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C H A P T E R
15
MODULE 2
Multiple-choice questions
R
1 In triangle PQR the length of PQ is closest to:
A 16.6
D 25.7
B 15.7
E 21.7
C 12.6
16.5
Q
2 In the diagram the size of ∠BCD is exactly:
◦
B 76
E 114◦
C 100
B 60◦
E 150◦
C 75◦
62°
A
B 54
E 140
D
C
C
3 cm
B
30°
3 cm
4 The area of triangle PQR, in square centimetres, is closest to:
A 24
D 70
P
42°
3 In triangle ABC the magnitude of ∠BAC is:
A 30◦
D 90◦
40°
B
◦
M
A 104
D 94◦
◦
SA
15.1
PL
E
Revision: Geometry
and trigonometry
A
Q
C 66
70°
10 cm
14 cm
P
R
5 The value of p is closest to:
A 0.82◦
D 50.7◦
B 35.1◦
E 54.9◦
C 39.3◦
11
9
p°
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Essential Further Mathematics — Module 2 Geometry and trigonometry
6 The length of AB, to the nearest millimetre, is:
A 12
D 47
B 16
E 51
A
C 21
63°
35 mm
B
42°
C
7 The magnitude of the largest angle in the triangle
shown, to the nearest degree, is:
B 39◦
E 148◦
8 cm
3 cm
C 71◦
7 cm
E
A 32◦
D 98◦
8 In triangle ABC, AB = AC. The length of AB and AC, in
centimetres, is given by:
5
A 10 sin 80◦ B 5 cos 80◦
C
cos 50◦
5
D
E 5 sin 40◦
sin 50◦
A
PL
80°
B
9 In this diagram PQ and SR are parallel and SR = SQ.
x and y satisfy the equation:
A x=y
C 2x + y = 42
E 2x − y = 42
B x + y = 138
D x = y + 42
P
S
M
D
1
2
x cm
C
10 cm
6 cm
D
x cm
B
6
F
C 4.5
15
9
G
3
4
1
4
5
x
C
A
C
D
6
E
B − 12
E
Q
B
12 For triangle ABC, cos =
A − 14
x°
42°
A
B 4
E 5
R
y°
11 In this figure, AB = 15, CD = 5, BF = 6, GD = 6 and
EG = 9. x is equal to:
A 3
D 4.75
C
10 cm
10 In this diagram, angles ACB and ADC are right angles.
If BC and AD each have a length of x cm, then x =
√
B 4
C 5
A 2 17
√
√
E 5 2
D 4 2
SA
Revision
432
θ
6
4
8
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Chapter 15 — Revision
433
B
C 9
D
2 cm
B
12 cm
B
3 cm
3 cm
C
16 A conical container is 40 cm tall. Water is poured into
the cone leaving a conical airspace of height 10 cm,
as shown in the diagram. The ratio of the volume of
the water to the volume of the airspace is:
B 27 : 1
E 64 : 1
6 cm
airspace
B 48 cm2
E 81 cm2
40 cm
B
X
C 54 cm2
18 Which one of the following equations gives the
correct value for x?
SA
10 cm
water
17 ABC is similar to AX Y. AX = 23 AB. The area
of ABC is 108 cm2 . The area of AXY is:
A 32 cm2
D 72 cm2
A
C 55 : 1
M
A 16 : 1
D 63 : 1
C
D
PL
D 10
E 11
15 In this figure the length of DB, in centimetres, is:
√
A 6
B 9
C 3 5
√
√
E 3 7
D 3 6
E
4 cm
+
9
2
3 cm
E
A 6
A
+
14 D and E are points on AB and AC respectively.
AD = 4 cm, D B = 2 cm, AE = 3 cm and
BC = 12 cm. The magnitude of ∠ADE = the
magnitude of ∠ACB. The length DE, in
centimetres, is:
A
C
Y
X
7m
130°
8m
◦
A x = 49 + 64 + 2 (7) (8) cos 50
J
K
xm
B x 2 = 49 + 64 + 2 (7) (8) cos 70◦
x
8
x
7
C
=
D
=
sin 130◦
sin 25◦
sin 130◦
sin 25◦
E x 2 = 49 + 64 − 2 (7) (8) cos 50◦
19 The height, h m, of a television tower can be calculated
by measuring the angles of elevation of the top of the
hm
tower from two points that are in line with the tower
25°
15°
but that are 100 m apart. Which one of the following
100 m
equations will give the correct value of h?
◦
◦
◦
100 sin 15 tan 25
100 sin 15 sin 25◦
100 sin 10◦ tan 25◦
A h=
B
h
=
C
h
=
sin 10◦
sin 10◦
sin 15◦
100 sin 10◦ tan 25◦
D h=
E h = 100 tan 15◦
sin 15◦
2
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Revision
13 The volumes of two similar solids are in the ratio 4 : 9. The ratio of their surface areas is:
√ √
√
√
√ √
3
3
A 2:3
B
2: 3
C
8 : 27
D 4:9
E
16 : 81
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Essential Further Mathematics — Module 2 Geometry and trigonometry
20 The diagram shows a right square pyramid of height 400 m
with its base a square of side 300 m. If is the angle
between a sloping face and the base, which one of the following
equations will give the correct value of ?
√
400
50 73
400
B tan =
√
A tan =
C tan =
√
150
300 2
150 2
400
400
E tan =
D tan =
150
300
21 In ABC, the length of AC in centimetres is
determined by evaluating:
√
√
100 + 96 cos 120◦
B
100 − 96 cos 120◦
A
√
√
C
100 − 96 cos 60◦
D
64 + 36 − 96
A
100 (1 + 2 cos 120◦ )
E
22 A rectangle is 8 cm long and 6 cm wide. The acute angle between its diagonals, correct to the nearest degree, is:
400 m
300 m
300 m
E
B
B 41◦
E 83◦
23 A vertical mast, AD, of height 20 m is supported by two
cables attached to the ground at C and B as shown. ∠CAB
is a right angle. Cable CD is of length 40 m and cable BD
is of length 30 m. The distance CB in metres is:
√
√
√
A
1700
B
2500
C
3300
√
√
√
√
2000 + 1300 E
1200 + 500
D
θ
6 cm
D
40 m
30 m
20 m
B
A
C
24 In the figure shown, the length of OD in centimetres is:
√
√
√
3
A
2
B
3
C
2
D 2
E 4
D
1 cm C
1 cm
B
1 cm
O
1 cm
A
3
25 An inverted right circular cone of capacity 80 cm is
filled with water to half of its depth. The volume of water
(in cm3 ) is:
√
√
3
A
80
B
80
C 10
D 20
E 40
C
8 cm
C 49◦
M
6 cm
120°
8 cm
PL
A 37◦
D 74◦
SA
Revision
434
h
h
2
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Chapter 15 — Revision
435
A
D
4
15
4
5
B
E
8
15
5
6
C
X
5
12
10
8
Z
27 The area of triangle LMN in cm2 is:
6
6
B
C 15 sin 40◦
sin 40◦
A
5 cos 40◦
5
D 30 cos 40◦ E 30 sin 40◦
β
α
Y
M
5 cm
L
E
40°
6 cm
N
B
3 cm
4 cm
120°
PL
28 In the diagram shown, the value of b is:
√
√
A
13
B 5
C
37
D 13
E 37
A
C
b cm
M
29 A right pyramid with a square base is shown. The
height of the pyramid is 3 m and the square base has
sides of length 8 m. The length of a sloping edge in
metres is:
√
√
√
41
B
52
C
73
A
√
√
80
E
137
D
8m
B
100°
C
30°
12 cm
A
R
S
U
T
θ°
P
°
31 In this figure, PQRS is a rectangle inclined at an angle
of 45◦ to the horizontal plane PQTU. The magnitude
of ∠PQS = 60◦ . Let ◦ be the angle of inclination of
QS to the horizontal plane. Then sin =
√
√
2
2
1
C
B
A
2
4
4
√
√
6
3
E
D
4
2
8m
45
SA
30 The length of AB in centimetres is equal to:
√
12
A
× sin 30◦
B
144 − 2 × 12 cos 30◦
◦
sin 100
C
302 + 1002 − 2 × 30 × 100 cos 50◦
12
12
E
× sin 100◦
D
× sin 50◦
◦
sin 50◦
sin 100
3m
Q
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26 Given that X Z = 8, X Y = 10 and sin = 23 , sin equals:
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Essential Further Mathematics — Module 2 Geometry and trigonometry
X
32 In the given triangles the length of XZ is 40% greater
than AC. The ratio of the areas is:
A 49 : 25
D 64 : 25
B 25 : 4
E 7:5
A
50°
60°
C 16 : 9
50°
C
60°
Y
B
Z
L
E
33 The area of triangle LMN in square centimetres is:
6
6
B
C 15 sin 40◦
A
sin 40◦
5 cos 40◦
5
D 24 cos 40◦ E 24 sin 40◦
6 cm
40°
N
C
PL
B
35 The area of triangle ABC is 20 cm2 . Triangle
XYZ is similar to triangle ABC. The area of
triangle XYZ, in square centimetres, is:
B 35
E 50
A
X
B 1:9
E 2:3
30°
9 cm
Y
100°
C
Z
45 cm
C 1 : 27
15 cm
10 cm
37 A right pyramid with a square base is shown. The square
base has sides of length 10 m. The length of each sloping
edge is also 10 m. The height of the pyramid in metres is:
√
√
√
A
40
B
50
C
60
√
√
200
E
1000
D
30 cm
10 m
height
10 m
10 m
38 In triangle ABC as shown, sin x = 37 . The value of sin y is:
A
1
7
B
9
28
D
4
7
E
3
4
A
C 40
36 Two right circular cones are shown. The ratio of the
volume of the smaller cone to the volume of the larger
cone is:
A 1:3
D 1 : 50
10 cm
15 cm
6 cm
B
100°
M
A 30
D 45
M
8 cm
34 The area of triangle ABC, in square centimetres, is:
A 15
B 37.5
C 75
D 90
E 150
SA
Revision
436
C
B
1
2
A
6 cm
8 cm
x
y
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Chapter 15 — Revision
437
A 50◦
D 130◦
B 70◦
E 140◦
10 km
M
P
5 km
6 km
N
C 120◦
A 2.5
B 5
C 7.5
A 5.5
D 8.4
D 10
E 15
Z
2.5 cm
PL
41 In right angled triangle XYZ, X Y = 8.0 cm and
Y Z = 2.5 cm as shown. The length of ZX in
centimetres, correct to one decimal place, is:
E
40 A hiker travels a distance of 5 km from point P to point Q on a bearing of 030◦ . She then
travels from point Q to point R on a bearing of 330◦ for 10 km. How far west of P is R in
kilometres?
B 7.6
E 10.5
Y
C 8.2
42 The contour map has scale 1 : 20 000. A path XY is represented
on the map by a straight line segment 4 cm long. Point X is
on the 100 metre contour and point Y on the 250 metre
contour. The average slope of the path XY is:
B 0.075
E 0.375
X
C 0.1875
M
A 0.03
D 0.3125
SA
B 60◦
E 98◦
Y
300
250
200
150
100
50
43 In the triangle shown angle PQR, correct to the nearest
degree, equals:
A 38◦
D 82◦
X
8.0 cm
Q
C 73◦
7 cm
5 cm
P
R
8 cm
44 The diameter of a large sphere is 4 times the diameter of a smaller sphere. It follows that
the ratio of the volume of the large sphere to the volume of the smaller sphere is:
A 4:1
B 8:1
C 16 : 1
D 32 : 1
E 64 : 1
45 QR is parallel to ST and P Q : Q S = 2 : 1. Given that the
area of triangle PST is 18 square centimetres, the area of
triangle PQR in square centimetres is:
A 2
D 9
B 6
E 12
P
Q
C 8
R
S
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Revision
39 A yacht follows a triangular course MNP as shown.
The largest angle between any two legs of the course is
closest to:
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15.2
Essential Further Mathematics — Module 2 Geometry and trigonometry
Extended-response questions
1 A yacht has two flat triangular sails as shown in
the diagram. The sail ABC is in the shape of a
right-angled triangle. The height AC is 10 metres
and the length AB is 3.6 metres.
C
F
10 m
8.3 m
PL
E
a Calculate angle ABC. Write your answer
2.7 m
3.6 m
130°
correct to the nearest degree.
B
A
D
E
b Calculate the length BC. Write your
answer in metres, correct to one
decimal place.
The sail DEF has side lengths D E = 2.7 metres and D F = 8.3 metres. The angle EDF
is 130◦ .
c Calculate the length EF. Write your answer in metres, correct to one decimal place.
d Calculate the area of the sail DEF. Write your answer in square metres, correct to one
decimal place.
N
2 A course for a yacht race is triangular in shape and is marked
T
by three buoys T, U and V. Starting from buoy V, the yachts sail
◦
5.4 km
5.4 kilometres on a bearing of 030 to buoy T. They then sail
30°
to buoy U and back to buoy V. The angle TVU is 72◦ and the
angle TUV is 48◦ .
V 72°
M
48°
a Determine the bearing of V from U.
U
b Determine the distance TU. Write your answer in kilometres,
correct to one decimal place.
c Determine the shortest distance to complete the race. Write your answer in kilometres,
correct to one decimal place.
3 A navigational marker XYZ is in the shape of an equilateral
triangle with side length of one metre. It is located in the
vicinity of a yacht race.
a Write down the size of angle XYZ.
SA
Revision
438
Point O is the centroid (centre) of the triangle. Points
M and N are the midpoints of sides XZ and YZ respectively.
Z
1m
X
1m
Y
1m
Z
b Calculate the shortest distance from point O to side XY.
Write your answer in metres, correct to three decimal places.
1m
M
O
X
N 1m
1m
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Chapter 15 — Revision
439
1m
1m
r
reflective
material
O
X
Y
1m
[VCAA 2000]
E
B
2.20 m
5 A paved area is constructed in the shape of a regular
octagon as shown.
a By calculation, show that the size of the angle GOH is
45◦ , where point O is the centre of the octagon.
b The length OG = OH = 2.30 metres. Calculate the area
of the octagonal paved area. Write your answer correct to
the nearest square metre.
M
C
8.21 m
A
PL
4 Jane is landscaping her garden. A piece of shade
cloth ABC has the dimensions as shown.
a Determine the length BC in metres. Write your
answer correct to two decimal places.
b Determine the angle ACB. Write your answer
correct to the nearest degree.
Z
[VCAA 2003]
G
O
H
A square herb garden EFGH is surrounded by four
regular octagonal paved areas as shown in the diagram.
K
SA
c Calculate the side length GH of the square herb
garden. Write your answer in metres, correct to two
decimal places.
d A straight wooden frame is to be built between points
O and K for hanging baskets.
i Calculate the length GK. Write your answer in
metres, correct to two decimal places.
ii Hence calculate the length OK. Write your
answer in metres, correct to two decimal places.
A second piece of shade cloth PQR is also
triangular and has dimensions as shown in the
diagram.
P
O
45°
2.30 m
G F
H E
R
35°
105°
3m
Q
(cont’d.)
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A piece of reflective material in the shape of a circle is
attached to the centre of the navigational marker at the
centroid O. The ratio of the area of the shaded region of
the navigational marker XYZ to the area of the reflective
material is 2 : 1.
c Determine the radius, r, of the circle. Write your answer
in metres, correct to three decimal places.
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Essential Further Mathematics — Module 2 Geometry and trigonometry
e Calculate the length PR. Write your answer in metres,
correct to two decimal places.
35°
105°
Q
3.5 m
3m
Z
2.7 m
Y
[VCAA 2003]
M
PL
f Calculate the length of the vertical pole
RZ. Write your answer correct to the
nearest centimetre.
R
E
The second piece of shade cloth PQR is
P
attached to three vertical poles located
3.5 m
at X, Y and Z as shown in the diagram.
Poles PX and QY are each 3.5 metres
X
long. The horizontal distance YZ is 2.7 metres.
SA
Revision
440
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