Scale Drawing Questions and Answers

Scale Drawing Questions

1.  Three mountains Mikai, Kembo and Chaka in a village are situated in such a way that Kembo is 900m on a bearing of 120⁰ from Mikai. Mt. Chaka is 1200m on a bearing of 030⁰ from Kembo.

1. Draw a sketch showing the position of the three mountains   (1 mk)
2. Calculate the distance of Mt. Chaka from Mt. Mikai (2 mks)

2.  Shopping centres XY and Z are such that Y is 12km south of X and Z is 15kn from X. Z is on a bearing of N30⁰W from Y. Calculate and give compass bearing of Z from X. (4mks)

3.  Four telephone posts PQR and S stand on a level ground such that Q is 28m on a bearing of 060⁰  from P. R is 20m to the south of Q and S is 16m on a bearing of 140⁰ from P.

 (a) Using a scale of 1cm represent 4m show the relative positions of the posts. (4mks)

 (b) Find the distance and bearing of R from S. (3mks)

 (c) If the height of post P is 15.6m. on a separate scale drawing, draw a diagram and determine the  angle of depression of post R from the top of post P. (3mks)

 (Same scale as above)

4.  Alice chepchumba on her cycling practice cycled on a bearing of 120^o for 5.5km, then on a bearing of 200^o for 8km finally he turned northwards for 13.5km, by scale drawing determine her final position from starting point. (4 marks)

5.  A surveyor recorded the measurement of field in a field book using lines AB = 260m as shown below.

 a) Use a suitable scale to draw the map of the field. (2 marks)

 b) Find the area of the field. (2 marks)

6. (a) In a Safari rally drivers are to follow route ABCGA. B is 250km from A on a bearing of 075⁰
from A. C is on a bearing of 110⁰ from A and 280km from B. the bearing of C from D is 140⁰ and at a distance of 300km. By scale drawing, show the position of the point A, B, C and D. (4 mks)

 (b) Determine

 (i) Distance of A from C (2 mks)

 (ii) The bearing of B from C  (1 mk)

 (iii) The distance and bearing of A from D (3 mks)

7.  Town X is 20km in the direction 060^o from Y and Z is 30km in the direction 150^o from Y. Using the scale 1cm represents 5km, find by scale drawing;

 (a)  the bearing of Y from Z

 (b)  the distance of X from Z (4mks)

8.  A field was surveyed and its measurements recorded in a field book as shown below.

(a) Using a scale of 1cm to represent 10m, draw a map of the field.  (4mks)

(b) Calculate the area of the field.

(i) in square metres.  (4mks)

(ii) in hectares.  (2mks)

9.  A plane leaves town P to town Q on a bearing of 130⁰ and a distance of 350km. it then flies to  town R 500km away and on a bearing 060⁰. Find by scale drawing the distance of R from P (3mks)

10.  A surveyor recorded the following information in his field book after taking measurements in  metres of a plot. The baseline is the straight line AH = 300m.

(a) Using a scale of 1cm to represent 20m, draw an accurate diagram of the plot. (5mks)

b) Use your diagram to calculate the actual area of the field in hectares (5mks)

11.  Three town P,Q and R are such that P is on a bearing of 120° and 20 km from Q. Town R is on bearing of 220^o and 12km from P

 a)  Using a scale of 1 cm to 2 km, draw and locate the positions of the three towns.  (3mks)

b)  Measure

i)  the distance between Q and R in kilometres. (2mks)  

ii)  the bearing of P from R.  (1mk)

iii)  the bearing of R from Q.  (2mks)

 c)  Calculate the area of the figure bounded by PQR. (2mks)

12.  The area of a forest on a map whose scale is 1:50,000 is 17cm². Calculate the area of the forest in hectares. (2 mks)

13.  Four towns P, Q, R and S are such that town Q is 120km due east of town P. Town R is 160km due North of town Q. Town S is on a bearing of 330^o from P and on a bearing 300 ^o from R. use a ruler and a pair of compasses only for all your constructions.

a)  Using a scale of 1cm to represent 50km, construct a scale drawing showing the positions P, Q, R and S. (6 mks)

b)  Use the scale to determine

1. The distance from town S to town P. (1 mk)

1. The distance from town S to town R. (1 mk)
2. The bearing of town S from town Q. (2 mks)

14.  The actual area of an estate is 3510 hectares. The estate is represented by a rectangle measuring 2.6cm by 1.5cm on the map whose scale is l:n. Find the value of n  (3 mks)

15.  The following measurements were recorded in a field book of a farm in metres (xy = 400m)

 a) Using a scale of 1cm representing 4000 cm, draw an accurate map of the farm.

 b) If the farm is on sale at Kshs.80,000.00 per hectare, find how much it costs.  (10 mks)

16.  Four points A, B, C and D are situated on a horizontal plane such that B is 250 m on a bearing of 070⁰ from A. C is 325 m on a bearing of 150⁰ from B. D is due west of C and on a bearing of 210⁰ from B.  (6 marks)

1. Using a scale of 1 cm to 50 m draw an accurate drawing to show the position of A, B, C and D.
2. Use your scale drawing fo find the :
   1. The distance between A and D (2 marks)
   2. The bearing of A from D (2 marks)

17.  Town X is 13.5km from town Y on a bearing of 028^o. A matatu leaves y at 7:35a.m

towards a bearing of 080^o. The matatu is at point Z due south of X at 8:55a.m

 (a) Calculate the average speed of the matatu from Y to Z

 (b) If the matatu continues on the same bearing, calculate the distance it covers from Z

when it is East of X  

18.  Three towns X, Y and Z are such that Y is 500km on a bearing of 315^o from X. Z is on

a bearing of 230^o from X. given that the distance between Y and Z is 800km.

 (a) using a scale of 1cm to represent 100km, draw a scale diagram to show the position

of the Towns  

 (b) Find the bearing of;

  (i) X from Z

  (ii) Z from Y  

 (c) Use the scale drawing to find the distance from X to Z

19.  Two aeroplanes S and R leave an airport at the same time. S flies on the bearing of 240^o

at 750Km/h while R flies due East at 600Km/hr..

 (a) (i) Calculate the distance of each aeroplane after 30minutes  

  (ii) Using a scale of 1cm to represent 50km make an accurate scale drawing to show

the positions of the aeroplanes after 30minutes

 (b) (i) Use the scale drawing to find the distance between the two aeroplanes after 30minutes

  (ii) If each aeroplane landed after 30minutes and S received a signal to join R in 45minutes.

Find its speed  

 (c) Determine the bearing of :

  (i) S from R  

  (ii) R from S

20.  The table below gives a field book showing the results of a survey of a section of a piece of land

between A and E. All measurements are in metres.

 (a) Draw a sketch of the land.

 (b) Calculate the area of this piece of land.  

21. Three towns A B and C are situated such that town A is 40km from B on a bearing of 280^o.

C is 60km from B on a bearing of 130^o. Another town D is only 10km from C on a bearing of 210^o.

(a) Drawing accurately and using a scale of 1cm to 10km find the:-

(b) Distance from A to C and the bearing of A from C

(c) (i) Distance of B from D

(ii) Distance of A from D

(iii) Bearing of A from D

(iv) Bearing of C from D

22.  A train left Naivasha for Nakuru at 1000hours. It traveled at an average speed of 45km/h

and reached Gilgil after 40minutes. It then covered the remaining 50km in 1½ hours. A second

train left Nakuru for Naivasha at 1015 hours and arrived at Gilgil at the same time as the first

train arrived at Nakuru.

 a) Using a scale of 1cm to represent 10minutes in the time axis and 1cm to represent 10km

on the distance axis, draw on the same axes the graphs to show the movement of the two

trains

b) use your graph to find;

  i) the distance between Naivasha and Nakuru

  ii) the time at which the train met

 c) calculate the average speed, in km/h of the second train

23.  On a certain map, a road 20km long is represented by a line 4cm long. Calculate the area

of a rectangular plot represented by dimensions 2.4cm by 1.5cm on this map – leaving

your answer in hectares

24.  A port B is on a bearings of 080^o from a port A and at a distance of 95km. a submarine is

stationed at a port D, which is on a bearing of 200^o from A, and a distance of 124km from B.

A ship leaves B and moves directly southwards to an island P, which is on a bearing of 140^o

from A. the submarine at D on realizing that the ship was heading for the island P, decides to

head straight for the island to intercept the ship.

  (a) Using a scale of 1cm to represent 10km draw a diagram to show the positions of A,B,D, and P

 (b) Hence;

  Determine

  (i) the distance from A to D

  (ii) the bearing of the submarine from the ship when the ship was setting off from B  

  (iii) the bearing of the island P from D  

  (iv) the distance the submarine had to cover to reach the island P

25.  Use a scale of 1cm represents 50km in these questions. Five towns A, B, C, D and E are

situated such that A is 200 km from B on a bearing of 050° from E. C is 300 km from B on

a bearing of 150° from B. D is 350km on a bearing of 240° from C. E is 200km from D and the bearing of D from E is 100°

 a) Draw the diagram representing the positions of the towns

 b) From the diagram, determine;

  i) The distance in km of A from E

  ii) The bearing of D from B

26.  Four towns P, Q, R & S are such that P is 280 km North of R, S is190 km from R on a

bearing of 310^o and Q is 240 km from P on a bearing of 105^o.

a) Using scale of 1 cm rep. 50 km, locate the four towns.

b) Find; (i) distance SQ.  

(ii) Bearing of S from Q.

  (iii) The shortest distance between P and side QR.

27.  Four ships are at sea such that a streamliner S is 150km on a bearing of 025° from a cargo

ship C. A trawler T is 300km on a bearing of 145° from the cargo ship and a yacht Y is due

West of C and on a bearing of 300° from T.

 a) Using a scale of 1cm= 50km, draw on accurate scale drawing showing the positions of S, C, T

and Y

b) By measurement from your scale drawing determine:

 i) The distance and bearing of Y from S

 ii) The distance ST

 iii) The distance YT

28.  A tea farm in Kakamega forest was surveyed and the results were recorded in the surveyors

note book as shown below. The measurements are in meters

Using a scale of 1: 25, draw the map of the plot and hence calculate the area of the plot in Hectares

29.  The information below shows the entries in a surveyor’s field book after a survey of a farm.

 XY = 280m is the baseline. All measurements are in metres

  (a) Use a scale of 1cm represents 20m to draw the map of the farm  

  (b) Estimate the area of the farm in hectares  

  (c) If the point Y lies due north of X, find correct to 1 decimal place, the :

  (i) Bearing of E from X

  (ii) Distance of E from X

30.  The measurements of a flower garden were recorded in a surveyor’s field book as shown.

Draw a sketch of the field and find its area. (Measurements are in m)

31.  A map has a scale 1:40,000:

 (a) Calculate the distance between two points on the ground if the corresponding distance

shown on the map is 3.25cm

 (b) Calculate the area in the map of woodland which occupies 36ha on the ground

32.  Three scouts John, Peter and Samwel stand on three adjacent peaks of equal altitude

on mountain range. The distance between John and Peter is 800metres and the bearing

of Peter from John is 020^o. The distance between John and Samwel is 1500metres, and the

bearing of Samwel from John is 320^o.

(a) Calculate the bearing of John from Peter  

(b) Calculate:- (i)
the distance

(ii) the bearing of Samwel from Peter

33.  The figure below represents a surveyor’s sketch of a plot of land. Calculate the area of the plot in

square metres given that XY = 50m, XK = 20m, XM = 25m, XL = 35m, KA = 40m, MD = 38m

and LB = YC = 60m.  

34.  Two boats P and Q are located 30km apart; P being due North of Q. An observer at P

spots a ship whose bearing he finds as S 56^oE from Q, the bearing of the same ship is 038^o. Calculate the distance of the ship from Q to 2 decimal places  

35.  A map is drawn to scale of 1:100,000. What area in km², is represented by a rectangle

measuring 4.5cm by 5.4 cm  

36.  Two places A and B are 900km apart on the earth’s surface. If A is due North of B and

given that the latitude of A is 5^oN. Find the latitude of B. (Take radius of the earth to be 6370km)

37.  A car starts from rest and build up a speed of 40m/s in 1min 40seconds. It then travels

at this steady speed for 5minutes. Brakes are then applied and the car is brought to rest

in 2minutes.

 (a) Draw a velocity-time graph to show the journey

 (b) Use your graph to find;

  (i) the initial acceleration

  (ii) the deceleration when the car is brought to rest

(iii) the distance traveled

38.  The diagram below represents two vertical watch-towers AB and CD on a level ground.

 P and Q are two points on a straight road BD. The height of the tower AB is 20m and

road BD is 200m

 (a) A car moves from B towards D. At point P, the angle of depression of the car from

point A is 11.3^o. Calculate the distance BP to 4 significant figures

 (b) If the car takes 5 seconds to move from P to Q at an average speed of 36km/.hr. Calculate

the angle of depression of Q from A to 2 decimal places

 (c) Given that QC = 50.9m, calculate;

(i) the height of CD in metres to 2 decimal places

  (ii) the angle of elevation of A from C to the nearest degree  

39.  Town B is 180 km on a bearing of 050⁰ from town A. Another town C is on a bearing of 110⁰

from town A and on a bearing of 150⁰ from town B. A fourth town D is 240 km on a bearing of

320⁰ from A. Without using a scale drawing, calculate to the nearest kilometer.

(a) The distance AC

1. The distance CD

Scale drawing Answers

13. Positions

Q  B1

R  B1

S  B1

 Const 300 B1

1. B1

Scale B1

b.) i.  7.8 x 50 = 390 km. B1

ii.  7.10 x 50 = 355 km  B1

iii.  320⁰  B2

10

17.  a)   YZ = 13.5

Sin 28^o sin 100^o

Duration of travel = 8:55a.m – 7.35a.m

= ⁴/₃

Speed = 6.436

    ⁴/₃

   = 4.827km/hr

(b)  13.5 = 6.436 + ZQ

Sin 10^o Sin 118^o

6.436 + ZQ = 13.5 x sin118^o= 68.659

ZQ= 68.659-6.436

= 62.223

18. 1cm rep 100km

b) i) 049  1

  ii) 190  1

c)  6.7 0.1

670  10

19.  a) (i) Distance covered by s

= (750 x ½ )km = 375 km

  Distance covered by R

= (600 x ½ ) km = 300 km

(b) (i) Distance between the two aeroplanes

  = 12.5 x 50  = 625 + 5 km

 (ii) Speed = 625 x 60 km/hr

45

= 833 ¹/₃ km /h

 (c) (i) Bearing of S from R = 225^o

  (ii) The bearing of R from S = 72^o

20.  

 Area A: ½ x 25 (33 + 21) = 675

  Area B: ½ x 40 (21 x 42) = 1260

  Area C: ½ x 30 x 42 = 630

 Area D: ½ x 25 x 40 = 500

  Area E: ½ x 5 (40 + 25) = 162.5

  Area F: ½ X 60 (25 + 36) = 1830

 Area G: ½ x 5 x 36 = 90 √

= 5,147.5m²

21.  A to C = 96 ± 1 km

Bearing = 300^o

1. 62  1km
   
2. 97  1 km
   
   1. 304^o
      

030^o

22.  Graph

b) i) 80 km

  ii) 11.06a.m

c) Average speed of the 2^nd train

  Time taken = 80  1¹¹/₁₂ = 80 x 12

23

= 41.74km/h

23.  L.S.F = 4 = 1

2000000 500000

A.S.F = 1
² = 1

  5 x 10⁵ 2.5 x 10¹¹

Area of rectangle = (2.4 x 1.5) cm²

 = 3.6cm²

Actual area = 3.6 x 2.5 x 10¹¹ ha

100 x 10000

= 9 x 10⁵

= 900,000ha

24.  a) Δ ABD  ly constructed

Δ ABP

  b) i) AD = 4.5 + 0.1cm

  Distance A + D

  = 4.5 X 10 = 45km

  ii) Bearing of (i) from B

  = 241 + 1

  iii) Bearing P from D

= 123 = 2

  iv) Dp = 12.9 + 0.2 am

 Distance D + P = 12.9 X 10

= 129 km

25. a)

b) i)  6.8 + 0.1cm

Distance Ae = 340 + 5 km

ii) 180 + 18 = 198 + 2

26.  a)

b)  (i) SP = 7.8 x 50 = 390 km + 5 km

(ii) S & Q = 255^o
+ 1^o

(iii) 4 x 50 = 200 km + 5 km

27.  (a) Scale = 50km

Drawing accuratelyo

o

o

Lines drawn //

(b)By measurement:

(i) Distance SY = 6.9 x 50 = 345 5km

Bearing Y For S = 360^o – 114 = 2461^o

(ii) distance ST = 7.9 x 50 = 39.5 5km

1. distance YT = 9.8 x 50 = 490 5km
   

XY = 250m

Area of A = ½ x 50 x 60 = 1500m²

B = ½ x 70 x 60 = 2100m²

C = ½ (60 +80) x 120=11050m²

D = ½ x 80 x 80 = 3200m²

F = ½ x 10 x 70 = 350m²

Total area = 26600m²

Ha = 26600 = 2.66ha

10,000

29.

(b) Total area = area (1) + (2) + (3) + (4) +(5) + (6) + (7)

Area (1)= ½ x 90 x 100 = 4500m²

(2) = (100 + 105)10 = 10250m²

2

(3) = ½ x 90 x 105 = 4725m²

(4) = ½ x 50 x 110 = 2750m²

(5) = ½ x (110 + 45)70 = 5425m²

(6) = (45 + 95) 120 8400m²

  2

(7) = ½ x 40 x 95 = 1900m²

Total area = 37,950m²

In hectares = (37950) ha = 3.795ha

  10,000

(c) (i) bearing of E from x is 0.25  1^o

`   (ii) Distance Ex = (12.8 0.1 x 20m) = 256  2m

30.   Area  A = ½ x 170 x 80 = 6800

 B = ½ x 80 x 80 = 3200

C = ½ x 10 x 70 = 350

 D = ½ x 170 x 130 = 11050

E = ½ x 70 x 60 = 2100

Total = 23,500 m²

31.  (a)   L.s.f = 1

  40,000

1 = 3.25

40,000 x

  x = 130,000cm

(b) A.s.f = 1 ²

40,000

1 ² = x

  40,000 36,000,000

x = 0.0225cm²

32.  

(a) bearing = 180 + 20 = 200^o

(b)  a² = 1500 +

a² = b² + c² – 2bc cos A

a² = 1500² + 800² – 2 x 1500 x 800cos 60

= 2250000 + 640000 – 1200000

= 1690000

a = 1300m

(c)  1300 = 1500

   Sin 60 sin c

1300 sin c = 1500 sin 60

  Sin c = 1500 sin 60

1300

  = 0.9993

c = 87.79^o

c = 87.80

33.

.  A of ∆ XYD = ½ x 50 x 38 = 950m²

 A of XBCY = ½ (50 + 15) 60

= ½ x 65 x 60

= 1950m²

Total A = (950 + 1950)m²

= 2900m²

34.  B1 for 86^o

30 = Q5

Sin 86^o Sin 56^o

QS = 30sin 56^o

Sin 86^o

= 24.93km

35.  1cm for 100000cm

1cm² = (100000cm)²

Area = 5.4 x 4.5 x 100000 cm²

= 5.4 x 4.5 x 100000 x 100000Km²

100000 x 100000

= 24.3km²

36.    x 22 x 6370 x 2 = 900

360 7

= 900 x 360 x 7

  22 x 6370 x 2

= 8.1^o

Latitude of B = 8.1^o – 5^o N

= 3.5^o S

37.  i) acc = 40 – 20

100 – 50

= ²⁰/₅₀ = 0.4m/s

ii) 20 – 40 = -20

  460 – 400 60 = 0.3333 m/s²

iii) Area = ½ (520 + 300) x 40 x ¹/₁₀₀₀ = 16.4 km

38.  a) Tan 11.3 = 200

  x

x = 200

  Tan 11.3 = 100.1m

b)  (36 x 1000) m/s

60 x 60

D = (10 x 5) 50m Tan  = 7.590

< of depression = 7.590

c) i) √ 50.9² – 49.9² = 10.04cm

  ii) Tan  = 10.04

200

 = 2.874°

= 3°

39.  a) Make a sketch to show positive of A, B, C and D

Use sine rule in ∆ABC

X
180
 x = 180 sin 80^o

Sin 80^o Sin 40^o sin 40^o

= 275.8

Hence AC = 276 km

(b) Use the cosine rule in ∆ AD when  DAC = 150^o

y² = 240² + 276² – 2 x 240 x 276 cos 150^o

= 576000 + 76180 – 132 480 (-cos 30^o)

= 133776 + 114731 = 248507

 y = 248507

  = 498.5

  Hence CD = 499 km

(c) Using sine rule in ∆ABC we have

   BC = 180

Sin 60^o sin 40^o

 BC = 180 sin 60

  Sin 40

 = 242.5

 = 243 km