POLYGONS€¢ Pythagoras (approx. 569-475 BC) • Euclid of Alexandria (approx. 325-265 BC) (often referred to as the "Father of modern geometry') • Archimedes of Syracus (approx

PASSPORT

PASSPORT

Poly

gons POLYGONSPOLYGONSPOLYGONS

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PolygonsMathletics 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 description.

Work through the book for a great way to do this

This booklet is about identifying and manipulating straight sided shapes using their unique properties

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) (often referred to as the "Father of modern geometry')

• Archimedes of Syracus (approx 287-202 BC)

• Apollonius of Perga (approx. 261-190 BC)

After an attack on the city of Alexandria, many of the works of these mathematicians were lost.

Look up these people sometime and read about their contribution to this subject.

New discoveries in geometry are still being made with the advent of computers, in particular fractal

geometry. The most famous of these being Benoit Mandelbrot Fractal pattern.

Q


2 12HSERIES TOPIC

How does it work?

Polygons

Polygons

Polygons are just any closed shape with straight lines which don’t 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 vertices and is not a side)

Exterior angle

Interior angleSide

Each corner is called a Vertex (vertices plural)

There are many basic types of polygons. Here are the ones we will be looking at in this booklet:

Here is another difference between convex and concave polygons.

Convex ConcaveA straight line drawn through the polygon can only cross a maximum of 2 sides

A straight line drawn through the polygon can cross more than two sides.

Convex polygon

All interior angles are 180c1

Equilateral polygonAll sides are the same length

Cyclic polygonAll vertices/corner points lie on the edge (circumference) of the same circle.

Equiangular polygon

All interior angles are equal

Regular polygon

All interior angles are equalAll sides are the same lengthThey are cyclic polygons

Concave polygon

Has an interior angle 180c2


312HSERIES TOPIC

How does it work? Polygons

Polygons

Any polygon can be named using Greek prefixes 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 Enneadecagon10 Decagon 20 Icosagon11 Hendecagon 30 Tricontagon12 Dodecagon 40 Tetracontagon13 Tridecagon 50 Pentacontagon14 Tetradecagon 60 Hexacontagon15 Pentadecagon 70 Heptacontagon16 Hexadecagon 80 Octacontagon17 Heptadecagon 90 Enneacontagon18 Octadecagon 100 Hectogon

Many of these polygons have more than one name.

Look them up sometime!

Nonagon Enneagon

9 sides

Polygon naming and classification chart

Sides Name Concave Convex Equilateral Equiangular Cyclic Regular

3 Triangle (Trigon) N/A

4Quadrilateral

(Tetragon)

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon


4 12HSERIES TOPIC

How does it work? PolygonsYour Turn

Identify which of these shapes are polygons or not.

Tick all the properties that each of these polygons have and then name the shape:

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Polygons

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g h

Polygon Not a polygon

Convex Concave Equilateral Equiangular Cyclic Regular













Draw and label:

A regular tetragon. A concave nonagon.

O


512HSERIES TOPIC


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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 tetragon which is not equilateral.

A concave, equilateral heptagon with two reflex angles ( angle180 360c c1 1 ).

Polygons

POLYGONS * POLYGO

NS * POLYG

ONS *

...../...../20....

How would you describe these polygons to someone drawing them in another room?


6 12HSERIES TOPIC


Transformations

Transformations are all about re-positioning shapes without changing any of their dimensions.

There are three main types:

Reflections (Flip) Reflecting an object about a fixed line called the axis of reflection.

Translations (Slide) This transformation involves sliding an object either horizontally, vertically or both. Every part of the object is moved the same distance.

Rotations (Turn) A transformation of turning an object about a fixed point counter-clockwise.

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Keep equal spacing from axis.

Horizontal reflection to the right.

3 cm translation horizontally to the right

Two translations: 2 cm horizontally right, and then 3 cm vertically up

Axis of reflection(or axis of dilation)

Vertical reflection up followed by a horizontal reflection left.

object(before)

object(before)

object(before)

image(after)

image(after)

image(after)

2nd

1st

co

unter-clockwise

A B

O

O

90c rotation (or 41 turn)



270c rotation (or 43 turn)180c rotation (or

21 turn)

3 cm

3 cm

2 cm

Centre of rotation (or centre of dilation)


712HSERIES TOPIC


Transformations

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object image

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Identify which type of transformation each of these playing cards has undergone:

Each of these objects has undergone two different transformations. Tick them both.

Draw the image on the grids below when each of these objects are reflected about the given axis.

Draw the image on the grids below when each of these objects are translated by the given amounts.

Reflection Translation Rotation






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objectobject

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image image

Five squares horizontally to the left.

Four squares vertically up. Eight squares to the right, then six squares down.

object image

Axis of dilation

centre of dilation


8 12HSERIES TOPIC


Transformations

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6

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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 transformations given.

One half turn (180c rotation).

Translate ten units to the right first then reflect down about the given axis of reflection.

Reflect about the given axis first, then tranlsate two units to the left.

Three quarter turn (270c rotation) first, then reflect about the given axis of dilation.

Three quarter turn (270c rotation).

One quarter turn (90c rotation).

O

O

O

O

O

Rotate 180c about the centre of rotation O, then translate six units up.


912HSERIES TOPIC


7 Earn yourself an awesome passport stamp with this one.The object (ABCODE) requires thirteen transformations to move along the white production line below. It needs to leave in the position shown at the exit for the next stage of production.

Describe the thirteen transformation steps used to navigate this object along the path, including the direction of transformation and the sides/points used as axes of dilation 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 production line path.• Any of the sides AB, BC, DE and AE can be used as an axis of reflection.• The vertex O is the only centre of rotation used at the two circle points along the path.


10 12HSERIES TOPIC


Transformations

8

a

b

c

d

For the diagram shown below, describe four different ways the final image of the object can be achieved using different transformations.

Method 1

Method 2

Method 3

Method 4

A

B

AB

...../...../20....

T

RA

NS

FORM

AT

IO

N S *


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 reflection splits a shape into two identical pieces, then that shape has reflection symmetry.

The axis of reflection 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).

SymmetricShape has reflection symmetry

AsymmetricShape does not have reflection symmetry

B

Y

A

X

C

Z AB = BC and XY = YZ

Regular Hexagon

There are 6 different ways this shape can be folded in half with both sides of the fold fitting over each other exactly.So we can say it has six-fold symmetry.

Axis of reflection = axis of symmetry

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12 12HSERIES TOPIC


Reflection symmetry

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Identify which of these shapes have reflection symmetry by ticking symmetric or asymmetric.

How many axes of reflection symmetry would these nature items have if perfectly symmetrical?

These shapes all have reflection symmetry. Calculate the distance between X and Y.

Draw all the axes of symmetry for those that do.

(i)

(ii)

Symmetric Asymmetric












Distance from X to Y = Distance from X to Y =

YX

XZ

Z

YZ = 5 cm XZ = 14 cm

Y


1312HSERIES TOPIC


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Answer these questions about the symmetric web below:

Complete these diagrams to produce an image with as many axes of reflective symmetry as indicated.

Reflection symmetry

How many axes of symmetry does the web have?

What pair of points are equidistant to LM?

Briefly 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

QH

G

REFLEC

TION SYM

METRY

REFLECTION SYM

METRY

...../...../

20....


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

Rotational Symmetry of order 2

i.e. it looks the same 2 times in one full rotation.

Rotational Symmetry of order 4

i.e. it looks the same 4 times in one full rotation.

When an object is rotated 360c (a full circle), it looks the same as it was before rotating.

If the object looks the same again before completing a full circle, it has rotational symmetry.

The number of times the object ‘repeats’ before completing the full circle tells us the order of rotational 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 often have both rotational 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 direction.

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.


1512HSERIES TOPIC


Rotational and point symmetry

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2

3

Identify which of these objects are rotationally symmetric or asymmetric.

All these propellers have rotational symmetry. Identify which ones also have point symmetry.

Describe the relationship between the number of blades and the point symmetry of these propellers.

Describe the relationship between the number of blades and the order of point symmetry for the symmetric blades.

Write the order of rotational symmetry each of these mathematical symbols have:

Rotationally symmetric Rotationally asymmetric

Has point symmetry No point symmetry











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a c

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b d

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f

(i)

(iv)

(ii)

(v)

(iii)

(vi)


16 12HSERIES TOPIC


Rotational and point symmetry

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5

Complete each of the half drawn shapes below to match the given symmetries.

Rotational symmetry of order 4 and also point symmetry.

Rotational symmetry of order 3 and no point symmetry.



(i)(ii)

Mark in the other vertices.Draw the boundary of the whole shape.

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P

VU

OO

OO

A

B

C

O

O

O

O

All the vertices shown below represent half of all the vertices of shapes which have point symmetry about the centre of rotation (O).


1712HSERIES TOPIC


Combo time: Reflection, rotation and point symmetry

Identify if these flags of the world have symmetry and what type.Include the number of folds or order of rotations for those flags with the relevant symmetry.

6

a

c

e

g

b

d

f

h

Reflection symmetry with folds

Rotational symmetry of order .

Point of symmetry. No symmetry






















* COMBO TIME: REFLECTION, R

OT

ATION AND POIN

T SYMMETRY...../...../20.... .....

/...../2

0...

.

Canada

India

Jamaica

South Africa

Malaysia

Australia

Pakistan

United States of America


18 12HSERIES TOPIC


Identify if these flags of the world have symmetry and what type.Include the number of folds or order of rotations for those flags with the relevant symmetry.

6

k

m

o

q

l

n

p

r

























Combo time: Reflection, rotation and point symmetry

Letter 'Y' signal flag

Letter 'D' signal flag

Georgia

Vietnam

Letter 'N' signal flag

Letter 'L' signal flag

New Zealand

United Kingdom


1912HSERIES TOPIC

PolygonsWhere does it work?

Special triangle properties

Determine what type of triangle is described from the information given.

(i)

(ii)

All internal angles are less than 90c , and it has one axis of reflection symmetry.

All internal angels are equal and it has point symmetry.

Isosceles triangles have one axis of reflection symmetry.

Identifying properties 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 different length.All three internal angles are a different size.

Two of the intenal angles have the same size. The two sides opposite the equal angles have equal lengths.

1-fold reflective symmetry. No rotational symmetry.

All of the internal angles have the same size of 60c . All sides have the same length.

3-fold reflective symmetry. Has rotational 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 different types, each with their own special features (properties) and names.

Here they are summarised in this table:


20 12HSERIES TOPIC

PolygonsWhere does it work? Your Turn

Special triangle properties

Classify what type of triangle is described from the information given in each of these:

All internal angles are less than 90cand it has no axes of reflection.

One internal angle is equal to 90cand two sides are equal in length.

One internal angle is obtuse and there is one axis of reflection.

Has rotational 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 properties shown.

1

2

a

a

c

b

b

d

c

d

e

SPECIAL TRIANGL

E PROPER

TIES

...../...../20....


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 different length.All four internal angles are a different 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 ratios.

Two pairs of equal internal angles with common arms. 1 axis of reflective 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 reflective symmetry. Rotational symmetry of order 2 and point symmetry at the intersection 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 reflective symmetry.Rotational symmetry of order 2 and point symmetry at the intersection of the diagonals O.

Quadrilaterals exist in many different forms, each with their own special properties and names.

Here they are summarised in this table:


22 12HSERIES TOPIC


SHAPE

Square

A regular quadrilateral

Rhombus


Kite


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 reflective symmetry. Rotational symmetry of order 4 and point symmetry at the intersection 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 reflective symmetry. Rotational symmetry of order 2 and point symmetry at the intersection of the diagonals O.

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 reflective symmetry. No Rotational 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


2312HSERIES TOPIC



Classify what special quadrilateral is being described from the information given in each of these:

Write down two differences between each of these special quadrilaterals:

A quadrilateral has been partially drawn below. Draw and name the three possible quadrilaterals this diagram could have been the start of according to the given information.

Two pairs of equal sides, all internal angles are right-angles and has 2-fold reflective 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 reflective symmetry.

Diagonals bisect each other and split all the internal angles into pairs of 45c .

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 pairof opposite equal sides.

Perpendicular diagonals and no rotational symmetry.

1

2

3

a

a

c

e

c

e

b

b

d

f

d

f

b ca

SPECIAL QUADRILATE

RAL

PROPER

TIES *

...../...../20

....

axis of symmetry

diagonal

A kite and an isosceles trapezium.


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 transformation used, then name and draw the two special quadrilaterals formed.

Draw all the different quadrilaterals that can be formed using these two identical right-angled scalene triangles.

1

2

3

These two identical trapeziums can be transformed and combined to make two special quadrilaterals. Explain the transformation used, and then name and draw the new quadrilateral formed.


2512HSERIES TOPIC

A

B C

D

F

E

A

B C

D

F

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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 vertices help us locate and move objects accurately.

Determine the new coordinates for the points after these translations

(i)

(ii)

The coordinates of ‘B’ after ABCD is reflected about the line x = 1.

The coordinates of ‘E’ after the shape ABCDEF is rotated 90cabout the origin (0,0).

-2

-2 -2

-21

1 1

12

2 2

2

2

1

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5

2 2

1 1

3 3

4 4

2

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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 for B are (-1.5, 2)

New coordinates for E are (-2, 4)

-4 0-2 2-3 1-1 3 4

4

3

2

1

-1

-2

-3

-4

y

x

object

Positive y direction

Translated 3 units in the positive x direction

Rotated one quarter turn 90c about the point ,2 1-^ h

Reflected about the y-axis

Negative y direction

Positive x directionNegative x direction

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 time including the new coordinates of important points.

D

A

C

B

D

A

C

B

ABCD

FE

x = 1


26 12HSERIES TOPIC

PolygonsYour TurnWhat else can you do?


1

a

c

e

b

d

f

All these images are reflections of the object.Choose whether the reflection was vertical (up/down), horizontal (right/left) or both (diagonally).

Reflected

Vertically Horizontally

Diagonally

Reflected


Diagonally

Reflected


Diagonally

Reflected


Diagonally

Reflected


Diagonally

Reflected


Diagonally

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

object

object

object

object

object

object

image

image

image

image

image

image


2712HSERIES TOPIC


2


a b

c d

e f

g h

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

y

x-4 -2 -1 1 2 3 4-3 0

-2

2

3

4

-4

-3

-1

1

object

obje

ct

object

object

object

object

imageimage

imageimage

imag

e

image

90c 180c 270c rotation 90c 180c 270c rotation




image

object

O

O

O

O

object

image O

O

O

O

All these images are rotations of the object. Choose whether the rotation is 90c , 180cor 270c about the given point of rotation labelled O.


28 12HSERIES TOPIC



3 (i) Draw the image for the requested transformations on the number planes below. (ii) Write down the new coordinates for the dot marked on each object.

a

c

e f

b

d

Reflect object about the line x = 1.

Rotate the object 180c about the ,0 0^ h.

Reflect object about the x-axis. reflect object about the given axis line, y = x.

Translate the object four units in the positive y direction.

Translate the object four units in the negative y direction.

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


2912HSERIES TOPIC



4 (i) Draw the image for the requested double transformations 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 positive x-direction and then reflect about the line y = 1.

Rotate object 270c about the point (-1, 1) and then reflect about the x-axis.

Reflect object about the y-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 negative y-direction.

Reflect the object about the y-axis, and then reflect about the line y = 1.

Translate the object 2.5 units in the negative y-direction and then reflect about the line y = -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 object

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


30 12HSERIES TOPIC



A player in a snow sports game can only use transformations to perform tricks and change direction 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 flags 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 after each transformation) a player can take to get maximum points from start to finish.

5

Start here

Finish here

TRANSFORMATION ON THE CART

ESIAN NUMBER

PLANE *

...../...../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


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 don’t cross. Like a square or triangle.

Transformations

Reflection Symmetry

Polygon? Polygon? Polygon? Polygon?

Exterior angle

Vertex

Interior angle

DiagonalSide

Parts of a polygonShapes which are/are not polygonsTypes 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 lengthThey are cyclic polygons

CyclicAll vertices/corner points lie on the edge (circumference) of the same circle.

object objectimage image

co

unter-clockwise

image

Reflections (Flip) Translations (Slide) Rotations (Turn)

object




Where an axis of reflection splits an object into two identical pieces.

The distances from the edge of the shape to the axis of symmetry are the same on both sides of the line.

Symmetric: Shape has reflection symmetry

Asymmetric: Shape does not have reflection symmetry

Y

A

X

C

Z AB = BC and XY = YZ

Axis of reflection = axis of symmetry

B

Sides Polygon name3 Trigon (triangle)

6 Hexagon

9 Nonagon

12 Dodecagon

Sides Polygon name4 Tetragon

7 Heptagon

10 Decagon

15 Pentadecagon

Sides Polygon name5 Pentagon

8 Octagon

11 Hendecagon

20 Icosagon


32 12HSERIES TOPIC

Cheat Sheet Polygons

Rotational Symmetry

Point Symmetry

Special Triangles and Quadrilaterals (summary of key sides and angle differences only)

If an object looks the same during a rotation before completing a full circle, it has rotational symmetry. The number of times the object ‘repeats’ before completing the full circle tells us the order of rotational symmetry.

Rotational Symmetry of order 4 as it looks the same four times within one full rotation.

(half turn)180c (three quarter turn)270c(quarter turn)90cO

O O

O

Point symmetry for one object

For both diagrams: AO = BO and OX = OY

XX

Y

YAA

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 SquareOpposite 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 90cand 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

TRANSFORMATION ON THE CART

ESIAN NUMBER

PLANE *

...../...../20....

SPECIAL TRIANGL

E PROPER

TIES

...../...../20....

* COMBO TIME: REFLECTION, ROTATI

ON AND POINT SYM

METRY

...../...../20....

...../...

../2

0........./...../20....

TR

A

NSFO

RM

AT

IO N S

*

POLYGONS * POLYG

ONS * POL

YGONS *

...../...../20....

Documents

POLYGONS€¢ Pythagoras (approx. 569-475 BC) • Euclid of Alexandria (approx. 325-265 BC) (often referred to as the "Father of modern geometry') • Archimedes of Syracus (approx