.
Give the reason.
A
B
PT
Q
AC
B
PT
Qx
y
A
B
PT
Q37°
68°
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CHAPTER 29 Circle geometry
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
The reason may be shortened to this.
Add 37° and 68°.
Subtract the sum from 180°.
Give the reason.
Add 37° and 68°.
Subtract the sum from 180°.
Give the reason.
The reason may be shortened to this.
AB
D
C
47°
36°
The reason may be shortened to this.
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.
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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
29.4 Circle theorems CHAPTER 29
A
B
T
P
38°
62°
28°
A
B
C
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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
CHAPTER 29 Circle geometry
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
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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
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484
CHAPTER 29 Circle geometry
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°
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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
Diagram NOTaccurately drawn
30°
132°
C
DA
P
B
Diagram NOTaccurately drawn
80°70°
x°
C
A
PST
B
Diagraaccura25°
Diagram NOTaccurately drawn
C
B
D A
ODiagram NOTaccurately drawn
2x°
x°
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CHAPTER 29 Circle geometry
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
Diagram NOTaccurately drawn
P
A
CB
Diagram NOTaccurately drawn
P Ox°
T
50 cm
92 cm
P Ox°
T
50 cm
92 cm