chapter 16 i angles of elevation & depressions enhance

8/3/2019 Chapter 16 I Angles of Elevation & Depressions ENHANCE

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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 =


OB

(iii)

(ii)

(i)

Example

P

Q

R

S


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(b)

P

Q

R

S




(c)

E

F

G

H




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


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:


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 '


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 '


P

Q

T

R

JEBAT

BLOCKTUAH

BLOCK

DIAGRAM 17

N

MJ L

K

80 m 60 m

DIAGRAM 8


15/21

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


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


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


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 '


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


chapter 16 i angles of elevation & depressions enhance

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