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PASSPO
RT
PASSPO
RT
Polygons POLYGONSPOLYGONS
www.mathlecs.co.nz
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Polygons
Mathletics Passport 3P Learning
112HSERIES TOPIC
Write down how you would describe this shape over the phone to a friend who had to draw it
accurately. Try it with a friend/family member and see if they draw this shape from your descripon.
This booklet is about idenfying and manipulang straight sided shapes using their unique properes
Many clever people contributed to the development of modern geometry including:
Thales of Miletus (approx. 624-547 BC)
Pythagoras (approx. 569-475 BC)
Euclid of Alexandria (approx. 325-265 BC) (oen referred to as the "Father of modern geometry')
Archimedes of Syracus (approx 287-202 BC)
Apollonius of Perga (approx. 261-190 BC)
Aer an aack on the city of Alexandria, many of the works of these mathemacians were lost.
Look up these people someme and read about their contribuon to this subject.
New discoveries in geometry are sll being made with the advent of computers, in parcular fractal
geometry. The most famous of these being Benoit Mandelbrot Fractal paern.
Q
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Polygons
Mathletics Passport 3P Learning
2 12HSERIES TOPIC
Howdoes it work?
Polygons
Polygons
Polygons are just any closed shape with straight lines which dont cross. Like a square or triangle.
All polygons need at least three sides to form a closed path.
Polygon?
- All sides are straight
- Shape is closed
Polygon?
- All sides are straight
- Shape is NOT closed
Polygon?
- All sides are NOT straight
- Shape is closed
Polygon?
- Sides cross
Parts of a polygon:
Diagonal (line that joins two verces and is not a side)
Exterior angle
Interior angleSide
Each corner is called a Vertex (verces plural)
There are many basic types of polygons. Here are the ones we will be looking at in this booklet:
Here is another dierence between convex and concave polygons.
Convex Concave
A straight line drawn through the polygon
can only cross a maximum of2 sides
A straight line drawn through the
polygon can cross more than two sides.
Convex polygon
All interior angles
are 180c1
Equilateral polygon
All sides are the
same length
Cyclic polygon
All verces/corner points lie
on the edge (circumference)of the same circle.
Equiangular polygon
All interior angles
are equal
Regular polygon
All interior angles are equal
All sides are the same lengthThey are cyclic polygons
Concave polygon
Has an interior
angle 180c2
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Polygons
Mathletics Passport 3P Learning
312HSERIES TOPIC
How does it work? Polygons
Polygons
Any polygon can be named using Greek prexes matching the number of straight sides it has.
= Hexa = Deca
= Tetradeca
= Penta = Nona
= Trideca
= Tetra = Octa
= Dodeca
= Trio = Hepta
= Hendeca
Here are some more polygon names.
Sides Polygon name Sides Polygon name
9 Nonagon 19 Enneadecagon
10 Decagon 20 Icosagon
11 Hendecagon 30 Tricontagon
12 Dodecagon 40 Tetracontagon
13 Tridecagon 50 Pentacontagon
14 Tetradecagon 60 Hexacontagon
15 Pentadecagon 70 Heptacontagon
16 Hexadecagon 80 Octacontagon
17 Heptadecagon 90 Enneacontagon
18 Octadecagon 100 Hectogon
Many of these polygons
have more than one name.
Look them up someme!
Nonagon Enneagon
9 sides
Polygon naming and classicaon chart
Sides Name Concave Convex Equilateral Equiangular Cyclic Regular
3Triangle
(Trigon)N/A
4Quadrilateral
(Tetragon)
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
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Polygons
Mathletics Passport 3P Learning
4 12HSERIES TOPIC
How does it work? PolygonsYour Turn
Idenfy which of these shapes are polygons or not.
Tick all the properes that each of these polygons have and then name the shape:
1
2
3
Polygons
a
a
d
a
e
b
b
e
b
f
c
c
f
d
g h
Polygon
Not a polygon
Convex
Concave
Equilateral
Equiangular
Cyclic
Regular
Convex
Concave
Equilateral
Equiangular
Cyclic
Regular
Convex
Concave
Equilateral
Equiangular
Cyclic
Regular
Convex
Concave
Equilateral
Equiangular
Cyclic
Regular
Convex
Concave
Equilateral
Equiangular
Cyclic
Regular
Convex
Concave
Equilateral
Equiangular
Cyclic
Regular
PolygonNot a polygon
Polygon
Not a polygon
PolygonNot a polygon
Polygon
Not a polygon
Polygon
Not a polygon
PolygonNot a polygon
PolygonNot a polygon
Draw and label:
A regular tetragon. A concave nonagon.
O
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Polygons
Mathletics Passport 3P Learning
512HSERIES TOPIC
How does it work? PolygonsYour Turn
4
5
6
a
a
c
b
b
d
Draw and label:
Explain why it is not possible to draw a cyclic, equilateral, concave octagon.
A convex, equilateral hexagon.
An equiangular, pentagon
which is not equilateral.
A convex, cyclic tetragonwhich is not equilateral.
A concave, equilateral heptagon with
two reex angles ( angle180 360c c1 1 ).
Polygons
POLY
GONS
*POLYGO
NS*PO
LYGO
NS*
...../...../20....
How would you describe these polygons to someone drawing them in another room?
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Polygons
Mathletics Passport 3P Learning
6 12HSERIES TOPIC
How does it work? Polygons
Transformations
Transformaons are all about re-posioning shapes without changing any of their dimensions.
There are three main types:
Reecons (Flip) Reecng an object about a xed line called the axis of reecon.
Translaons (Slide) This transformaon involves sliding an object either horizontally, vercally or both.
Every part of the object is moved the same distance.
Rotaons (Turn) A transformaon of turning an object about a xed point counter-clockwise.
A
A
B
B
A
A A
A
A
B
B B B
A
B
A
BA
B
B
A
B
Keep equal spacing from axis.
Horizontal reecon to the right.
3 cm translaon horizontally
to the right
Two translaons: 2 cm horizontally
right, and then 3 cm vercally up
Axis of reecon
(or axis of dilaon)
Vercal reecon up followed by a
horizontal reecon le.
object
(before)
object
(before)
object
(before)
image
(aer)
image
(aer)
image
(aer)
2nd
1st
coun
ter-clockwise
A B
O
O
90c rotaon (or4
1 turn)
90c rotaon (or4
1 turn)
180c rotaon (or2
1 turn)
270c rotaon (or4
3 turn)180c rotaon (or2
1 turn)
3 cm
3 cm
2 cm
Centre of rotaon (or centre of dilaon)
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Polygons
Mathletics Passport 3P Learning
8 12HSERIES TOPIC
How does it work? PolygonsYour Turn
Transformations
5
6
a
a
c
b
b
d
c
Draw the image on the grids below when each of these objects are rotated by the given amounts.
Draw the image on the grids below when each of these objects undergo the transformaons given.
One half turn(180c rotaon).
Translate ten units to the right rst then
reect down about the given axis of reecon.
Reect about the given axis rst, then
tranlsate two units to the le.
Three quarter turn (270c rotaon) rst, then
reect about the given axis of dilaon.
Three quarter turn(270c rotaon).
One quarter turn(90c rotaon).
O
O
O
O
O
Rotate 180c about the centre of rotaon O,
then translate six units up.
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Polygons
Mathletics Passport 3P Learning
912HSERIES TOPIC
How does it work? PolygonsYour Turn
7 Earn yourself an awesome passport stamp with this one.
The object (ABCODE) requires thirteen transformaons to move along the white producon line
below. It needs to leave in the posion shown at the exit for the next stage of producon.
Describe the thirteen transformaon steps used to navigate this object along the path, including the
direcon of transformaon and the sides/points used as axes of dilaon where appropriate.
Transformations
(i)
(iii)
(v)
(vii)
(ix)
(xi)
(xiii)
(ii)
(iv)
(vi)
(viii)
(x)
(xii)
ENTRY EXIT
The object must not overlap the shaded part around the producon line path.
Any of the sidesAB,BC,DEandAEcan be used as an axis of reecon.
The vertex O is the only centre of rotaon used at the two circle points along the path.
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Polygons
Mathletics Passport 3P Learning
10 12HSERIES TOPIC
How does it work? PolygonsYour Turn
Transformations
8
a
b
c
d
For the diagram shown below, describe four dierent ways the nal image of the object can be
achieved using dierent transformaons.
Method 1
Method 2
Method 3
Method 4
A
B
AB
...../...../20....
TR
A
N
S
FOR
MA
T
I
ON
S *
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Polygons
Mathletics Passport 3P Learning
1112HSERIES TOPIC
How does it work? Polygons
Reflection symmetry
There are many types of symmetry and in this booklet we will just be focusing on three of them.
If the axis of reecon splits a shape into two idencal pieces, then that shape has reecon symmetry.
The axis of reecon is then called the axis of symmetry.
The distances from the edge of the shape to the axis of symmetry are the same on both sides of the line.
This shape has only one axis of symmetry. When this happens, we say the shape has bilateral symmetry.
Many animals/plants or objects in nature have nearly perfect bilateral symmetry.
Other shapes can have more than one axis of symmetry (axes of symmetry for plural).
Symmetric
Shape has reecon symmetry
Asymmetric
Shape does not have reecon symmetry
B
Y
A
X
C
Z AB=BC and XY=YZ
Regular Hexagon
There are 6 dierent ways this shape can be folded in half
with both sides of the fold ng over each other exactly.
So we can say it has six-fold symmetry.
Axis of reecon = axis of symmetry
5
4
3
21
6
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Polygons
Mathletics Passport 3P Learning
12 12HSERIES TOPIC
How does it work? PolygonsYour Turn
Reflection symmetry
1
2
3
a
e
i
a
a
b
f
j
b
c
g
k
c
b
d
h
l
d
Idenfy which of these shapes have reecon symmetry by cking symmetric or asymmetric.
How many axes of reecon symmetry would these nature items have if perfectly symmetrical?
These shapes all have reecon symmetry. Calculate the distance between Xand Y.
Draw all the axes of symmetry for those that do.
(i)
(ii)
Symmetric
Asymmetric
Symmetric
Asymmetric
Symmetric
Asymmetric
Symmetric
Asymmetric
Symmetric
Asymmetric
Symmetric
Asymmetric
Symmetric
Asymmetric
Symmetric
Asymmetric
Symmetric
Asymmetric
Symmetric
Asymmetric
Symmetric
Asymmetric
Symmetric
Asymmetric
Distance fromXto Y= Distance fromXto Y=
YX
XZ
Z
YZ=5 cm XZ= 14 cm
Y
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Polygons
Mathletics Passport 3P Learning
1312HSERIES TOPIC
How does it work? PolygonsYour Turn
4
5
a
a
d
b
b
e
c
c
f
Answer these quesons about the symmetric web below:
Complete these diagrams to produce an image with as many axes of reecve symmetry as indicated.
Reflection symmetry
How many axes of symmetry does the web have?
What pair of points are equidistant toLM?
Briey explain below how you decided this was the
correct answer.
Psst: equidistant means the same distance
Bilateral symmetry.
Two axes of symmetry.
Two fold symmetry.
Five-fold symmetry.
(show the other four axes)
Three axes of symmetry.
Eight-fold symmetry.
(show the other seven axes)
X
Y
L M
AJ
B
KP
Q
H
G
REFLEC
TION
SYM
METRY
REFLEC
TION
SY
MME
TRY
.....
/.....
/20
....
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Polygons
Mathletics Passport 3P Learning
14 12HSERIES TOPIC
How does it work? Polygons
Rotational symmetry
Point symmetry
(half turn)180c
(half turn)180c
(three quarter turn)270c
(quarter turn)90c
O
O
O
O
O
O
Rotaonal Symmetry of order 2
i.e. it looks the same 2 mes in one full rotaon.
Rotaonal Symmetry of order 4
i.e. it looks the same 4 mes in one full rotaon.
When an object is rotated 360c (a full circle), it looks the same as it was before rotang.
If the object looks the same again before compleng a full circle, it has rotaonal symmetry.
The number of mes the object repeats before compleng the full circle tells us the order of
rotaonal symmetry.
Point symmetry for one object Point symmetry for a picture with two objects
For both diagrams:AO=BO and OX=OY
Objects and pictures can oen have both rotaonal and point symmetry.
X
X
Y
Y
AA
BB
OO
This is when an object has parts the same distance away from the centre of symmetry in the opposite
direcon.
A straight line through the centre of symmetry will cross at least two points on the object.
Each pair of points crossed on opposite sides of the centre of symmetry are an equal distance away from it.
These both have point symmetry because for every point on them, there is another point opposite the
centre of symmetry (O) the same distance away.
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Polygons
Mathletics Passport 3P Learning
1512HSERIES TOPIC
How does it work? PolygonsYour Turn
Rotational and point symmetry
1
2
3
Idenfy which of these objects are rotaonally symmetric or asymmetric.
All these propellers have rotaonal symmetry. Idenfy which ones also have point symmetry.
Describe the relaonship between the number of blades and the point symmetry of
these propellers.
Describe the relaonship between the number of blades and the order of point symmetry for
the symmetric blades.
Write the order of rotaonal symmetry each of these mathemacal symbols have:
Rotaonally symmetric
Rotaonally asymmetric
Has point symmetry
No point symmetry
Has point symmetry
No point symmetry
Has point symmetry
No point symmetry
Has point symmetry
No point symmetry
Has point symmetry
No point symmetry
Has point symmetry
No point symmetry
Rotaonally symmetric
Rotaonally asymmetric
Rotaonally symmetric
Rotaonally asymmetric
Rotaonally symmetric
Rotaonally asymmetric
Rotaonally symmetric
Rotaonally asymmetric
Rotaonally symmetric
Rotaonally asymmetric
a
a c
a
b
c
d
b
b d
e
c
f
(i)
(iv)
(ii)
(v)
(iii)
(vi)
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Polygons
Mathletics Passport 3P Learning
16 12HSERIES TOPIC
How does it work? PolygonsYour Turn
Rotational and point symmetry
a
a
c
c
b
b
d
d
4
5
Complete each of the half drawn shapes below to match the given symmetries.
Rotaonal symmetry of order 4 and alsopoint symmetry.
Rotaonal symmetry of order 3 and no
point symmetry.
Rotaonal symmetry of order 2 and alsopoint symmetry.
Rotaonal symmetry of order 2 and also
point symmetry.
(i)
(ii)
Mark in the other verces.
Draw the boundary of the whole shape.
W
T
K
J
S
RQ
P
V
U
OO
OO
A
B
C
O
O
O
O
All the verces shown below represent half of all the verces of shapes which have point symmetry
about the centre of rotaon (O).
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Polygons
Mathletics Passport 3P Learning
1712HSERIES TOPIC
How does it work? PolygonsYour Turn
Combo time: Reflection, rotation and point symmetry
Idenfy if these ags of the world have symmetry and what type.
Include the number of folds or order of rotaons for those ags with the relevant symmetry.
6
a
c
e
g
b
d
f
h
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
*COMB
OTIME:
REF
LECTION,ROT
ATI
ONA
ND
POI
NTSY
MMETRY
...../...../
20....
.....
/.....
/
20.
...
Canada
India
Jamaica
South Africa
Malaysia
Australia
Pakistan
United States of America
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Polygons
Mathletics Passport 3P Learning
18 12HSERIES TOPIC
How does it work? PolygonsYour Turn
Idenfy if these ags of the world have symmetry and what type.
Include the number of folds or order of rotaons for those ags with the relevant symmetry.
6
k
m
o
q
l
n
p
r
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Reecon symmetry with folds
Rotaonal symmetry of order .
Point of symmetry.
No symmetry
Combo time: Reflection, rotation and point symmetry
Leer 'Y' signal ag
Leer 'D' signal ag
Georgia
Vietnam
Leer 'N' signal ag
Leer 'L' signal ag
New Zealand
United Kingdom
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Polygons
Mathletics Passport 3P Learning
1912HSERIES TOPIC
PolygonsWhere does it work?
Special triangle properties
Determine what type of triangle is described from the informaon given.
(i)
(ii)
All internal angles are less than 90c , and it has one axis of reecon symmetry.
All internal angels are equal and it has point symmetry.
Isosceles triangles have one axis of reecon symmetry.
Idenfying properes and naming shapes that match is called classifying.
` It is an acute angled isosceles triangle.
` It is an equilateral triangle.
90c1
90c=
90 180c c1 1:
O
SHAPE
TRIANGLES
Scalene
Isosceles
Equilateral
Acute angled triangle
Right angled triangle
Obtuse angled triangle
PROPERTIES
Three straight sides and internal angles.
All three sides have a dierent length.
All three internal angles are a dierent size.
Two of the intenal angles have the same size.
The two sides opposite the equal angles have equal lengths.
1-fold reecve symmetry.
No rotaonal symmetry.
All of the internal angles have the same size of60c .
All sides have the same length.
3-fold reecve symmetry.
Has rotaonal symmetry of order 3.
All of the interal angles are smaller than 90c .
One of the internal angles is equal to 90c
(i.e. one pair of sides are perpendicular to each other).
One of the internal angles is between 90c and 180c .
Triangles come in a number of dierent types, each with their own special features (properes) and names.
Here they are summarised in this table:
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Polygons
Mathletics Passport 3P Learning
20 12HSERIES TOPIC
PolygonsWhere does it work? Your Turn
Special triangle properties
Classify what type of triangle is described from the informaon given in each of these:
All internal angles are less than 90cand it has no axes of reecon.
One internal angle is equal to 90c and two sides are equal in length.
One internal angle is obtuse and there is one axis of reecon.
Has rotaonal symmetry and all internal angles equal to 60c .
No internal angles are the same size and one side is perpendicular to another.
Classify what type of triangle has been drawn below with only some properes shown.
1
2
a
a
c
b
b
d
c
d
e
SPEC
I
AL
TRIANG
LEP
ROPERT
IES
...../...../20....
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Polygons
Mathletics Passport 3P Learning
2112HSERIES TOPIC
PolygonsWhere does it work?
Special quadrilateral properties
O
O
SHAPE
QUADRILATERAL
Scalene
A convex or concave
quadrilateral
Trapezium
A convex
quadrilateral
Isosceles
Trapezium
Parallelogram
A convex
Qaudrilateral
Rectangle
A convex, equiangular
quadrilateral
PROPERTIES
Four straight sides and internal angles.
All four sides have a dierent length.
All four internal angles are a dierent size.
No symmetry.
At least one pair of parallel sides.
No symmetry.
Non-parallel sides are the same length.
Diagonals cut each other into equal raos.
Two pairs of equal internal angles with common arms.1 axis of reecve symmetry.
Opposite sides are parallel.
Opposite sides are equal in length.
Diagonally opposite internal angles are equal.
Diagonals bisect each other (cut each other exactly in half).
No axis of reecve symmetry.
Rotaonal symmetry of order 2 and point symmetry at the
intersecon of the diagonals O.
Opposite sides are parallel.
Opposite sides are equal in length.
All internal angles =90c .
Diagonals are equal in length.
Diagonals bisect each other (cut each other exactly in half).
2-fold reecve symmetry.
Rotaonal symmetry of order 2 and point symmetry at the
intersecon of the diagonals O.
Quadrilaterals exist in many dierent forms, each with their own special properes and names.
Here they are summarised in this table:
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Polygons
Mathletics Passport 3P Learning
22 12HSERIES TOPIC
PolygonsWhere does it work?
SHAPE
Square
A regular
quadrilateral
Rhombus
A convex
quadrilateral
Kite
A convex
quadrilateral
PROPERTIES
Opposite sides are parallel.Opposite sides are the same length.
All internal angles =90c .
Diagonals bisect each other.
Diagonals bisect each internal angle.
Diagonals cross at right angles to each other (perpendicular).
4-fold reecve symmetry.
Rotaonal symmetry of order 4 and point symmetry at the
intersecon of the diagonals O.
Opposite sides are parallel.
All sides are the same length.
Diagonally opposite internal angles are the same.
Diagonals bisect each other.
Diagonals bisect each internal angle.
Diagonals cross at right angles to each other (perpendicular).
2-fold reecve symmetry.
Rotaonal symmetry of order 2 and point symmetry at the
intersecon of the diagonalsO.
Two pairs of adjacent, equal sides.
Internal angles formed by unequal sides are equal.
Shorter diagonal is bisected by the longer one.
Longer diagonal bisects the angles it passes through.
Diagonals are perpendicular to each other.
1-fold reecve symmetry.
No Rotaonal symmetry.
Special quadrilateral properties
Quadrilateral Square
Kite Rhombus
Rectangle
Parallelogram
O
O
This diagram shows how each quadrilateral relates to the previous one which shares one
similar property.
Trapezium
Isosceles Trapezium
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Polygons
Mathletics Passport 3P Learning
2312HSERIES TOPIC
PolygonsWhere does it work? Your Turn
Special quadrilateral properties
Classify what special quadrilateral is being described from the informaon given in each of these:
Write down two dierences between each of these special quadrilaterals:
A quadrilateral has been parally drawn below. Draw and name the three possible quadrilaterals
this diagram could have been the start of according to the given informaon.
Two pairs of equal sides, all internalangles are right-angles and has 2-fold
reecve symmetry.
A square and a rectangle.
A parallelogram and a rhombus.
A rhombus and a square.
Two pairs of equal internal angles
with the diagonals the only axes of
reecve symmetry.
Diagonals bisect each other and split
all the internal angles into pairs of45c .
One pair of parallel sides and one pairof opposite equal sides.
A rectangle and a parallelogram.
A rhombus and a kite.
One pair of parallel sides and one pair
of opposite equal sides.
Perpendicular diagonals and no
rotaonal symmetry.
1
2
3
a
a
c
e
c
e
b
b
d
f
d
f
b ca
SPECI
AL
QUADRILATE
RAL
PROPERTI
ES*
...../...../
20....
axis of symmetry
diagonal
A kite and an isosceles trapezium.
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Polygons
Mathletics Passport 3P Learning
24 12HSERIES TOPIC
PolygonsWhere does it work? Your Turn
Combo time! Special quadrilateral and triangles
These two equal isosceles triangles can be transformed and combined to make two special
quadrilaterals. Explain the transformaon used, then name and draw the two special
quadrilaterals formed.
Draw all the dierent quadrilaterals that can be formed using these two idencal right-angled
scalene triangles.
1
2
3
These two idencal trapeziums can be transformed and combined to make two special quadrilaterals.
Explain the transformaon used, and then name and draw the new quadrilateral formed.
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Polygons
Mathletics Passport 3P Learning
2512HSERIES TOPIC
A
B C
D
F
E
A
B C
D
F
E
D
A
C
B
PolygonsWhat else can you do?
Transformations on the Cartesian number plane
Just as grids were used earlier to help transform shapes, the number plane can also be used.
The coordinates of verces help us locate and move objects accurately.
Determine the new coordinates for the points aer these translaons
(i)
(ii)
The coordinates of B aerABCD is reected about the line x=1.
The coordinates of E aer the shape ABCDEFis rotated 90c about the origin (0,0).
-2
-2 -2
-21
1 1
12
2 2
2
2
1
3
4
5
2 2
1 1
3 3
4 4
2
1
3
4
5
3
3 3
34
4 4
4-1
-1 -1
-10
0 0
0
y
y y
y
x
x x
x
-1 -1
New coordinates forB are (-1.5, 2)
New coordinates forEare (-2, 4)
-4 0-2 2-3 1-1 3 4
4
3
2
1
-1
-2
-3
-4
y
x
object
Posivey direcon
Translated 3 units in the posivex direcon
Rotated one quarter turn 90c about
the point ,2 1-^ h
Reected about the y-axis
Negave y direcon
Posivex direconNegave x direcon
object
object
image
image
image
,2 1-^ h
,1 3-^ h
,4 2-^ h ,1 2-^ h
,1 3- -^ h
Same methods apply as before, this me including the new coordinates of important points.
D
A
C
B
D
A
C
B
AB
CD
FE
x=1
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Polygons
Mathletics Passport 3P Learning
2712HSERIES TOPIC
PolygonsYour TurnWhat else can you do?
2
Transformations on the Cartesian number plane
a b
c d
e f
g h
object
object
object
object
object
object
imageimage
image
image
image
image
90c 180c 270c rotaon 90c 180c 270c rotaon
90c
180c
270c
rotaon 90c
180c
270c
rotaon
90c 180c 270c rotaon 90c 180c 270c rotaon
90c 180c 270c rotaon 90c 180c 270c rotaon
image
object
O
O
O
O
object
imageO
O
O
O
All these images are rotaons of the object.
Choose whether the rotaon is 90c , 180c or 270c about the given point of rotaon labelled O.
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Polygons
Mathletics Passport 3P Learning
28 12HSERIES TOPIC
PolygonsYour TurnWhat else can you do?
Transformations on the Cartesian number plane
3 (i) Draw the image for the requested transformaons on the number planes below.
(ii) Write down the new coordinates for the dot marked on each object.
a
c
e f
b
d
Reect object about the line x=1.
Rotate the object 180c about the ,0 0^ h.
Reect object about thex-axis. reect object about the given axis line, y=x.
Translate the object four units in the posive
y direcon.
Translate the object four units in the negave
y direcon.
y
y
y y
y
y
x
x
x x
x
x
-4
-4
-4 -4
-4
-4
-2
-2
-2 -2
-2
-2
-1
-1
-1 -1
-1
-1
1
1
1 1
1
1
2
2
2 2
2
2
3
3
3 3
3
3
4
4
4 4
4
4
-3
-3
-3 -3
-3
-3
0
0
0 0
0
0
-2
2
3
4
-4
-3
-1
1
-2
2
3
4
-4
-3
-1
1
-2
2
3
4
-4
-3
-1
1
-2
2
3
4
-4
-3
-1
1
-2
2
3
4
-4
-3
-1
1
-2
2
3
4
-4
-3
-1
1
object
object
object
object
object
object
New coordinates for dot =
New coordinates for dot =
New coordinates for dot = New coordinates for dot =
New coordinates for dot =
New coordinates for dot =
x=
1
( , )
( , )
( , ) ( , )
( , )
( , )
y=x
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Polygons
Mathletics Passport 3P Learning
2912HSERIES TOPIC
PolygonsYour TurnWhat else can you do?
Transformations on the Cartesian number plane
4 (i) Draw the image for the requested double transformaons on the number planes below.
(ii) Write down the new coordinates for the dot marked on each image.
a
c
e
b
d
f
Translate object 3 units in the posive
x-direcon and then reect about the
liney=1.
Rotate object 270c about the point (-1, 1)
and then reect about the x-axis.
Reect object about they-axis then rotate
180c about the origin ,0 0^ h.
Rotate the object one quarter turn about the
point (-1, 3) then translate 2.5 units in the
negave y-direcon.
Reect the object about the y-axis, and then
reect about the line y=1.
Translate the object 2.5 units in the
negave y-direcon and then reect about
the liney=-x.
y
y
y
x
x
x
-4 -2 -1 1 2 3 4-3 0
-4 -2 -1 1 2 3 4-3 0
-4 -2 -1 1 2 3 4-3 0
-2
-2
-2
2
2
2
3
3
3
4
4
4
-4
-4
-4
-3
-3
-3
1
1
1
object
object
object obje
ct
New coordinates for dot = New coordinates for dot =
New coordinates for dot =
New coordinates for dot =
y=1
( , ) ( , )
( , )
( , )
New coordinates for dot =( , )
New coordinates for dot =( , )
-1
-1
-1
y
y
y
x
x
x
-4 -2 -1 1 2 3 4-3 0
-4 -2 -1 1 2 3 4-3 0
-4 -2 -1 1 2 3 4-3 0
object
object
-2
2
3
4
-4
-3
1
-1
-2
2
3
4
-4
-3
1
-1
-2
2
3
4
-4
-3
1
-1
y=1
y=-x
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Polygons
Mathletics Passport 3P Learning
30 12HSERIES TOPIC
PolygonsYour TurnWhat else can you do?
Transformations on the Cartesian number plane
A player in a snow sports game can only use transformaons to perform tricks and change direcon
to get through the course marked by trees.
Points are deducted if trees are hit. Points are awarded when the corner dot marked A passes
directly over coordinates marked with ags on the course.
The dimensions of the player are a square with sides two units long.
Write down the steps (including the coordinates of point A aer each transformaon) a player can
take to get maximum points from start to nish.
5
Start here
Finish here
TRANSF
ORMATIO
NO
NTHECART
ESIA
NNU
MBE
RPLA
NE*
...../...../20....
A
B C
D
A
BC
D
-6 1-5 2-4 3-3 4-2 50-1 6
6
5
4
3
2
1
-1
-2
-3
-4
y
x
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Polygons
Mathletics Passport 3P Learning
3112HSERIES TOPIC
Cheat Sheet Polygons
Here is what you need to remember from this topic on polygons
Polygons
Polygons are just any closed shape with straight lines which dont cross.Like a square or triangle.
Transformaons
Reecon Symmetry
Polygon? Polygon? Polygon? Polygon?
Exterior angle
Vertex
Interior angle
Diago
nal
Side
Parts of a polygonShapes which are/are not polygons
Types of polygons:
All polygons need at least three sides to form a closed path.
Convex
Concave
All interior angles are 180c1
Has an interior angle 180c2
Equilateral
Equiangular
All sides are the same length
All interior angles are equal
RegularAll interior angles are equal
All sides are the same
length
They are cyclic polygons
CyclicAll verces/corner
points lie on the edge
(circumference) of the
same circle.
object objectimage image
coun
ter-clock
wise
image
Reecons (Flip) Translaons (Slide) Rotaons (Turn)
object
90c rotaon (or4
1 turn)
270c rotaon (or4
3 turn)
180c rotaon (or2
1 turn)
Where an axis of reecon splits an
object into two idencal pieces.
The distances from the edge of theshape to the axis of symmetry are
the same on both sides of the line.
Symmetric: Shape has reecon
symmetry
Asymmetric: Shape does not have
reecon symmetry
Y
A
X
C
Z AB=BC and XY=YZ
Axis of reecon
= axis of symmetry
B
Sides Polygon name
3 Trigon (triangle)
6 Hexagon
9 Nonagon
12 Dodecagon
Sides Polygon name
4 Tetragon
7 Heptagon
10 Decagon
15 Pentadecagon
Sides Polygon name
5 Pentagon
8 Octagon
11 Hendecagon
20 Icosagon
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Polygons
Mathletics Passport 3P Learning
32 12HSERIES TOPIC
Cheat Sheet Polygons
Rotaonal Symmetry
Point Symmetry
Special Triangles and Quadrilaterals (summary of key sides and angle dierences only)
If an object looks the same during a rotaon before compleng a full circle, it has rotaonal symmetry.
The number of mes the object repeats before compleng the full circle tells us the order of
rotaonal symmetry.
Rotaonal Symmetry of order 4 as it looks the same four mes within one full rotaon.
(half turn)180c (three quarter turn)270c(quarter turn)90c
O
O O
O
Point symmetry for one object
For both diagrams:
AO=BO and OX=OY
X
X
Y
Y
AA
BB
OO
Point symmetry for two object
Scalene
No equal sides or angles. At least 1 pair of parallel sides. At least 1 pair of parallel sides.
Non-parallel sides equal in length.
Parallelogram Rectangle Square
Opposite sides equal in length and
parallel to each other.
Opposite sides equal in length and
parallel to each other.
All internal angles =90c .
All sides equal in length and opposite
sides parallel to each other.
All internal angles =90c .
Rhombus Kite
All sides equal in length and opposite
sides parallel to each other. Diagonally
opposite internal angles equal.
Two pairs of adjacent equal sides.
Angles opposite short diagonal equal.
Acute
All internal angles 90c1
Obtuse
One internal angle between 90c and 180c
Scalene
No equal sides or angles
Isosceles
1 pair of equal sides & angles
Equilateral
All sides and angles equal
Right angled triangle
1 internal angle =90c
Triangles
Quadrilaterals
For a more detailed summary, see pages 19, 21 and 22 of the booklet.
These objects have point symmetry because for every point on them, there is another point opposite the
centre of symmetry (O) the same distance away.
Trapezium Isosceles Trapezium
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TRANSFORMATIO
NO
NTHECAR
TES
IA
NNU
MBERPLA
NE*
...../...../20....
SPE
CIAL
TRIANG
LEPR
OPERT
IES
...../...../20.
...
*COMBOT
IME:
REFLEC
TION,ROTAT
IONA
NDPOINT
SYM
METR
Y...../...../20....
.....
/.....
/20....
...../...../20....
T
R
A
NS
FOR
M
A
T
IO N S
*
POL
YGONS
*POLYGO
NS*P
OLYG
ONS*
...../...../20....