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Chapter 16: ANGLES OF ELEVATION AND DEPRESSION
16.1 Important concepts
6.1.1 The angle of elevation of a pointA from a lower point O is the acute angle measuredup from the horizontal line OB through point O to the line of sight, OA.
Exersice:
For each of the following situations, complete the diagram by stating
i. Line of sight
ii. Angle of elevation
iii. Horizontal line
OB
(i)
(ii)
(iii)
(a) i.
ii.iii.
OB
(i)
(ii)
(iii)
(b) i.
ii.iii.
Angles of Elevation and Depression 229
O
A
Angle of elevation
Horizontal line
(object)
lineof
sight
B
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OB
(i)
(ii)
(iii)
(c) i.ii.
iii.
OB
(i)
(ii)
(iii)
(d) i.
ii.
iii.
6.1.2 The angle of depression of a point P from a higher point O is the acute angle measured
down from the horizontal line, OR through point O to the line of sight, OP
Exercise:
For each of the following situations, complete the diagram by statingi.Line of sight
ii. Angle of depression
iii.Horizontal line
O B
(iii)
(ii)
(i)
(a) i.
ii.iii.
O
B
(iii)
(ii)
(i)
(b) i.
ii.iii.
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OB
(iii)
(ii)
(i )
(c) i.
ii.
iii.
(d) i.
ii.iii.
16.2 Teaching and Learning (Progressive Practices)
16.2.1 Identify the horizontal line, the angle of elevation and the angle of depression
1. For each of the following situations, identify
(i) the horizontal line
(ii) the angle of elevation
(iii) the angle of depression
(i) Horizontal line = PR
(ii) Angle of elevation = QPR(iii) Angle of depression = RPS
(a)
K
L
M
J
(i) Horizontal line =
(ii) Angle of elevation =
(iii) Angle of depression =
Angles of Elevation and Depression 231
OB
(iii)
(ii)
(i)
Example
P
Q
R
S
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(b)
P
Q
R
S
(i) Horizontal line =
(ii) Angle of elevation =
(iii) Angle of depression =
(c)
E
F
G
H
(i) Horizontal line =
(ii) Angle of elevation =
(iii) Angle of depression =
16.2.2 Draw a diagram to represent each of the following situations.
Example
Nurul is watching a plane in the sky. The plane
is 1500m above the ground. Nurul is 1.75 m
tall and the angle of elevation of the plane from
her eye level is 270.
Example
Effendi is standing at the balcony of his house. He
observes that the angle of depression of a car,
which is 7 m away, is 540.
540
7m
(a) A car is 18m away from a 5-level building. The angle of elevation of thetop of the building from the car is 520.
(b) Zainab is standing in the middle of twotrees which are 10 m apart. Zainab is
1.57m tall and the angles of elevation of
the tops of trees are 380 and 320
respectively.
(c) Azizah is 155 m tall. The angle ofdepression of a cat from her eye level is
(d) Hassan and Ganesh stay at different floorsof a building. Their friend, John is 11.5 m
Angles of Elevation and Depression 232
1500m
1.75m27
0
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360.away from the building. The angles of
depression of John from Hassans flat and
Ganeshs flat are 490 and 380
respectively.
16.2.3 Solve each of the following problems
Exercise:
Angles of Elevation and Depression 233
Example:
Rashid is 1.7 m tall and Syarif is 1.05 m tallThey are standing 5 m apart. Calculate theangle of depression of Syarif from Rashid.
0
0
5 m
1.7 m
1.05 m
Solving:
RT = 1.7 1.05
= 0.65 m
5
65.0tan =
5
65.tan
1 o=
041.7=
The angle of depression of Syarif
from Rashid is 7.41 0
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(a) Zainal is sitting at his desk. His formteacher, Mr. Gan, who sits in front of
Zainal is 1.68m tall and 4.5m away from
Zainal. Given that the height from the
ground level to Zainals eye level when
he is sitting down is 1 m, find the angleof elevation of the top of Mr. Gans head
from Zainals eye level.
(b) Based on the diagram, calculate the height
of the flagpole if the angle of elevation of
the top of the flagpole from the lady's eye
level is 200.
20 m
1.6m
p
0.7m
2.8m
(c) Based on the diagram, calculate the value of
p if the angle of elevation of the top of the
traffic light from Alis eye level is 80.
M
P
Q R
(d) In the diagram,P, Q and R are three
points on a horizontal ground. Given thatMQ = 3 m,NR = 5 m and the angle ofdepression ofR from M is 220, find
(i) the length of QR(ii) the angle of elevation ofNfrom M
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1.5 m
10 m
P
Q
(e) In the diagram,PQ is a vertical pole. Theangle of elevation ofP from Azmis eyelevel is 190.
(i) Calculate the angle of depression of
Q from Azmis eye level.(ii) Find the height of the pole
(f) Raj and Kumar stay at the 15th and the 18thfloor of a building respectively. The angles
of depression of a car from Rajs flat and
Kumars flat are 480 and 61.20
respectively. Given that Rajs flat is 55 m
high from the horizontal ground. Calculate
(i) the distance between the building
and the car
(ii) the height between the 15th floor
and the 18th of the building
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Questions According to Examination Format
P Q
T
1. In the diagram, TQ is a vertical flagpolestanding on a horizontal ground. Given that
PQ = 20 m and TQ = 6 m, the angle ofelevation of T and P is
A 16.70 C 72.50
B 17.50 D 73.30
55 m
100 m
2. Based on the diagram, find the angle of
depression of the ball from the top of the
building.
A 24.20 C 61.20
B 28.80 D 65.80
E F30 m
3. In the diagram, the angle of elevation of the
top of the tree fromF is 220. The height ofthe tree in metres is
A 11.24 C 21.08B 12.12 D 27.82
Q
R
P
h
6.2 m
4. In the diagram, angle of depression of point
P from pointR is 150. Find the value of h
A 16.6 m C 21.5 m
B 18.9 m D 23.1 m
S
P RQ5 m 2 m
5. In the diagram, SQ is a vertical polestanding on a horizontal ground. Given that
the angle of elevation of S from P is 300
,find the angle of elevation of S from R
A 130 C 55017
B 34043 D 770
P
Q 16 m
R
S
2 m
6. In the diagram,PQ andRS are two vertical
poles standing on a horizontal ground. Given
that the angle of depression ofR from P is100, the height of PQ in metres is
A 3.96 C 4.82
B 4.69 D 4.98
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M
N
U
V
2.5 m
4.8 m
7. In the diagram, MN and UV are two
vertical poles standing on a horizontal
ground. Given that the angle of depression of
M from U is 19.50NV in metres is
A 6.2 C 7.1
B 6.5 D 7.3
E
F
G
P
9 m8 m
10 m
8. In the diagram, E, Fand G are threepoints on a horizontal ground. PE is a
vertical poleand the angle of elevation ofPfrom F is 15.50. Find the angle ofdepression of G from P
A 12.50 C 14.50
B 13.80 D 14.80
R
S
P
Q
6 m
3.3 m
T
9. In the diagram,PQ and TS are two
vertical poles on a horizontal plane QRS.Given that the angle of elevation of Pfrom R is 380, the angle of elevation ofPfrom T is
A 18.20 C 19.40
B 18.80 D 19.80
P
E
Q
F
R
G
2 m4.2 m
3.8 m
10.
100R from E
280Q from G
220P from F
Value of angleAngle of e levation of...
In the diagram,E, F and G are three points ona horizontal ground. PE, QF and RG arethree vertical poles. Based on the
information in the table, calculate theperimeter of the triangle EFG
A 25.8 m C 27.6 mB 26.7 m D 28.6 m
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R
P Q
15 m
11. In the diagram, QR is a vertical pole on ahorizontal ground. The angle of elevation
ofR from P is 280. Calculate the height,in m, of the pole QRA 7.04 C 13.24
B 7.98 D 28.21
R
PQ
6 m
12. In the diagram,PR is a vertical wall 6mhigh. The angle of depression of Q from Ris 250. Calculate the distance, in m, from Pto Q
A 2.80 C 12.87
B 8.57 D 22.39
10 m
N
ML
13. In the diagram, MN represents a tower
standing on horizontal ground. The angle
of depression ofL from N is 520.Calculate the height, in m, of the tower
MN
A 7.88 C 12.80B 9.00 D 18.81
P Q
R
S
25 m
18 m
14. In the diagram, QRS represents a verticalwall of height 18 m. The angle of elevation
of R fromPis 260. Calculate the height, inm, from R to S
A 4.47 C 6.81B 5.81 D 12.19
S
R
P Q20 m15. In the diagram, QRS represents a
building. The angle of elevation ofR from Pis 200 while the angle of depression ofP from
S is 300. Calculate the distance, in m, from Rto S
A 3.53 C 18.83
B 4.27 D 27.36
Q
P
S
R
5 m8 m
16. In the diagram,PQ and RS are two
vertical poles. The angle of elevation of
Q fromR is 220. Calculate the distancePR in metres
A 8.60 C 10.88
B 12.38 D 22.90
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T
P Q R
S
10 m
4 m
17. In the diagram, P , Q and R are threepoints on horizontal ground.PTand RS aretwo verticalpoles.The angleof elevation of T
from Q is 450 whereas the angle of
depression of Q from S is 350. Find the
distance, in m, between the two poles
A 8.19 C 11.00
B 9.74 D 12.19
18. A man, 1.7 m tall, is standing 20 m from a
monument. The angle of elevation of the top
of the monument from the eyes of the man is
230. The height, in m, of the monument is
A 8.49 C 10.19
B 10.03 D 45.42
P
Q R S
20 m
5 m
19. In the diagram,PQ is a vertical pole. Theangle of elevation ofP from S is 450.
Calculate the angle of depression of Rfrom P
A 1402 C 41025
B 36052 D 5308
S
P QR
10m
4 m
20. In the diagram,PS is a vertical pillar. Theangle of depression ofR from S is 200.Calculate the distance, in m, from Q to R
A 5.40 C 7.64
B 7.42 D 9.40
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SPM Past Year Questions
SPM 2003
1. In Diagram, TS and UV are two vertical poles standing on horizontal ground. The angle of
elevation of V from S is .42 0
V
S
3 m h m
T U
Calculate the value of h.
A 10.4 C 14.1
B 12.0 D 17.9
SPM 2004 (July)
2. In Diagram , S, M and T arecollinear points on a horizontal plane. SM = MT and TU is a
vertical pole.
A wire is tied from the vertex U to the point S. Calculate the angle of elevation of vertex Ufrom the point M
A 22 37 ' B 50 12 '
C 39 48 ' D 67 23 '
Angles of Elevation and Depression 240
U
TS
M
25 m10 m
DIAGRAM 10
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SPM 2004
In Diagram , MN and PQR are two vertical poles on a horizontal plane.
P
M Q
N R
The angle of elevation of point P from point M is
A MPQ
B PMQ
C PMR
D PNR
4. In Diagram, PQ and RS are two vertical poles on a horizontal plane.R
P 25 m
Q 20 m S
The angle of depression of peak P from peak R is 34 0 . Calculate the height, in m, of the pole
PQ.
A 4.65
B 8.42
C 11.51
D 13.82
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5. SPM 2005 (July)
Diagram 17 shows two flats adjacent to each other on a horizontal plane. Four friends are seating
at balconiesP, Q, R and T respectively. Which of the following is the angle of elevation of Pfrom R?
A PRQ
B PRT
C RPQ
D PTP
6. In Diagram 8,J, K, L and Mare four points on a horizontal ground. MN is a vertical pole withthe height of 20 m,JM = ML , 90JKL =
Calculate the angle of elevation of vertexN from the point L
A 66 25 '
Angles of Elevation and Depression 242
P
Q
T
R
JEBAT
BLOCKTUAH
BLOCK
DIAGRAM 17
N
MJ L
K
80 m 60 m
DIAGRAM 8
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B 26 34 '
C 33 41'
D 21 48 '
SPM 2005
7. Diagram below shows two vertical poles, PS and QR, on a horizontal plane.
P
Q
25 m
S R
The angle of depression of vertex Q from vertex P is 42 0 . Calculate the angle of elevation of
vertex Q from S.
A 32 0
B 41 0 59 '
C 57 0 59 '
D 68 0 11 '
8. SPM Year 2006 (July)
Diagram below shows the position of a rescuer and the position of an occupant who is trapped onthe 3rd floor of a tower.
Tower
Occupant
Rescuer 12 m
Te angle of elevation of the occupant from the rescuer is 35. Calculate the distance, in m, of therescuer from the bottom of the tower.
A 14.85
B 17.14
C 18.13
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10 m
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D 20.92
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9. SPM Nov 2006, Q16
In Diagram 9,JKis a vertical flag pole.KL is horizontal.
The angle of elevation of J fromL is o42 . The height, in m, of the flag pole is
A 4223
B 0126
C 5131
D 8738
10. SPM Jun 2007, Q17
Diagram 10 shows two towers on a horizontal plane.
Pand Q are two points on the top of the towers as shown. R is a point vertically below Q.
The angle of elevation ofQ fromPis o55 and the angle of elevation ofR fromPis o40 .
Calculate the distance, in m, from Q toR.
A 3435
B 5029
C 5315
D 5810
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DIAGRAM 9
J
K 35 m L
P
Q
R
60 m
DIAGRAM 10
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11. SPM Nov 2007, Q15
In Diagram 9, PQ and RS are two vertical poles on a horizontal plane. The height of RS
is twice the height of PQ. The angle of depression of P from vertex R is 53 .
Calculate the height, in m, of RS.
A 15.07B 26.54
C 30.14
D 53.08
12. SPM June 2008, Q16
In Diagram X, Q and S are two points on a horizontal. R is the top of a vertical flagpole
The angle of elevation of R from Q is 52 o. The distance between Q and S is 14 m.
Calculate the height, in m, of the flagpole SR.
A 10. 94B 11.03
C 15.59
D 17.92
Angles of Elevation and Depression 246
P
Q
R
S20 m
R
S
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13. SPM Nov 2008, Q16
Diagram 9 shows a tower PQ. The points P, R and S lie on a horizontal plane.
The angle of depression of R from Q is 45 . Find the angle of elevation of Q from S.
A 22 37 '
B 35 13 '
C 54 47 '
D 67 23 '
Angles of Elevation and Depression 247
Q
P R S
5 m
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ANSWERS
Chapter 16: Angles of Elevation and Depression
16.1.1
a) i. Line of sightii Angle of elevation
iii.Horizontal line
b) i. Line of sightii Angle of elevation
iii.Horizontal line
c) i. Line of sightii Angle of elevation
iii.Horizontal line
16.1.2
a) i. Horizontal lineii Angle of depression
iii.Line of sight
b) Horizontal line
ii Angle of depression
iii.Line of sight
c) Horizontal line
ii Angle of depression
iii.Line of sight
16.2.1
(a) i. JKii. JK
iii. JM
(b) i. QSii. PS
iii. RS
(c) i. EGii. EF
iii. EH
16.2.2
(a)
52
0
18 m
(b)
3238
10 m 10 m
1.57 m
0 0
(c)
1.55m
36 0
(d)
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John
Hassan
Ganesh11.5 m
38
490
0
16.2.3(a) 8.59(b) 8.88 m
(c) 14.9 m
(d) (i) 7.43 m (ii) 15.10
(e) (i) 8.530 (ii) 4.94 m
(f) (i) 49.52 m (ii) 35.08 m
Questions According to Examination
Format
1. A
2. C
3. B
4. D
5. C6. C
7. B
8. A
9. C
10. D
11. B
12. C
13. C
14. B
15. B
16. B
17. D18. C19. D
20. A
Past Year Questions
1. B
2. B3. B
4. C
5. A6. D7. C
8. B
9. C10. A
11. B
12. D13. B
Angles of Elevation and Depression 249