Download Chapter 29 - Maths Area

29
Circle geometry
CHAPTER
29.1 Parts of a circle
C
e
i
Tangent
Radius
u
e
Diameter
c
Chord
m
f
n
c
r
r
e
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 triangles
Triangles 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.
Example 1
A and B are points on the circumference of a circle, centre O.
Angle OAB 40°.
O
Calculate the size of angle AOB.
Give reasons for your answer.
40°
Solution 1
B
A
At each step, mark the new information on
the diagram.
O
100°
40°
40°
B
A
OA OB
Angle OBA 40°
In an isosceles triangle, the angles opposite
the equal sides are equal.
472
OA and OB are radii.
Triangle OAB is isosceles.
Give the reason.
Angle AOB 180° (40° 40°)
100°
Add the equal angles and subtract the sum from 180°.
The angle sum of a triangle is 180°.
Give the reason.
29.3 Tangents and chords
CHAPTER 29
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
15°
b
110°
O
a
O c
O
25°
4
5
6
70°
270°
e
30°
O
O
O
d
42°
f
g
In Questions 7–9, give reasons for your answers.
7
8
9
j
O
k
i
O
h
O
136°
38°
42°
29.3 Tangents and chords
Here 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°
O
P
T
Q
● Tangents from an external point to a circle are equal in length.
PA PB
A
P
O
B
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Circle geometry
CHAPTER 29
A
● A line drawn from the centre of a circle perpendicular
to a chord bisects the chord.
AM BM
The converse (opposite) of this statement is also true.
M
● A line drawn from the centre of a circle to the midpoint
of a chord is perpendicular to the chord.
Example 2
O
B
U
PT is a tangent at T to a circle, centre O.
TU is a chord of the circle.
Angle PTU 54°.
P
O
Find the size of angle TOU.
Give reasons for your answer.
54°
T
Solution 2
U
36°
108° O
P
54°
36°
T
Angle OTU 90° 54°
36°
Subtract 54° from 90°.
Tangent is perpendicular to the radius.
Give the reason.
Angle OUT 36°
OT OU.
In an isosceles triangle, the angles
opposite the equal sides are equal.
Give the reasons.
Angle TOU 180° (36° 36°)
108°
Add the equal angles and
subtract the sum from 180°.
Angle sum of a triangle is 180°.
Give the reason.
A
Example 3
PA and PB are tangents to a circle.
Angle APB 68°.
P
68°
Calculate the size of angle PAB.
Give reasons for your answer.
B
Solution 3
A
56°
P
68°
56°
B
474
PA PB
Tangents from an external point to a circle Give the reason.
are equal in length.
180° 68°
Angle PAB 2
56°
Subtract 68° from 180°
and divide the result by 2.
Triangle PAB is isosceles.
The angle sum of a triangle is 180° and in
an isosceles triangle the angles opposite
the equal sides are equal.
Give the reasons.
29.3 Tangents and chords
CHAPTER 29
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.
O
37°
a
P
T
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 P
b
P
y
40°
A
A
O
x
O
20°
B
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
b P
A
y
34°
118°
B
O
O
P
x
A
A
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.
M
B
64°
O
A
P
61°
B
475
Circle geometry
CHAPTER 29
6 PA and PB are tangents to a circle, centre O.
Find the size of angles x and y.
a
b
A
A
y
P
x
64°
O
O
78°
P
B
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.
A
72°
P
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.
B
D
B
152°
O
P 71°
A
29.4 Circle theorems
C
Theorem 1 – the angle subtended by an arc at the
centre of a circle is twice the angle subtended at
the circumference
O
Angle AOB 2 angle ACB
Proof
Draw the line CO and produce it to D.
OA OB OC (radii).
A
C
x y
x
A
O
2x 2y
D
y
B
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.
476
B
29.4 Circle theorems
CHAPTER 29
Example 4
R
P, Q and R are points on a circle, centre O.
Angle PRQ 41°.
41°
Work out the size of angle POQ.
Give a reason for your answer.
O
Solution 4
Angle POQ 2 41°
82°
Q
P
Double angle PRQ.
The angle at the centre of a
circle is twice the angle at the
circumference.
The reason may be shortened to this.
C
Theorem 2 – the angle in a semicircle is a right angle
Angle ACB 90°
B
Proof
The angle subtended at O, the centre of the circle, by the
arc AB is 180°, that is, angle AOB 180°.
O
A
C
B
O
A
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°
Example 5
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.
C
B
58°
A
Solution 5
Angle ACB 90°
State the size of angle ACB.
The angle in a semicircle is a right angle.
Give the reason.
Angle ABC 180° (90° 58°)
180° 148°
32°
Add 90° and 58°.
Subtract the sum from 180°.
The angle sum of a triangle 180°.
Give the reason.
477
Circle geometry
CHAPTER 29
Theorem 3 – angles in the same segment are equal
Q
P
Angle APB angle AQB
Proof
Angle APB 12 angle AOB (angle at the centre of a circle
is twice the angle at the circumference).
Similarly, angle
AQB 12
O
B
angle AOB.
A
So angle APB angle AQB.
Example 6
A, B, C and D are points on a circle.
Angle ADB 63°.
D
C
63°
Find the size of angle ACB.
Give a reason for your answer.
A
B
Solution 6
Angle ACB 63°
State the size of angle ACB, which
is equal in size to angle ADB.
The angles in the same segment are equal.
Give the reason.
Cyclic quadrilaterals
A 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.
S
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°.
R
P
Q
Proof
PQRS is a cyclic quadrilateral whose vertices lie on a circle, centre O.
S
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 the
circumference).
Similarly, y 2b.
O
P
a
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°).
478
b
y x
Q
R
29.4 Circle theorems
CHAPTER 29
Example 7
D
ABC is a straight line.
B, C, D and E are points on a circle.
Angle ABE 81°.
E
C
Work out the size of angle CDE.
Give a reason for each step in your working.
81°
B
A
Solution 7
Angle CBE 180° 81°
99°
Subtract 81° from 180°.
The sum of angles on a straight line 180°.
Give the reason.
BCDE is a cyclic quadrilateral
so angle CDE 180° 99°
81°
Subtract 99° from 180°.
The sum of opposite angles of a cyclic quadrilateral 180°.
Give the reason.
Notice that angle ABE angle CDE.
Angle ABE is an exterior angle of the cyclic quadrilateral and it is the same size as the opposite
interior angle.
A
Theorem 5 – the angle between a chord and the
tangent at the point of contact is equal to the
angle in the alternate segment
B
Angle PTB angle BAT
Proof
PTQ 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.
P
Q
T
C
A
y
Let angle PTB x and angle BAT y.
B
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).
x
P
T
In triangle CBT, 90° 90° x angle BCT 180° (angle sum of triangle).
So angle BCT x.
Q
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.
Example 8
A
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.
B
P
68°
37°
T
Q
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Circle geometry
CHAPTER 29
Solution 8
Method 1
Angle PTB angle BAT
37°
Alternate segment theorem.
The reason may be shortened to this.
Angle ABT 180° (37° 68°)
75°
Add 37° and 68°.
Subtract the sum from 180°.
The angle sum of triangle 180°.
Give the reason.
Method 2
Angle ATQ 180° (37° 68°)
75°
Add 37° and 68°.
Subtract the sum from 180°.
The sum of angles on a straight line 180°.
Give the reason.
Angle ATQ angle ABT
75°
Alternate segment theorem.
The reason may be shortened to this.
Example 9
ABCD is a cyclic quadrilateral.
Angle ADB 36°. Angle BDC 47°.
ii angle ABC.
a Find the size of
i angle BAC
Give reasons for your answers.
b Is AC a diameter? Explain your answer.
C
D
47°
36°
A
B
Solution 9
a 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°.
480
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 statement
and an explanation.
29.4 Circle theorems
CHAPTER 29
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
a
48°
b
c
74°
4
5
6
e
d
36°
f
77°
g
47°
7
8
i
9
j
64°
h
117°
k
51°
m
l
A
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.
n
B
38°
T
P
11 A, B and C are points on a circle.
Angle ABC 28°.
Angle BAC 62°.
Is AB a diameter? Explain your answer.
C
28°
B
62°
A
481
Circle geometry
CHAPTER 29
12 A, B, C and D are points on a circle.
Angle ABC 76°.
Angle ADB 31°.
D
C
31°
ii angle CAB.
Work out the size of
i angle BDC
Give a reason for each step in your working.
A
76°
B
13 a Is a rectangle a cyclic quadrilateral? Explain your answer.
b Is this quadrilateral cyclic? Explain your answer.
75°
98°
104°
14 A, B, C and T are points on the circle.
PTQ is a tangent to the circle.
Angle PTC 51°.
Angle BAC 23°.
B
C
P
23°
Work out the size of angle BCT.
Give a reason for each step in your working.
51°
A
T
Q
15 A, Q and R are points on the circle.
PQ and PR are tangents to the circle.
Angle QPR 48°.
R
A
P
Work out the size of angle QAR.
Give a reason for each step in your working.
48°
Q
Chapter summary
You should know the meaning of:
● circumference
c
f
e
n
e
r
482
radius
m
● tangent
diameter
u
● radius
c
● diameter
tangent
i
r
● chord
c
e
ord
ch
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
a
b 2a
b
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°
d
c
a
b
483
Circle geometry
CHAPTER 29
the angle between a chord and the tangent at the point of
contact is equal to the angle in the alternate segment.
Chapter 29 review questions
The 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.
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.
A
O
P
O
116°
B
47°
S
Q
T
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.
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.
B
O
P
54°
39°
P
T
A
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.
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.
D
38°
O
O
57°
C
Q
P
484
M
A
B
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.
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.
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.
A
B
O
Diagram NOT
accurately drawn
24°
D
C
(1384 June 1995)
C
Diagram NOT
accurately drawn
B
132° D
30°
A
T
(1385 June 2000)
B
P
C x°
80°
D
70°
A
B
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.
C
AT AP.
AB is parallel to TC.
a Calculate the size of angle APT.
Give reasons for your answer.
S
b Calculate the size of angle BTS.
Give reasons for your answer.
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
ii angle AOD
iii angle ABD.
i angle BDA
Diagram NOT
accurately drawn
Diagram NOT
accurately drawn
Diagra
accura
25°
A
P
T
(1384 June 1997)
C
2x°
B
Diagram NOT
accurately drawn
O
x°
D
A
(1385 November 1998)
485
Circle geometry
CHAPTER 29
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.
A
Diagram NOT
accurately drawn
16 cm
O
12 cm
B
C
(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.
6 cm
Give your answer correct to three
40°
significant figures.
P
Diagram NOT
accurately drawn
O
36°
Q
T
(4400 November 2004)
A
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 APC
are congruent.
B
Angle BPA 10°.
b Find the size of angle ABC.
C
Diagram NOT
accurately drawn
(1387 June 2004)
P
T
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.
92 cm
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.
P
T
92 cm
P
486
x°
50 cm
x°
50 cm
O
O