28. [Location / Transformation] Skill 28.1

Follow the directions one at a time. Hints: A compass showing North will allow you to find your bearings. Clockwise from North, “Never Eat Sea Weed” is one way to remember the 4 point compass.

Q. At Homebush, in which pic

6

e Av rd

ve tA

E

St

2

4 3

rah

k rac

Du

e. Av

8

10 Hom

SW

SE

E S 8 Golf Driving Range

9 sh Bay Drv

N

b) From Montrésor castle, which direction

do you have to drive to reach Loches castle? ire Chambord LOIRE VALLEY CASTLES Loire East - FRANCE

Lo

Blois

N

Cheverny To Sydney

Young Street

Hume Street

Montlouis

Amboise

N

Smollett Street

Hume Street

Ouchamps

Chenonceau Young Street

You are here

Chaumont

Pontlevoy

Swift Street

Dean Street Townsend Street

Smollett Street

Clive Street

Albury - Australia Stanley Street

Tours

Kiewa Street

Dean Street

Wodonga Place

Clive Street

NE

W

Swift Street

David Street

Townsend Street

Wodonga Place

You are here

Kiewa Street

From where you are, travel east until you reach David Street. Then walk north. If you take the second turn left, what street are you in? To Sydney Stanley Street

N NW

Bicentennial Park

ebu

5

Albury - Australia

Olympic Stadium 1

ri

r

nF

Daw

Sh irl ey

Ave

e r Av H ase

A. NW Focus on the relevant information.

d

llio

erb

Sa

a)

7

va

ule

Bo

k Flac

1

1 Olympic Stadium 2 Athletic Centre 3 Warm-up Arena 4 Aquatic Centre 5 Hockey Centre 6 Baseball Stadium 7 Railway Station 8veGolf Driving Range n d a l A ck 9 Tennis Centre 10 Sports Centre Benne long R

ve ia A tral Aus

ym Ol

in Edw

direction is the Olympic Stadium from the Golf Driving Range?

Homebush - Sydney

David Street

•

MM4.2 1 1 2 2 3 3 4 4 MM5.1 1 1 2 2 3 3 4 4

Following directions and using compass bearings to describe location on a map.

Indre

Montbazon

Cormery Montpoupon

N

Cher

To Melbourne

Loches

Montrésor

To Melbourne

From the northern most bridge over Rio Kusichaca you travel south-east on the Inca Trail until the T intersection. Then you turn right and follow the Inca Trail to the Inca Steps. How many more bridges do you cross? Aguas Calientes

INCA TRAIL train station

Piaza

START 4 day trek

aca Rio K usich

Paca may o Rio ][

Runkuracay 3800 m

Pa 360 caymay 0m o

Phuyupatamarca 3600 m

Cusco

d) Using the closest tunnel entrance to

building 58, take the first turn right, then turn left. Turn right and walk to the end of the tunnel. If you turn left again, which building are you facing? Northeastern University - Boston, MA Tunnel Map

40 41 HUNTINGTON AVENUE 42 42 43 43 41 52 48 50 40 51 51 48 52 55 Tunnel 53 Entrances 53 Tunnel Entrance 55 50 57 57 58 58 59 59

Barletta Natatorium Cabot Physical Education Centre Richards Hall Dodge Hall Mugar Life Sciences Building Curry Student Centre Blackman Auditorium Ell Hall Hayden Hall Forsyth Building Dana Research Centre Snell Engineering Centre Snell Library

][

][

Inca Steps

Sayaqmarca 3600 m

Llactapata ] [

Wa Pas rmiwa s 42 nus Llul 00 m ca luch apa mp a

][

Winay Wayna 2700 m

[

Chachabamba

][

a mb ba Uru

Intu Punku 2400 m

]

Rio

Machu Picchu 2400 m

START 2 day trek

page 281

N

] [

bus to ruins

to Machu Picchu

LEGEND River Rail line Inca ruin Camp Inca Trail Bridge

][

c)

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Skill 28.2 •

MM4.2 1 1 2 2 3 3 4 4

Identifying and classifying symmetry in two-dimensional shapes.MM5.1 1 1 2 2 3 3 4 4

Imagine a line along which the shape can be folded to have one part fit exactly over the other part.

Q. Draw the axes of symmetry for these

A.

shapes. Circle the shapes that are both horizontally and vertically symmetrical.

vertical & horizontal

✔

a)

c)

e)

oblique

vertical

vertical & horizontal

✔

Draw all the axes of symmetry for this shape. How many axes of symmetry does this shape have?

b) Draw all the axes of symmetry for this

Draw all the axes of symmetry for this shape. How many axes of symmetry does this shape have?

d) Draw all the axes of symmetry for this

Draw the axes of symmetry for these shapes. Circle the shapes that have horizontal symmetry.

f)

g) Draw the axes of symmetry for these

shapes. Circle the shapes that have vertical symmetry.

page 282

shape. How many axes of symmetry does this shape have?

shape. How many axes of symmetry does this shape have?

Draw the axes of symmetry for these shapes. Circle the shapes that are both horizontally and vertically symmetrical.

h) Draw the axes of symmetry for these

shapes. Circle the shapes that are both horizontally and vertically symmetrical.

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Skill 28.3 • • •

MM4.2 1 1 2 2 3 3 4 4 MM5.1 1 1 2 2 3 3 4 4

Using a scale to calculate distance on a map.

Place a piece of paper against the scale matching the starting points. Slide the paper across the length of the scale marking the start and end points as you go. Add together the scale lengths covered.

Q. You walk from the Inspiration Point to

A. 0.5 + 0.5 + 0.5

There are 2 distances to be measured.

Grand View, along the marked path. = 1.5 km What distance did you travel in kilometres? Canyon Village Visitor Centre Post Office

Showers - Laundry Canyon Lodge

0.5

Services & Facilities

Yellowstone National Park Upper Falls View

Mark the start of the first distance and the turning point on paper. Rotate the paper to match the second distance and then mark the end.

Amphitheatre

0.5

km Grand View

0.5 Inspiration Point

Ranger station Campground Lookout Point

Lodging Grand View

Check the paper against the scale.

Food service

Inspiration Point

Picnic area Store

Uncle Tom’s Trail

Slide the paper along the scale as necessary.

Gas station

N

0

Self-guiding trail

0.5 km

Horse rental

a)

How far is it from Central Station, b) Using the scale, what is the marked along Hope St. to the Glasgow Royal distance on this map of Antarctica? Concert Hall? SOUTHERN OCEAN le ATLANTIC Glascow Royal Concert Hall (GRCH) Circ OCEAN

Syowa (Japan)

INDIAN OCEAN

An t

Bellingshausen (Russia)

tic arc

Halley (U.K.)

Palmer (U.S.) Rothera (U.K.)

Mawson (Australia)

RONNE ICE SHELF

Davis (Australia)

AMERY ICE SHELF

South Pole

Amundsen-Scott (U.S.)

Central Station

0

Queen Street Station

5 × 250 =

m

Using the scale, what is the marked distance from the University via the High Court to the Homiman Circle Gardens? MUMBAI - India Veer

m 0

page 283

100

SOUTHERN OCEAN

Nariman Rd

distance from the ranger station closest to Lake Hotel to Fishing Bridge? Fishing Bridge, Lake Village & Bridge Bay Yellowstone National Park 0 0

0.5 km

Fishing Bridge

0.5 mi Visitor Centre

Homiman Circle Gardens

Lake Village

Post Office

Lake Lodge

hR

d

St Thomas Cathedral

Sin g ag

Ice Bridge Marina

Bh

i St Dala

Lake Hotel

at

hi Rd Ambalal Dos

id

University of Mumbai

Mahatma Gandhi Rd

High Court

1500 km

d) Using the scale, what is the marked

Bay

YELLOWSTONE LAKE

Gull Point

Sh ah

c)

Dumont d'Urville (France)

km

250 m North

Casey (Australia)

McMurdo (U.S.) Scott (New Zealand)

Year-round research station

GRCH

0

Vostok (Russia)

ROSS ICE SHELF

PACIFIC OCEAN

m www.mathsmate.co.nz

Services & Facilities Ranger station Lodging Food service Picnic area Store Gas station Boat launch

N

km © Maths Mate 4.2/5.1 Skill Builder 28

Skill 28.4

Q. According to the compass, you are

N

A. 45°

facing north-west. How many degrees clockwise must you turn to face north?

NW

ckw

45°

ise

N

E N

W

E N

N W

clo

N N W

MM4.2 1 1 2 2 3 3 4 4 MM5.1 1 1 2 2 3 3 4 4

Describing rotations of two-dimensional shapes.

W

E

E

45°

Find the North direction.

SW

SE

Calculate the number of degrees by picturing a circle.

SE

SW

S

S

a)

By how many degrees must this shape be rotated to first match the original position? same shape

b) By how many degrees must this shape

be rotated to first match the original position?

rotation 180°

d) By how many degrees must this shape

By how many degrees must the big hand of this clock rotate to show exactly 11:05?

f)

be rotated to first match the original position?

1112 1 10 2 9 3 8 4 7 6 5

By how many degrees must the big hand of this clock rotate to show exactly 2:00? 1112 1 10 2 9 3 8 4 7 6 5

g) This compass shows that you are facing

south-west. How many degrees clockwise must you turn to face north?

h) This compass shows that you are facing

south. How many degrees anticlockwise must you turn to face north-west?

N

N

W

E

E

SE

SW

SW

S

S

How many degrees must the big hand of this clock turn to show exactly 9:45?

N E N

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According to the compass, you are facing south-east. How many degrees clockwise must you turn to face west?

W

E SW

1112 1 10 2 9 3 8 4 7 6 5

j)

N W

i)

E

W

N

W

N E N

W

N

SE

e)

By how many degrees must this shape be rotated to first match the original position?

SE

c)

S

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Skill 28.5

Drawing translations, reflections and rotations of objects on a grid (1).

MM4.2 1 1 2 2 3 3 4 4 MM5.1 1 1 2 2 3 3 4 4

Translation (slide) • Move the shape up (positive, vertically), down (negative, vertically), left (negative, horizontally) or right (positive, horizontally) on the grid, without flipping, turning or changing its size. Reflection (like in a mirror) • Draw a perpendicular line to the mirror line from each vertex of the shape. • Extend the perpendicular line beyond the mirror line by the same distance. • Plot and join the reflected points. Rotation (turning about a point or centre of rotation) • Rotate each vertex by the given angle, in the given direction. • Plot and join the rotated points. Hint: The resulting shapes are always congruent to the original shapes (same size and shape). Q. Redraw this shape after reflecting it in

A.

the horizontal dotted line and then translating it 10 units to the right. 10 units

a)

Redraw this shape after translating it 10 units to the right and 4 units down.

b) Redraw this shape after reflecting it in

the vertical dotted line.

10 4

c)

Redraw this shape after translating it 3 units up and 4 units to the right.

d) Redraw this shape after rotating it 180°

about the point O.

O

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Skill 28.5 e)

Drawing translations, reflections and rotations of objects on a grid (2).

Redraw this diagram after reflecting it in the horizontal dotted line.

g) Redraw this diagram after reflecting it

in the horizontal dotted line.

i)

f)

MM4.2 1 1 2 2 3 3 4 4 MM5.1 1 1 2 2 3 3 4 4

Redraw this diagram after reflecting it in the horizontal dotted line.

h) Redraw this diagram after reflecting it

in the horizontal dotted line.

Redraw this diagram after reflecting it in the horizontal dotted line.

j)

Redraw this shape after rotating it 180º about point O and then translating it 2 units up.

O

k)

Redraw this shape after rotating it 90º clockwise about point O and then reflecting it in the vertical dotted line.

l)

Redraw this shape after reflecting it in the horizontal dotted line and then translating it 9 units to the left.

O

m) Redraw this quadrilateral after

reflecting it in the vertical dotted line and then translating it 2 units to the right.

page 286

n) Redraw this rhombus after reflecting it

in the horizontal dotted line and then translating it 2 units to the left.

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Skill 28.6 • • •

Drawing enlargements and reductions on a Cartesian plane.

Multiply or divide the x- and y-coordinates of the vertices of the given shape. Plot the new points. Join these points to form a new shape. Hint: The resulting shape is always similar to the original shape (same shape, but different size).

Q. Redraw the shape after multiplying the

A.

coordinates of its vertices by 3.

Y

8 7 6 5 4 (0,3)3 ×3 2 (0,1)1

Y 8 7 6 5 4 3 2 1

0 1 2 (3,0) 3 4 5 6 7 8 (9,0) 9 10 11 12 13 14 15 16 17 18 X ×3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

a)

Redraw the triangle after doubling the coordinates of its vertices.

b) Redraw the parallelogram after halving

the coordinates of its vertices.

Y

Y

8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

c)

Redraw the kite after multiplying the coordinates of its vertices by 3.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

d) Redraw the shape after halving

the coordinates of its vertices.

Y

Y

8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

e)

MM4.2 1 1 2 2 3 3 4 4 MM5.1 1 1 2 2 3 3 4 4

Redraw the shape after halving the coordinates of its vertices.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

f)

Y

Redraw the shape after doubling the coordinates of its vertices. Y

8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

© Maths Mate 4.2/5.1 Skill Builder 28

Skill 28.7

Drawing translations, reflections and rotations of objects on a Cartesian plane (1).

Q. Redraw this triangle after rotating it 90°

A.

clockwise about the point of coordinates (2,4). Y 8 7 6 5 4 3 2 1

Move each vertex of the triangle by 90° clockwise.

Y 8 7 6 5 4 3 2 1

MM4.2 1 1 2 2 3 3 4 4 MM5.1 1 1 2 2 3 3 4 4

90°

Plot the new points.

(2,4) 90°

01 2 3 4 5 6 7 8 X

The point of coordinates (2,4) does not move. Join the new points.

01 2 3 4 5 6 7 8 X

a)

Redraw this triangle after reflecting it in the x-axis.

b) Redraw this trapezium after reflecting it

in the y-axis.

Y

Y

6 5 4 3 2 1

6 5 4 3 2 1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 -4 -5 -6

c)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 -4 -5 -6

Redraw this rectangle after subtracting 4 units from the coordinates of its vertices.

d) Redraw this rhombus after adding

3 units to the coordinates of its vertices.

Y

Y

8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

e)

Redraw this shape after subtracting 5 units from the coordinates of its vertices. Y

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

f)

Redraw this shape after adding 5 units to the coordinates of its vertices. Y

8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

© Maths Mate 4.2/5.1 Skill Builder 28

Skill 28.7

Drawing translations, reflections and rotations of objects on a Cartesian plane (2).

g) Redraw this triangle after subtracting

5 units from the x-coordinates and 6 units from the y-coordinates of its vertices.

h) Redraw this triangle after adding 4 units

to the x-coordinates and 7 units to the y-coordinates of its vertices.

Y

Y

6 5 4 3 2 1

6 5 4 3 2 1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 -4 -5 -6

i)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 -4 -5 -6

Redraw this shape after rotating it 180° about the point of coordinates (9,4).

j)

Redraw this shape after rotating it 180° about the point of coordinates (13,4). Y

Y

8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

k)

Redraw this shape after rotating it 180° about the point of coordinates (7,5). Y

l)

Redraw this shape after rotating it 90° clockwise about the point of coordinates (9,1). Y

8 7 6 5 4 3 2 1

8 7 6 5 4 3 2 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

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MM4.2 1 1 2 2 3 3 4 4 MM5.1 1 1 2 2 3 3 4 4

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

© Maths Mate 4.2/5.1 Skill Builder 28

Skill 28.8 • • •

The transformation is a translation if the two shapes have the same size and orientation. The transformation is a reflection if the two shapes have the same size and are symmetrical about a vertical or horizontal line. The transformation is a rotation if the two shapes have the same size, different orientation and are not symmetrical about a vertical or horizontal line.

Q. Which transformation has moved the

triangle? A) a translation of −4 along the x-axis B) a reflection in the line x = 2 C) a rotation of 180º Y 6 5 4 3 2 1

A. A) the shapes have different orientation ⇒ not a translation B) the shapes are symmetrical about a vertical line ⇒ a reflection C) the shapes are symmetrical about a vertical line ⇒ not a rotation The answer is B.

01 2 3 4 5 6 X

-6 -5 -4 -3 -2 -1

a)

Which transformation has moved the shape? A) a translation of −7 along the x-axis B) a reflection in the y-axis C) a rotation of 180º

b) Which transformation has moved the

trapezium? A) a translation of 4 along the x-axis B) a reflection in the line x = 1 C) a rotation of 180º

Y

6 5 4 3 2 1

Y

6 5 4 3 2 1

-7

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 -4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 -4 -5 -6

-5 -6

c)

MM4.2 1 1 2 2 3 3 4 4 MM5.1 1 1 2 2 3 3 4 4

Describing transformations on a Cartesian plane.

Which transformation has moved the shape? A) a translation of −6 along the y-axis B) a reflection in the line x = −1 C) a rotation of 90º anticlockwise

d) Which transformation has moved the

triangle? A) a translation of −3 along the y-axis B) a reflection in the line y = −2 C) a rotation of 90º clockwise

Y

Y

6 5 4 3 2 1

6 5 4 3 2 1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 -4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 -4 -5 -6

-5 -6

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Skill 28.9

MM4.2 1 1 2 2 3 3 4 4 MM5.1 1 1 2 2 3 3 4 4

Drawing reflections of shapes in lines of given equations on a Cartesian plane.

Reflection (like in a mirror) • Draw a perpendicular line to the mirror line from each vertex of the shape. • Extend the perpendicular line beyond the mirror line by the same number of units. • Plot and join the reflected points. Hint: The resulting shapes are always congruent to the original shapes (same size and shape). Q. Redraw this triangle after reflecting it in

the line of equation y = −1

A.

6 5 4 3 2 1

Y

6 5 4 3 2 1

Redraw this shape after reflecting it in the line of equation x = 7 Y

y = −1

b) Redraw this triangle after reflecting it in

the line of equation x = 10 Y

8 7 6 5 4 3 2 1

c)

{ {

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 4 -4 -5 -6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 -4 -5 -6

a)

4

8 7 6 5 4 3 2 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 X

Redraw this trapezium after reflecting it in the line of equation y = 2

d) Redraw this triangle after reflecting it in

the line of equation y = −2

Y

Y

6 5 4 3 2 1

6 5 4 3 2 1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 -4 -5 -6

page 291

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 X -1 -2 -3 -4 -5 -6

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