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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....