2695_21852150.NZL_H_Polygons_NZL

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RT

PASSPO

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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


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


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


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


Polygon

Not a polygon

Polygon

Not a polygon



Draw and label:

A regular tetragon. A concave nonagon.

O


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Polygons


512HSERIES TOPIC


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


6 12HSERIES TOPIC


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


8 12HSERIES TOPIC


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


912HSERIES TOPIC


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


10 12HSERIES TOPIC


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


1112HSERIES TOPIC


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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12 12HSERIES TOPIC


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


1312HSERIES TOPIC


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


14 12HSERIES TOPIC


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


1512HSERIES TOPIC


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











a

a c

a

b

c

d

b

b d

e

c

f

(i)

(iv)

(ii)

(v)

(iii)

(vi)


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Polygons


16 12HSERIES TOPIC


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


1712HSERIES TOPIC


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



Point of symmetry.

No symmetry



Point of symmetry.

No symmetry



Point of symmetry.

No symmetry



Point of symmetry.

No symmetry



Point of symmetry.

No symmetry



Point of symmetry.

No symmetry



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


18 12HSERIES TOPIC


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



Point of symmetry.

No symmetry



Point of symmetry.

No symmetry



Point of symmetry.

No symmetry



Point of symmetry.

No symmetry



Point of symmetry.

No symmetry



Point of symmetry.

No symmetry



Point of symmetry.

No symmetry



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


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.


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


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


2112HSERIES TOPIC


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 equal in length.

All internal angles =90c .

Diagonals are equal in length.

Diagonals bisect each other (cut each other exactly in half).




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


22 12HSERIES TOPIC


SHAPE

Square

A regular

quadrilateral

Rhombus

A convex

quadrilateral

Kite

A convex

quadrilateral

PROPERTIES

Opposite sides are parallel.Opposite sides are the same length.


Diagonals bisect each other.

Diagonals bisect each internal angle.

Diagonals cross at right angles to each other (perpendicular).





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).



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.


No Rotaonal symmetry.


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


2312HSERIES TOPIC



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


24 12HSERIES TOPIC


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


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


28/36


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Polygons


2712HSERIES TOPIC

PolygonsYour TurnWhat else can you do?

2


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



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


28 12HSERIES TOPIC



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 =



x=

1

( , )

( , )

( , ) ( , )

( , )

( , )

y=x


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Polygons


2912HSERIES TOPIC



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 =



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


30 12HSERIES TOPIC



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


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


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 sides equal in length and opposite

sides parallel to each other.


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


35/36


36/36

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....

2695_21852150.NZL_H_Polygons_NZL

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