Chapter 16 I Angles of Elevation & Depressions ENHANCE
Chapter 16 I Angles of Elevation & Depressions ENHANCE
Chapter 16 I Angles of Elevation & Depressions ENHANCE
16.1 Important concepts 6.1.1 The angle of elevation of a point A from a lower point O is the acute angle measured up from the horizontal line OB through point O to the line of sight, OA.
A (object)
eo lin f ht sig
Exersice: For each of the following situations, complete the diagram by stating i. Line of sight ii. Angle of elevation iii. Horizontal line
(i)
(i)
229
(i)
(i)
B (ii) (iii) O
(ii) O (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 stating i. Line of sight ii. Angle of depression iii. Horizontal line
(i) O
(ii i)
(ii)
O
(ii i)
(i) (ii)
(a) i. ii.
iii.
230
(i) (ii)
(i) (ii)
(iii )
i) (ii
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
Example Q
(a)
K J
P R
(i) Horizontal line = PR (ii) Angle of elevation = QPR (iii) Angle of depression = RPS
(i)
Horizontal line =
231
(b)
(c)
P E Q
F G S
R
(i) (ii) (iii) Horizontal line = Angle of elevation = Angle of depression =
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 Example
Nurul is watching a plane in the sky. The plane Effendi is standing at the balcony of his house. He is 1500m above the ground. Nurul is 1.75 m observes that the angle of depression of a car, tall and the angle of elevation of the plane from which is 7 m away, is 540. 0 her eye level is 27 . 540
27 1.75m
1500m
7m
(a)
A car is 18m away from a 5-level building. The angle of elevation of the top of the building from the car is 520.
(c)
Azizah is 155 m tall. The angle of depression of a cat from her eye level is
360.
away from the building. The angles of depression of John from Hassans flat and Ganeshs flat are 490 and 380 respectively.
Example: Rashid is 1.7 m tall and Syarif is 1.05 m tall They are standing 5 m apart. Calculate the angle of depression of Syarif from Rashid. 0 0 1.05 m 5m 1.7 m
= tan 1
= 7.410
Exercise:
233
(a)
Zainal is sitting at his desk. His form teacher, 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 angle of 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.
1.6m 20 m
N
2.8m 0.7m p
(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.
P M
(d) In the diagram, P, Q and R are three points on a horizontal ground. Given that MQ = 3 m, NR = 5 m and the angle of depression of R from M is 220, find (i) the length of QR (ii) the angle of elevation of N from M
234
(f) Raj and Kumar stay at the 15th and the 18th
floor 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) (ii) the distance between the building and the car the height between the 15th floor and the 18th of the building
1.5 m
10 m (e) In the diagram, PQ is a vertical pole. The angle of elevation of P from Azmis eye level is 190.
(i) Calculate the angle of depression of Q from Azmis eye level. (ii) Find the height of the pole
235
T
100 m
Q
55 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
1. In the diagram, TQ is a vertical flagpole standing on a horizontal ground. Given that PQ = 20 m and TQ = 6 m, the angle of elevation of T and P is A B 16.70 17.50 C 72.50 D 73.30
R 6.2 m P E 30 m F h
4. In the diagram, angle of depression of point P from point R is 150. Find the value of h A B 16.6 m 18.9 m C 21.5 m D 23.1 m 3. In the diagram, the angle of elevation of the top of the tree from F is 220. The height of the tree in metres is A 11.24 B 12.12 C 21.08 D 27.82
P R 2m
5m
2m
16 m
5. In the diagram, SQ is a vertical pole standing 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 B 130 340 43 C 55017 D 770
6. In the diagram, PQ and RS are two vertical poles standing on a horizontal ground. Given that the angle of depression of R from P is 100, the height of PQ in metres is A B 3.96 4.69 C 4.82 D 4.98
236
U M 2.5 m N V
4.8 m
E 8m F
10 m G 9m
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 19.50 NV in metres is A B 6.2 6.5 C 7.1 D 7.3
8. In the diagram, E, F and G are three points on a horizontal ground. PE is a vertical pole and the angle of elevation of P from F is 15.50. Find the angle of depression of G from P A B 12.50 13.80 C 14.50 D 14.8
0
P 6m Q
R
9. In the diagram, PQ and TS are two vertical poles on a horizontal plane QRS. Given that the angle of elevation of P from R is 380 , the angle of elevation of P from T is A B 18.20 18.80 C 19.40 D 19.8
0
E
10. Angle of elevation of... P from F Q from G R from E
Value of angle 22 0 28 0 10 0
In the diagram, E, F and G are three points on a horizontal ground. PE, QF and RG are three vertical poles. Based on the information in the table, calculate the perimeter of the triangle EFG A B 25.8 m 26.7 m C 27.6 m D 28.6 m
237
R 6m
15 m
11. In the diagram, QR is a vertical pole on a horizontal ground. The angle of elevation of R from P is 280. Calculate the height, in m, of the pole QR A 7.04 C 13.24 B 7.98 D 28.21
12. In the diagram, PR is a vertical wall 6m high. The angle of depression of Q from R is 250. Calculate the distance, in m, from P to Q A B 2.80 8.57 C 12.87 D 22.39
S R 18 m
10 m
25 m
13. In the diagram, MN represents a tower standing on horizontal ground. The angle of depression of L from N is 520. Calculate the height, in m, of the tower MN A 7.88 C 12.80 B 9.00 D 18.81
14. In the diagram, QRS represents a vertical wall of height 18 m. The angle of elevation of R from P is 260. Calculate the height, in m, from R to S A B 4.47 5.81 C 6.81 D 12.19
S R Q 5m P P 20 m Q 15. In the diagram, QRS represents a building. The angle of elevation of R from P is 200 while the angle of depression of P from S is 300. Calculate the distance, in m, from R to S A 3.53 C 18.83 B 4.27 D 27.36
S 8m R
16. In the diagram, PQ and RS are two vertical poles. The angle of elevation of Q from R is 220. Calculate the distance PR in metres A 8.60 B 12.38 C 10.88 D 22.90
238
S T 10 m 4m P
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 B 8.49 10.03 C 10.19 D 45.42
17. In the diagram, P , Q and R are three points on horizontal ground. PT and RS are two vertical poles. The angle of elevation of T 0 from Q is 45 whereas the angle of 0 depression of Q from S is 35 . Find the distance, in m, between the two poles A 8.19 B 9.74 C 11.00 D 12.19
S
10 m
20 m
R 5m
4m
19. In the diagram, PQ is a vertical pole. The angle of elevation of P from S is 450. Calculate the angle of depression of R from P A B 1402 36052 C 41025 D 5308
20. In the diagram, PS is a vertical pillar. The angle of depression of R from S is 200. Calculate the distance, in m, from Q to R A B 5.40 7.42 C 7.64 D 9.40
239
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 3m T Calculate the value of h. A 10.4 B 12.0 SPM 2004 (July) 2. In Diagram , S, M and T are collinear points on a horizontal plane. SM = MT and TU is a vertical pole.
U 25 m
hm U
C 14.1 D 17.9
10 m
T M DIAGRAM 10
A wire is tied from the vertex U to the point S. Calculate the angle of elevation of vertex U from the point M A C
B D
240
SPM 2004 In Diagram , MN and PQR are two vertical poles on a horizontal plane. P M N R Q
3.
A B C D 4.
P MQ M P Q M P R N PR
The angle of depression of peak P from peak R is 34 0 . Calculate the height, in m, of the pole PQ.
A B C D
241
5. SPM 2005 (July) Diagram 17 shows two flats adjacent to each other on a horizontal plane. Four friends are seating at balconies P, Q, R and T respectively. Which of the following is the angle of elevation of P from R?
P JEBAT BLOCK T Q R TUAH BLOCK
DIAGRAM 17
A B C D
PRQ PRT
RPQ
PTP
6. In Diagram 8, J, K, L and M are four points on a horizontal ground. MN is a vertical pole with the height of 20 m, JM = ML , JKL = 90
N
80 m DIAGRAM 8 K
60 m
6625'
242
B C D
SPM 2005
7. Diagram below shows two vertical poles, PS and QR, on a horizontal plane. P Q 25 m S
10 m
The angle of depression of vertex Q from vertex P is 42 0 . Calculate the angle of elevation of vertex Q from S.
A B C D
32 0 41 0 59 57 0 59 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 on the 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 the rescuer from the bottom of the tower. 14.85 A 17.14 B 18.13 C
Angles of Elevation and Depression 243
20.92
244
35 m
DIAGRAM 9
60 m
DIAGRAM 10
P and Q are two points on the top of the towers as shown. R is a point vertically below Q. The angle of elevation of Q from P is 55 o and the angle of elevation of R from P is 40 o . Calculate the distance, in m, from Q to R. A B C D
35 34 29 50 15 53 10 58
245
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 . R
20 m
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 R
S 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 B C D 10. 94 11.03 15.59 17.92
246
13. SPM Nov 2008, Q16 Diagram 9 shows a tower PQ. The points P, R and S lie on a horizontal plane. Q
5m
The angle of depression of R from Q is 45 . Find the angle of elevation of Q from S. 2237 ' A 3513' B 5447 ' C 6723' D
247
ANSWERS
Chapter 16: Angles of Elevation and Depression 16.1.1
a) i. Line of sight ii Angle of elevation iii.Horizontal line b) i. Line of sight ii Angle of elevation iii.Horizontal line c) i. Line of sight ii Angle of elevation iii.Horizontal line
52
38 18 m 10 m
32
1.57 m 10 m
(c)
36 0 1.55m
(d)
248
49
John
38
Hassan
Ganesh
11.5 m
16.2.3
(a) (b) (c) (d) (e) (f)
8.59
8.88 m 14.9 m (i) 7.43 m (i) 8.530 (i) 49.52 m (ii) 15.10 (ii) 4.94 m (ii) 35.08 m
Questions According to Examination Format 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. A C B D C C B A C D B C C B B B D C D
20. A
249