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)
www.mathsmate.co.nz
© Maths Mate 4.2/5.1 Skill Builder 28
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.
www.mathsmate.co.nz
© Maths Mate 4.2/5.1 Skill Builder 28
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
page 284
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
www.mathsmate.co.nz
© Maths Mate 4.2/5.1 Skill Builder 28
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
page 285
www.mathsmate.co.nz
© Maths Mate 4.2/5.1 Skill Builder 28
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.
www.mathsmate.co.nz
© Maths Mate 4.2/5.1 Skill Builder 28
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
page 287
www.mathsmate.co.nz
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
page 288
www.mathsmate.co.nz
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
page 289
MM4.2 1 1 2 2 3 3 4 4 MM5.1 1 1 2 2 3 3 4 4
www.mathsmate.co.nz
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
page 290
www.mathsmate.co.nz
© Maths Mate 4.2/5.1 Skill Builder 28
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
www.mathsmate.co.nz
© Maths Mate 4.2/5.1 Skill Builder 28
page 292
www.mathsmate.co.nz
© Maths Mate 4.2/5.1 Skill Builder 28