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Acknowledgment

First and foremost I would like to thank God that finally, I have succeeded in

finishing this project work. I would like to thank my beloved Additional

Mathematics teacher, Puan Nur Nadiah Lani for all the guidance she had

provided me during the process of finishing this work.

Next, I would like to give a thousand appreciations to both En Miswan Sairi

and Pn Fatimah Embong as my parent who had gave me full support in this

project, financially and mentally. They gave me moral support when I needed

it.

I also would like to give my thanks to my fellow friends such as Nik Husni Nik

Rusdi and Muhammad Syahmi Masli who had helped me in finding the

information that I’m clueless about, and the time we spent together in study

groups on finishing this project. Last but not least I would like to express this

appreciation towards all who gave me the possibility to complete this

coursework.

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Objectives

The aims of carrying out this project work are:

  to apply and adapt a variety of problem-solving strategies

tosolve problems.

to improve thinking skills.

to promote effective mathematical communication.

to develop mathematical knowledge through problem solving in a way that increases students interest and confidence.

to use the language of mathematics to express mathematical ideas precisely.

to provide learning environment that stimulates and enhanceseffective learning.

to develop positive attitude towards mathematics

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Introduction

In geometry a polygon ( /ˈpɒlɪɡɒn/ ) is a flat shape consisting of straight lines that are

joined to form a closed chain or circuit.

A polygon is traditionally a plane figure that is bounded by a closed path, composed of a

finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are

called its edges or sides, and the points where two edges meet are the polygon's vertices (singular:

vertex) or corners. An n-gon is a polygon with n sides. The interior of the polygon is sometimes

called its body. A polygon is a 2-dimensional example of the more general polytope in any

number of dimensions.

The basic geometrical notion has been adapted in various ways to suit particular purposes.

Mathematicians are often concerned only with the closed polygonal chain and with simple

polygons which do not self-intersect, and may define a polygon accordingly. Geometrically two

edges meeting at a corner are required to form an angle that is not straight (180°); otherwise, the

line segments will be considered parts of a single edge – however mathematically, such corners

may sometimes be allowed. In fields relating to computation, the term polygon has taken on a

slightly altered meaning derived from the way the shape is stored and manipulated in computer

graphics (image generation).

Polygons have been known since ancient times. The regular

polygons were known to the ancient Greeks, and the pentagram, a non-convex

regular polygon (star polygon), appears on the vase of Aristophonus, Caere, dated

to the 7th century B.C.Non-convex polygons in general were not systematically

studied until the 14th century by Thomas Bradwardine. In 1952, Geoffrey Colin

Shephard generalized the idea of polygons to the complex plane, where each real

dimension is accompanied by an imaginary one, to create complex polygons.

Historical image of polygons (1699)

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Numerous regular polygons may be seen in nature. In

the world of geology, crystals have flat faces, or facets,

which are polygons.Quasicrystals can even have regular

pentagons as faces. Another fascinating example of regular

polygons occurs when the cooling of lava forms areas of

tightly packed hexagonal columns of basalt, which may be

seen at the Giant's Causeway in Ireland, or at the Devil's

Postpile in California.

The most famous hexagons in nature are found in the animal kingdom. The

wax honeycomb made by bees is an array of hexagons used to store honey and pollen, and as a

secure place for the larvae to grow. There also exist animals who themselves take the approximate

form of regular polygons, or at least have the same symmetry. For example, sea stars display the

symmetry of a pentagon or, less frequently, the heptagon or other polygons. Other echinoderms,

such as sea urchins, sometimes display similar symmetries. Though

echinoderms do not exhibit exact radial symmetry, jellyfish and comb

jellies do, usually fourfold or eightfold.

Radial symmetry (and other symmetry) is also widely observed in

the plant kingdom, particularly amongst flowers, and (to a lesser extent)

seeds and fruit, the most common form of such symmetry being

pentagonal. A particularly striking example is the starfruit, a slightly tangy

fruit popular in Southeast Asia, whose cross-section is shaped like a

pentagonal star.

Moving off the earth into space, early mathematicians doing calculations

using Newton's law of gravitation discovered that if two bodies (such as the sun and the earth) are

orbiting one another, there exist certain points in space, called Lagrangian points, where a smaller

body (such as an asteroid or a space station) will remain in a stable orbit. The sun-earth system has

five Lagrangian points. The two most stable are exactly 60 degrees ahead and behind the earth in

its orbit; that is, joining the center of the sun and the earth and one of these stable Lagrangian

points forms an equilateral triangle.

The Giant's Causeway, in Northern Ireland

Starfruit, a popular fruit in Southeast Asia

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Part 1

a)pictures of polygons

The Egyptian pyramids are ancient pyramid-shaped mansory structure located in Egypt

The Kaaba is a cuboid-shaped structure located in Mecca

A hexagonal building, possibly government purpose, located on the corner of Sydney Ave and National Circuit in Australia. 

Pentagon ,an American federal pentagon-shaped building located in The United States of America

Unique Modern Home Office Sphere-DesignRectangular-shaped brick wall

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b) definition and brief history about polygon

A polygon is traditionally a plane figure that is bounded by a closed path, composed of a

finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are

called its edges or sides, and the points where two edges meet are the polygon's vertices (singular:

vertex) or corners. An n-gon is a polygon with n sides. The interior of the polygon is sometimes

called its body. A polygon is a 2-dimensional example of the more general polytope in any

number of dimensions.

The word "polygon" comes from Late Latin polygōnum (a noun), from Greek πολύγωνον

(polygōnon/polugōnon), noun use of neuter of πολύγωνος (polygōnos/polugōnos, the masculine

adjective), meaning "many-angled". Individual polygons are named (and sometimes classified)

according to the number of sides, combining a Greek-derived numerical prefix with the suffix -

gon, e.g. pentagon, dodecagon. The triangle, quadrilateral or quadrangle, and nonagon are

exceptions. For large numbers, mathematicians usually write the numeralitself, e.g. 17-gon. A

variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula.

Some special polygons also have their own names; for example the regular star pentagon is also

known as the pentagram.

History

Polygon have been known since ancient times. The regular pentagon were known to the ancient

Greeks, and the Pentagram , a non-convex regular polygon (star polygon), appears on the base of

Aristophonus, Caere, dated to the 7th century B.C.Non-convex polygons in general were not

systematically studied until the 14th century by Thomas Bredwardine.

In 1952,Shephard generelized the idea of polygons to the complex plane, where each real

dimension is accompanied by an imaginary one, to create complex polygons

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c) methods to find the area of a triangle

METHOD 1

If you have two sides (say a and b) and the angle between them(say C), then the area can be

calculated from the formula as :

By simple trigonometry we can replace h by : 

Which can be written in term of the any two sides and the angle between them. i.e. 

 

example: 

Lets say two sides of the triangle are b=4 and c=3 and the common angle A(90) 

 

Hence the Area of the triangle after plugging all the values in the formula:

 

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METHOD 2

This method can be used if only all the side lengths are given . No information about the angles of

the triangle. 

Then the area of the triangle is given by

( where s is the semi-perimeter)

This formula is called Heron's formula. 

example: 

Lets say side lengths are a=5, b=4 and c=3

Hence the Area of the triangle after plugging all the values in the formula 

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METHOD 3

If you have two angles (say B and C) and the side common to them(say a), then the area can be

calculated as 

 

Where the third angle can be calculated by 

 

Example: 

Lets say two angles of the triangle are B=30 and C=60 and the common side length a= 5

so to apply the above formula we need third Angle of the triangle.

Angle  

Hence the Area of the triangle after plugging all the values in the formula 

 

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METHOD 4

If the coordinates of the vertices are given as 

 ,   ,   

then the Area is calculated as

,

where det denotes the determinant, i.e. 

 

example: 

Lets say the coordinates of the given triangle are as follows:

A(1,1), B(3,4) and C(4,1) 

Hence the Area of the triangle after plugging all the values in the formula

 

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p m θ q m

Part 2

A farmer wishes to build a triagular-shape herb garden on a piece of land,

where

one of its sides is 100 m in length. The garden has to be fenced with a 300 m

fence. The cost of fencing the garden is RM 20 per metre.

a) Calculate the cost needed to fence the garden

RM 20 x 300 = RM 6000

b) Complete table 1 by using various values of p, the corresponding value of q and θ

By using cosine rule and method 1 before

p(m) q(m) θ(degree) Area(m2)

50 150 0 0

60 140 38.2145 2598.15

65 135 44.1837 3057.92

70 130 49.5826 3464.10

80 120 55.7711 3968.63

85 115 57.6881 4130.67

90 110 58.9924 4242.64

95 105 59.7510 4308.42

99 101 59.9901 4329.26

100 100 60 4330.13

Table 1

100m

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c) Based on your findings in (b), state the dimension of the herb garden so

that the enclosed area is maximum.

The herb garden should be an equilateral triangle of sides 100 m with a maximum area of

4330.13 m2

d) (i)Only certain values of p and of q are applicable in this case. State the

range of values of p and of q.

50<p<150, 50<q<150

(ii) By comparing the lengths of p, q and the given side, determine the

relation between them.

p+q >100

(iii) Make generalisation about the lengths of sides of a triangle. State

the name of the relevant theorem.

The sum of the lengths of any two sides of a triangle is greater than the length of the third

side. This is theorem is known as The Triangle Inequality Theorem.

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Part 3 If the length of the fence remains the same, explore at least 5 various other

shapes of thegarden that can be constructed so that the enclosed area is

maximum.

Round off your answers correct to two decimal places.

Draw a conclusion from your exploration.

A. Quadrilateral

y 2x + 2y = 300 m2

x m x + y = 150 m2

y m

x y Area= xy m2

10 140 1400

20 130 2600

30 120 3600

40 110 4400

50 100 5000

60 90 5400

70 80 5600

75 75 5625

The maximum area is 5625 m2

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B. Regular Pentagon

5a =300

a=60

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tan54 °= t

30

t = 30tan54 ° = 41.2915 m

Area = 12

(41.2915 x 69) x 5 = 6139.73 m2

C. A Semicircle

2r +πr = 300

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r = 3002+π

Area =12 ( 300

2+π )2

π = 5247.55 m2

D. A Circle

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2 π r=300

r = 150π

Area = 12 ( 150

π )2

(2 π ) = 7161.97 m2

E. Regular Hexagon

6a = 300

a = 50

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Area = 12

x 50 x 50 x sin 60 ° x 6 = 6495.19 m2

Conclusion:

Circle is the best shape to use for the garden as it gives a maximum enclosed area among the other

shapes.

2r +πr = 300

r = 3002+π

Area =12 ( 300

2+π )2

π = 5247.55 m2

Further exploration

(a) The demand of herbs in the market has been increasing nowadays.

Suggest three types of local herbs with their scientific names that the farmer

can plant in the herb garden to meet the demand. Collect pictures and

information of these herbs.

1) Agrimony

Agrimony has an old reputation as a popular, domestic medicinal herb, being a simple well

known to all country-folk. It belongs to the Rose order of plants, and its slender spikes of yellow

flowers, which are in bloom from June to early September, and the singularly beautiful form of its

much-cut-into leaves, make it one of the most graceful of our smaller herbs.

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Agrimony was one of the most famous vulnerary herbs. The Anglo-Saxons, who called it

Garclive, taught that it would heal wounds, snake bites, warts, etc. In the time of Chaucer, when

we find its name appearing in the form of Egrimoyne, it was used with Mugwort and vinegar for 'a

bad back' and 'alle woundes': and one of these old writers recommends it to be taken with a

mixture of pounded frogs and human blood, as a remedy for all internal haemorrhages. It formed

an ingredient of the famous arquebusade water as prepared against wounds inflicted by an

arquebus, or hand-gun, and was mentioned by Philip de Comines, in his account of the battle of

Morat in 1476. In France, the eau de arquebusade is still applied for sprains and bruises, being

carefully made from many aromatic herbs. It was at one time included in the London Materia

Medica as a vulnerary herb, but modern official medicine does not recognize its virtues, though it

is still fully appreciated in herbal practice as a mild astringent and tonic, useful in coughs,

diarrhoea and relaxed bowels. By pouring a pint of boiling water on a handful of the dried herb -

stem, leaves and flowers - an excellent gargle may be made for a relaxed throat, and a teacupful of

the same infusion is recommended, taken cold three or four times in the day for looseness in the

bowels, also for passive losses of blood. It may be given either in infusion or decoction.

Agrimony contains a particular volatile oil, which may be obtained from the plant by

distillation and also a bitter principle. It yields in addition 5 per cent of tannin, so that its use in

cottage medicine for gargles and as an astringent applicant to indolent ulcers and wounds is well

justified. Owing to this presence of tannin, its use has been recommended in dressing leather.

2) Eurycoma Longifolia ( Tongkat Ali )

Eurycoma longifolia (commonly called tongkat ali or pasak bumi) is a flowering plantin

the family Simaroubaceae, native to Indonesia, Malaysia, and, to a lesser extent,

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Thailand, Vietnam, and Laos. It is also known under the names penawar pahit,penawar

bias, bedara merah, bedara putih, lempedu pahit, payong ali, tongkat baginda, muntah bumi, petala

bumi (all the above Malay); bidara laut (Indonesian);babi kurus (Javanese); cay ba

binh (Vietnamese) and tho nan (Laotian). Many of the common names refer to the plant's

medicinal use and extreme bitterness. "Penawar pahit" translates simply as "bitter charm" or

"bitter medicine". Older literature, such as a 1953 article in the Journal of Ecology, may cite only

"penawar pahit" as the plant's common Malay name.

The root and bark of Eurycoma longifolia are used for treating erectile dysfunction(ED)

and increasing interest in sex. Eurycoma longifolia is also used for fever, malaria, ulcers, high

blood pressure, tuberculosis, bone pain, cough, diarrhoea, headache, syphilis, and cancer.

3) Ficus Deltoidea ( Mas Cotek )

Mas Cotek is a small perennial herb, growing up to about 2 m tall. The different shapes of

the leaves represent different varieties some having a rounded shape and others having an

elongated egg shape. The color at the top of the leaf is shining green while underneath, the surface

color is golden yellow with black spots in between the leaf veins. Mas Cotek plant species are

male and female. The leaves of female species are big and round in shape, while the male species

are small, round and long in shape.

Mas Cotek plants grow wild in eastern peninsular Malaysia and it is popular among

traditional medical practitioners. The trees can be found in the jungle

D’herbs22

in Kelantan, Terengganu, Sabah, Sarawak and Kalimantan. It also can be found grew in the oil

palm plantation in Perak, Selangor and Johor.

Mas Cotek, also known as "mistletoe fig", has been scientifically researched by various

institutions, including University of Malaya,Universiti Putra Malaysia, Universiti Sains Malaysia,

the Forest Research Institute of Malaysia, the Malaysian Agriculture Research And Development

Institute (MARDI). Research results show that Mas Cotek possesses triterpenoids and natural

phenols (flavonoids and proanthocyanins, a type of condensed tannins).

Traditionally used as a postpartum depression treatment to help in contracting the muscles

of the uterus and in the healing of the uterus and vaginal canal, it is also used as a libido booster

by both men and women. The leaves of male and female plants are mixed in specific proportions

to be taken as an aphrodisiac. Among the traditional practices, Mas Cotek has been used for

regulating blood pressure, increasing and recovering sexual desire, womb contraction after

delivery, reducing cholesterol, reducing blood sugar level, treatment of migraines, toxin removal,

delay menopause, nausea, joints pains, piles pain and improving blood circulation.

Mas Cotek products are formulated and sold in the form of extracts, herbal drinks, coffee

drinks, capsules, and massage oil.

(b) These herbs will be processed for marketing by a company. The design of

the packaging plays an important role in attracting customers. The company

wishes to design an innovative and creative logo for the packaging. You are

given thetask of designing a logo to promote the product. Draw the logo on a

piece of A4 paper. You must include at least one polygon shape in the logo.

D’herbs

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Natural care solution

Conclusion

Polygons tend to be studied in the first few years of school,as an introduction to basic

geometry and mathematics. But afterdoing research, answering questions, completing table and

someproblem solving, I found that the usage of polygons is important inour daily life.It is not just

widely used in fashion design but also in otherfields especially in building of modern structures.

The triangle, forinstance, is often used in construction because its shape makes

itcomparatively strong. The use of the polygon shape reduces thequantity of materials needed to

make a structure, so essentiallyreduces costs and maximizes profits in a business

environment.Another polygon is the rectangle.

The rectangle is used in anumber of applications, due to the fact our field of vision

broadlyconsists of a rectangle shape. For instance, most televisions arerectangles to allow for easy

and comfortable viewing. The samecan be said for photo frames and mobile phones screens.In

conclusion, polygon is a daily life necessity. Without it, man‟s creativity will be limited.

Therefore, we should be thankfulof the people who contribute in the idea of polygon

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Reflection

While I conducting this project, a lot of information that I found. I have learnt

how polygons appear in our daily life. Apart from that, this project encourages the

student to work together and share their knowledge. It is also encourage student to

gather information from the internet, improve thinking skills and promote effective

mathematical communication.

Not only that, I had learned some moral values that I practice. This project had

taught me to responsible on the works that are given to me to be completed. This project

also had made me felt more confidence to do works and not to give easily when we

could not find the solution for the question.

I also learned to be more discipline on time, which I was given about a month to

complete this project and pass up to my teacher just in time. I also enjoy doing this

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project I spend my time with friends to complete this project and it had tighten our

friendship.

Last but not least, I proposed this project should be continue because it brings a

lot of moral value to the student and also test the students understanding in Additional

Mathematics.