unit 1 anglesmodule 3: spaces in the environment ccslc 3 cxc (b) what angle does john turn through...

Module 3: Spaces in the Environment

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MODULE 3: Spaces in the

Unit 1 AnglesThe objectives of this section are to

• recognise the relationship between turns and angles

• understand how to measure angles and how to work out their size inparticular problems.

Angles are an important building block in geometry and trigonometry as you willsee later. In this unit you will see how turns are related to angles, how to measurethem and how to work out their size in particular problems.

1.1 Angles and TurnsYou will need to understand clearly what the terms such as turn, half turn, etc.mean in terms of angles. There are 360 ° in one complete turn, so the followingare true.

You also need to refer to compass points: (north (N), south (S), east (E), west (W),northeast (NE), southeast (SE), southwest (SW) and northwest (NW).

Turning from N to S is 180 °clockwise or anticlockwise

Turning from NE to SEis 90 ° clockwise (or 270 ° anti-clockwise)

Turning clockwise from NE to Eis 45 ° (or 315 ° anticlockwise)

Environment

1 turn34

turn12

turn14

turn

is 360 ° is 270 ° is 180 ° is 90 °

N

NE

E

SE

S

SW

W

NW


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Example

What angle do you turn through if you turn

(a) from NE to NW anticlockwise,

(b) from E to N clockwise?

Solution

(a) You can see that this is 90 ° (or 14

turn).

(b) This is a 34

turn, i.e. 270 °.

Exercises1. John is standing on a hill. The church is north of the point where he stands.

ChurchTV

Mast

Chimney John Farm

OakTree

Bridge

(a) In what direction is he facing if he looks at:

(i) the chimney, (ii) the bridge,

(iii) the TV mast, (iv) the farm,

(v) the oak tree?

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N

NE

E

SE

S

SW

W

NW


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(b) What angle does John turn through if he turns clockwise from looking at:

(i) the church to the farm,

(ii) the oak tree to the bridge,

(iii) the TV mast to the oak tree,

(iv) the bridge to the TV mast,

(v) the TV mast to the church?

(c) What would the angles be for question (b) if John turned anti-clockwise instead of clockwise?

2. In a game, you spin a pointer and let it stop.

What angle does the pointer turn through if it completes:

(a) 1 turn, (b) 2 turns, (c)34

turn,

(d) 114

turns, (e) 134

turns, (f) 214

turns?

3. What angle do you turn through if you turn clockwise from facing:

(a) N to E, (b) W to NW, (c) SE to NW,

(d) NE to N, (e) W to NE, (f) S to SW,

(g) S to SE, (h) SE to SW, (i) E to SW?

4. What angle do you turn through if you turn anticlockwise from facing:

(a) N to SW, (b) S to SW,

(c) W to NW, (d) E to S?

5. In what direction will you be facing if you turn:

(a) 180 ° clockwise from NE,

(b) 180 ° anticlockwise from SE,

(c) 90 ° clockwise from SW,

(d) 45 ° clockwise from N,

(e) 225 ° clockwise from SW,

(f) 135 ° anticlockwise from N,

(g) 315° clockwise from SW?

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6. Noravo stands on a low hill. The diagram below shows some of the things hecan see. Using information from the diagram, answer the following questions.

(a) What is NE of Noravo?

(b) What is SE of Noravo?

(c) Noravo turns from looking at the Old Fort to look at the ship. Whatangle does he turn through?

Explain why there is more than one answer to this question.

(d) What angle does Noravo turn through if:

(i) he turns clockwise from looking at the ship to the crane,

(ii) he turns anticlockwise from looking at the radio mast to thefactory,

(iii) he turns anticlockwise from looking at the factory to the ship?

(e) Noravo starts looking at the factory. What does he end up looking atif he turns:

(i) 135 ° clockwise, (ii) 270 ° anticlockwise,

(iii) 225 ° clockwise, (iv) 405 ° clockwise?

7. Use the diagram on the next page to answer these questions.

(a) What is N of the shop?

(b) What is W of the church?

1.1

Lighthouse

FactoryRadio Mast

ChurchTower

OldFort

CraneShip

W

S

E

N

NigelNoravo


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Caribbean Sea

DavisCove

Lucea

Grange Hill

Banbury

LittleLondon

SilverSpring

Savannah la MarBrighton

Sheffield

Negril

N

GreenIsland

1.1

WindmillChurch

Tower

Beach

Shop

Lighthouse

Big Rock

N

(c) What is E of the church?

(d) What is E of Big Rock and NE of the shop?

(e) What is SW of Big Rock?

(f) In what direction should you walk from the beach to get to the tower?

(g) In what direction should you walk from the beach to get to the church?

(h) If you walk SE from the windmill, will you get to the tower? Explainyour answer.

8. The sketched map below shows towns and villages in west Jamaica.


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(a) What is east of Silver Spring?

(b) What is north of Grange Hill?

(c) What is SE of Grange Hill?

(d) What is NW of Little London?

(e) What is north east of Green Island and north west of Banbury?

(f) What is north west of Little London and north of Sheffield?

9. The sails of a windmill complete one full turn every 40 seconds.

(a) How long does it take the sails to turn through:

(i) 180 ° (ii) 90 ° (iii) 45 °?

(b) What angle do the sails turn through in:

(i) 30 seconds, (ii) 15 seconds, (iii) 25 seconds?

10. The diagram shows the positions of Jason and Nadina. The arrow shows thedirection of north.

N

Nadina

Jason

(a) Copy or trace the diagram.

(b) Karen is west of Nadina and north of Jason. Mark Karen's positionon your diagram.

(c) Jenny is east of Jason and southeast of Nadina. Mark Jenny's positionon your diagram.

(d) Wendy is west of Jenny and southeast of Karen. Where is Wendy inrelation to Nadina?

(e) Jai is north of Jason and south of Karen. Describe where he could bestanding.

1.1


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1.2 Measuring AnglesThe objectives of this section are to

• understand how to use a protractor to measure angles

• be able to draw angles, using a protractor

• understand how to interpret data shown in pie charts

• be able to use pie charts to illustrate data.

Note

The angle around a complete circle is is 360 °.

The angle around a point on a straight line is 180 °.

A right angle is 90 °.

Example 1

Measure the angle CAB in the triangle shown.

Solution

Place a protractor on the triangleas shown.

The angle is measured as 47 °.

100

2030

40

50

C

BA

180o

360o

90˚

C

B

A


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Example 2

Measure this angle.

Solution

Using a protractor, the smaller angleis measured as 100 °.

Sorequired angle = 360 100° − °

= 260 °

Example 3

Draw angles of

(a) 120 ° (b) 330 °.

Solution

(a) Draw a horizontal line.

Place a protractor on top ofthe line and draw a mark at120 °.

Then remove the protractorand draw the angle.

1.2

100

20

3040

50

100

110

9080

7060

100

2030

40

50

100110

9080 70

60120

120 ˚


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(b) To draw the angle of 330 °, first subtract

330 ° from 360 °:

360 330 30° − ° = °

Draw an angle of 30 °.

The larger angle will be 330 °.

Exercises1. For each of the following angles, first estimate the size of the angle and

then measure the angle to see how good your estimate was.

(a) (b)

(c) (d)

(e)

(f)

1.2

30�˚

330�˚330°

30°


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(g)

(h)

2. Estimate and measure the size of each of these reflex angles.

(a) (b)

(c) (d)

1.2


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3. (a) Measure each of the angles in this pie chart.

ManchesterUnited

Other

Chelsea

Newcastle

Arsenal

(b) Explain how you can tell that Manchester United is the mostpopular of these teams.

(c) Which is the second most popular team?

4. Draw the following angles:

(a) 20 ° (b) 42 ° (c) 80 ° (d) 105 °

(e) 170 ° (f) 200 ° (g) 275 ° (h) 305°

5. In which of these polygons are the angles all the same size?

Find all the angles in each polygon. (You may need to copy the shapes on topaper and extend the lines.)

(a) (b)

1.2

B

C

DE

A

B C

D

E

A

F

Favourite UK footballteams for students in

Grade 7


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(c) (d)

6. (a) Draw the shape below, where O is the centre of the circle. Make theradius of your circle 6 cm.

O

C

A

B

100˚95˚

(b) Measure the distances between AB, BC and AC.

7. Ravinder finds out the favourite sportsfor members of his class. He works outthe angles in the list shown opposite fora pie chart.

Draw the pie chart.

1.2

B C

D

E

A

FG

H

B

C

D

E

A

F

G

Sport Angle

Football 110 °

Swimming 70 °

Tennis 80 °

Rugby 40 °

Hockey 30 °

Badminton 10 °

Other 20 °


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1.3 Classifying Angles

The objectives of this section are to

• understand how to identify acute, obtuse and reflex angles

• be able to draw shapes to include the various angles.

Angles of less than 90 ° are acute angles

Angles between 90 ° and 180 ° are obtuse angles

Angles between 180 ° and 360 ° are reflex angles

So you can easily identify the three types of angle.

Here are some examples.

Obtuse

Acute

Obtuse

Reflex

Acute

Acute

Reflex

Reflex

Exercises1. Is each of the following angles acute, obtuse or reflex?

(a) (b)


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(c) (d)

(e) (f)

2. For each shape below state whether the angle at each corner is acute, obtuseor reflex.

(a) (b)

3. (a) Draw a triangle with one obtuse angle.

(b) Draw a triangle with no obtuse angles.

4. Draw a four-sided shape with:

(a) one reflex angle, (b) two obtuse angles.

1.4 Angles on a Line and Angles at a PointThe objectives of this section are to

• understand that angles on a straight line add up to 180 °

• understand that angles at a point add up to 360 °

• calculate the size of unknown angles using these two facts.

Remember that

(a) angles on a line add up to 180 °

1.3

A

B

C

D

A

B

C

D

E

F

180˚


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and

(b) angles at a point add up to 360 °.

These are two important results which help when finding the size of unknownangles.

Example 1

What is size of the angle marked ?

Solution

45 55 100° + ° = °

So angle = ° − °180 100

= °80

Example 2

What is the size of the angle marked ?

Solution

70 150 220° + ° = °

So angle = ° − °360 220

= °140

Exercises1. Calculate the unknown angle in each of the following diagrams.

(a) (b)

1.4

360˚

?45˚ 55˚

70˚150˚?

85˚?70˚ ?


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1.4

(c) (d)

(e) (f)

2. Calculate the unknown angle in each diagram.

(a) (b)

(c) (d)

(e) (f)

80˚60˚?

55˚ ?112˚

22˚

?126˚

62˚27˚

?45˚

?

255˚30˚

?

278˚

?92˚124˚

?

21˚57˚

41˚

80˚60˚

110˚30˚

?

?

30˚

118˚

62˚


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3. Some of the angles in the pie chart havebeen calculated:

Red 90 °

Blue 95 °

Purple 50 °

What is the angle for yellow?

4. The picture shows a tipper truck.

70˚

80˚b

a

(a) Find the angles marked a and b.

(b) The 80 ° angle decreases to 75 ° as the tipper tips further. Whathappens to angle b?

5. The diagram shows a playground roundaboutviewed from above. Five metal bars are fixedto the centre of the roundabout as shown. Theangles between the bars are all the same size.

(a) What size are the angles?

(b) What size would the angles be if therewere 9 metal bars instead of 5?

6.

A boy hangs a punchbag on a washing line.

Find the unknown angles if both angles arethe same size.

1.4

RED

YELLOWPURPLE

BLUE

142˚


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7. The diagram shows two straight lines.

Find the angles a, b and c.

What do you notice?

8. In the diagram the large angle is 4 timesbigger than the smaller angle.

Find the two angles.

9. The picture shown a jack that can beused to lift up a car.

Find the angles marked a and b.

10. The diagram shows a regular hexagon.

(a) Find the size of each of the anglesmarked at the centres of the hexagon.

(b) What would these angles be if thepolygon was a decagon (10 sides).

(c) If the angles were 30 °, how manysides would the polygon have?

1.5 Finding Angles in TrianglesThe objectives of this section are to

• understand that the interior angles of a triangle always sum to 180 °

• recognise whether a triangle is isosceles, equilateral or scalene

• find unknown interior and exterior angles of a triangle.

1.4

28˚

a

b

c

126˚

56˚b

b

a a


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1.5

The interior angles of any triangle willalways sum (add up) to 180 °.

Example

Find the angle marked a in the diagram opposite.

Solution

70 50 120° + ° = °

So 180 120 60° − ° = °

and a = °60

The final part of this section deals with the classification of triangles.

ISOSCELES TRIANGLE

EQUILATERAL TRIANGLE

SCALENE TRIANGLE

a c

b

interiorangles

exteriorangle

exteriorangle

exteriorangle

50˚70˚

a

a b c+ + = °180

Sides ofequal length

Equal angles

All sides have differentlengths.

All angles are of differentsizes.

All sides arethe same length

All angles are 60˚


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1.5

Exercises1. Find the unknown angle in each triangle.

(a) (b)

(c) (d)

(e) (f)

2. Find the unknown angles in each of the following triangles.

(a) (b)

(c) (d)

50˚

80˚

?35˚ 120˚

?

?

94˚ 23˚ 81˚

62˚

?

?38˚

17˚

91˚

?41˚

80˚

ab

5 cm 5 cm

a

b

85˚

8 cm

8 cm

4 cm4 cm

4 cma

b

c

122˚

a

b

3 cm

3 cm


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1.5

3. State whether each triangle below is isosceles, equilateral or scalene.

(a) (b)

(c) (d)

4. For each triangle below, find the unknown interior angle and the markedexterior angle.

(a) (b)

(c) (d)

45˚

45˚5 cm 7 cm

8 cm

20˚

80˚ 60˚

60˚

71˚ 75˚

ab a b120˚

18˚

63˚

62˚

a b

42˚

a b


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5. Explain how to find the exterior angle without having to calculate aninterior angle.

Find the exterior angles marked on these triangles.

(a) (b)

(c) (d)

6. Find the total of the 3 exterior angles for this triangle.

b

c

a61˚

65˚

Do you think you will get the same answer for different triangles? Explainyour answer.

7. For each of the following triangles, draw in the exterior angles and findtheir total.

(a) (b)

1.5

47˚

?

61˚

41˚

?

30˚

20˚

65˚

65˚

50˚

?

25˚

108˚

18˚

124˚

?


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(c) (d)

Comment on your results.

8. Find the unknown angle or angles marked in each of the followingdiagrams.

(a) (b)

(c) (d)

(e) (f)

1.5

38˚

58˚

62˚

52˚

a 121˚

b

a

c

111˚

a

152˚

dc

b

a130˚

cb d

a110˚

140˚

130˚ab


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9. Part of a roof is made out of 4 similar isosceles triangles.

40˚

Copy the diagram and mark the sides that have the same lengths.

On your diagram, write in the size of all the marked angles.

10. (a) For this isosceles triangle, find the other two interior angles.

10˚

(b) Find the other angles if the 10 ° increases to 20 ° and then to 30 °.

(c) What do you think will happen if the 10 ° is increased to 40 °?

11. One angle of an isosceles triangle is 70 ° . What are the other angles?(There is more than one solution!)

1.5


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1.6 Parallel and Intersecting LinesThe objectives of this section are to

• understand and be able to apply the rules for calculating unknownangles.

When a line intersects (or crosses) a pairof parallel lines, there are some simplerules that can be used to calculateunknown angles.

The arrows on the lines indicate that theyare parallel.

a b c d e f= = =( )and and , These are called vertically opposite angles.

a c b d= =( )and These are called corresponding angles.

b c= These are called alternate angles.

a e+ = °180 , because adjacent angles on a straight line add up to 180 °.These are called supplementary angles.

Note also, that c e+ = °180 (allied or supplementary angles)

Example 1

In the diagram opposite, find the unknownangles if a = °150 .

Solution

To find b:

a b+ = 180 ° (angles on a straight line, supplementary angles)

150 ° + b = 180 °

b = 30 °

dc

bea f

ed

cab


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1.6

To find c:

c = b (vertically opposite angles or angles on a straight line)

c = 30 °

To find d:

d = a (corresponding angles)

d = 150 °

To find e:

e = c (corresponding angles)

e = 30 °

Example 2

Find the size of the unknown angles inthe parallelogram shown in this diagram:

Solution

To find a:

a + °70 = 180 ° (allied or supplementary angles)

a = 110 °

To find b:

b a+ = 180 ° (allied or supplementary angles)

b + °110 = 180 °

b = 70 °

To find c:

c + °70 = 180 ° (allied or supplementary angles)

c = 110 °or

c = 360 70° − + + °( )a b (angle sum of a quadrilateral)

= 360 250° − °

= 110 °or

c = a (opposite angles of a parallelogram are equal)

a

b c

70˚


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Exercises1. Which angles in the diagram

are the same size as:

(a) a,

(b) b ?

2. Find the size of each of the angles marked with letters in the diagramsbelow, giving reasons for your answers:

(a) (b)

(c) (d)

3. Find the size of the three unknownangles in the parallelogram opposite:

4. One angle in a parallelogram measures 36 °. What is the size of each of theother three angles?

5. One angle in a rhombus measures 133 ° . What is the size of each of theother three angles?

6. Find the sizes of the unknown anglesmarked with letters in the diagram:

1.6

d cb

e

a

fgh

ba

70˚

ba

99˚

c db

a

c

110˚d

b

c

a

65˚

ba c

40˚

dcb e

a50˚ 30˚


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7. (a) In the diagram opposite, find thesizes of the angles marked in thetriangle. Give reasons for youranswers.

(b) What special name is given to thetriangle in the diagram?

8. The diagram shows a bicycle frame.Find the sizes of the unknown anglesa, b and c.

9. BCDE is a trapezium.

(a) Find the sizes of all theunknown angles, giving reasonsfor your answers.

(b) What is the special name givento this type of trapezium?

1.7 BearingsThe objectives of this section are to

• understand what is meant by a 'bearing'

• be able to calculate and measure bearings.

Bearings are a measure of direction, with north taken as a reference. If you aretravelling north, your bearing is 000 °.

If you walk from O in the direction shown in thediagram, you are walking on a bearing of 110 °.

1.6

143˚

37 a

c

b

B C

A

72˚

DE

126˚

46˚

20˚

62˚

a

bc

N

O110˚


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Bearings are always measured clockwise from north,and are given as three figures, for example:

Bearing 060 ° Bearing 240 ° Bearing 330 °

Example 1

On what bearing is a ship sailing if it is heading:

(a) E, (b) S, c) W,

(d) SE, (e) NW ?

Solution

(a) (b)

Bearing is 090 °. Bearing is 180 °.

(c)

Bearing is 270 °.

(d) (e)

Bearing is 135 ° Bearing is 315 °

1.7

N

60˚

N

240˚

N

330˚

N

S

EW

NENW

SW SE

90˚

N

E180˚

N

S

270˚

N

W

N

315˚

NW135˚

N

SE


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Example 2

A ship sails from A to B on a bearing of 060 °. On what bearing must it sail if itis to return from B to A?

SolutionThe diagram shows the journey fromA to B.

Extending the line of the journey allowsan angle of 60 ° to be marked at B.

Bearing of A from B = 60 180° + °

= 240 °

and this is called a back bearing or areciprocal bearing.

Exercises1. What angle do you turn through if you turn clockwise from:

(a) N to S, (b) E to W,

(c) N to NE, (d) N to SW,

(e) W to NW ?

2. Copy and complete the table:

3. The map of an island is shown below:

What is the bearing from the tower, of each place shown on the map?

1.7

N

N

A60˚

60˚

B

Direction Bearing

N

NE

W

SW

Tower

Quay

Beach

Lighthouse

Mine

Church

N


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4. The diagram shows the positions of twoships, A and B.

(a) What is the bearing of ship A fromship B ?

(b) What is the bearing of ship B fromship A ?

5. The diagram shows 3 places, A, B and C.

Find the bearing of:

(a) A from C,

(b) B from A,

(c) C from B,

(d) B from C.

6. In Canada, an aeroplane flies from Victoria to Edmonton on a bearing of044 °. On what bearing should the pilot fly, to return to Edmonton fromVictoria?

7. On four separate occasions, a plane leaves Kingston to fly to a differentdestination. The bearings of these destinations from Norman ManleyAirport, Kingston are given below.

Copy and complete the diagram to showthe direction in which the plane flies toeach destination.

1.7

A

B

N

N

N

A

N

C

N

B

Destination Bearing

Paris, France 077 °

Nassau, Bahamas 356 °

Santiago de Cuba 036 °

Cartagena, Colombia 162 °

N

Paris, France

Kingston


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8. A ship sails NW from a port to take supplies to an oil rig. On what bearingmust it sail to return from the oil rig to the port?

9. If A is north of B, C is southeast of B and on a bearing of 160 ° from A, findthe bearing of:

(a) A from B, (b) A from C,

(c) C from B, (d) B from C.

10. If A is on a bearing of 300 ° from O, O is NE of B, and the bearing of B

from A is 210 °, find the bearing of:

(a) A from B, (b) O from A, (c) O from B.

1.8 Scale DrawingsThe objectives of this section are to

• construct scale drawings, using bearings

• use these scale drawings to solve problems.

Example 1A ship sails 20 km NE, then 18 km S, and then stops.

(a) How far is it from its starting point when it stops?

(b) On what bearing must it sail to return to its starting point?

SolutionThe path of the ship can be drawn usinga scale of 1 cm for every 2 km, as shownin the diagram.

1.7

Scale: 1 cm = 2 km B

O

N

A 180˚

45˚

20 km

18 km


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(a) The distance BO can be measured on the diagram as 7.3 cm whichrepresents an actual distance of 14.6 km.

(b) The bearing of O from B can be measured as 285 °.

Note: Remember to always put the scale on the diagram.

Example 2

A man walks 750 m on a bearing of 030 °. He then walks on a bearing of 315 °until he is due north of his starting point, and stops.

(a) How far does he walk on the bearing of 315 ° ?

(b) How far is he from his starting point when he stops?

Solution

A scale drawing can be produced, using a scale of 1 cm to 100 m.

315˚A

N

750 m

30˚

N

B

O

(a) The distance AB can be measured as 5.4 cm, which represents an actualdistance of 530 m.

(b) The distance OB can be measured as 10.2 cm, representing an actualdistance of 1020 m.

1.8


34CCSLC CXC

Exercises1. A girl walks 80 m north and then 200 m east.

(a) How far is she from her starting position?

(b) On what bearing should she walk to get back to her starting position?

2. Andrew walks 300 m NW and then walks 500 m south and then stops.

(a) How far is he from his starting position when he stops?

(b) On what bearing could he have walked to go directly from his startingposition to where he stopped?

3. An aeroplane flies 400 km on a bearing of 055 ° It then flies on a bearing

of 300 °, until it is due north of its starting position. How far is theaeroplane from its starting position?

4. A captain wants to sail his ship from port A toport B, but the journey cannot be made directly.Port B is 50 km north of A.

The ship sails 20 km on a bearing of 075 °.

It then sails 20 km on a bearing of 335 ° and thendrops anchor.

(a) How far is the ship from port B when itdrops anchor?

(b) On what bearing should the captain sail theship to arrive at port B?

5. Julia intended to walk 200 m on a bearing of 240 °. Her compass did not

work properly, so she actually walked 200 m on a bearing of 225 °. Whatdistance and on what bearing should she walk to get to the place sheintended to reach?

6. A hot air balloon is blown 5 km NW. The wind then changes direction andthe balloon is blown a further 6 km on a bearing of 300 ° before landing.

How far is the balloon from its starting point when it lands?

7. Ronaldo and Jade set off walking at the same time. When they start,Ronaldo is 6 km NW of Jade. Jade walks 3 km on a bearing of 350 ° and

Ronaldo walks 4 km on a bearing of 020 °. How far apart are they now?

1.8

N

B

A

SeaLand


35CCSLC CXC

8. An aeroplane flies 200 km on a bearing of 335 ° . It then flies 100 km on a

bearing of 170 ° and 400 km on 280 °, and then lands.

(a) How far is the aeroplane from its starting point when it lands?

(b) On what bearing could it have flown to complete its journey directly?

9. Billy is sailing on a bearing of 135 °. After his boat has travelled 20 km, herealises that he is 1 km north of the port that he wanted to reach.

(a) On what bearing should he have sailed?

(b) How far from his starting point is the port that he wanted to reach?

10. A pilot knows that to fly to another airport he needs to fly 500 km on abearing of 200 °. When he has flown 400 km, he realises that he is 150 kmfrom the airport.

(a) On what bearing has the pilot been flying?

(b) On what bearing should he fly to reach the airport?

(Note that there are two answers.)

11. A Jamaican cellphone company needs to calculate the bearings anddistances from its phone mast in Ewarton to several other towns andvillages on the island.

The bearings and the distances from Ewarton to these locations are given inthe table below.

Using a scale of 1 cm to represent 5 km, draw a map showing the positionsof the five locations.

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Location Bearing Distance

Oracabessa 030 ° 30 km

Runaway Bay 318 ° 40 km

Hayes 208 ° 36.5km

Linstead 127 ° 6.5 km

unit 1 anglesmodule 3: spaces in the environment ccslc 3 cxc (b) what angle does john turn through...

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