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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 Im 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.

Objectives
The aims of carrying out this project work are:

to promote effective mathematical communication.

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

to develop positive attitude towards mathematics

Introduction

In geometry a polygon ( /pln/ ) 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)

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
The Giant's Causeway, in Northern Ireland

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
Starfruit, a popular fruit in Southeast Asia

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.

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 Pentagon ,an American federal pentagon-shaped purpose, located on the corner of Sydney Ave building located in The United States of America and National Circuit in Australia.

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 polygnum (a noun), from Greek (polygnon/polugnon), noun use of neuter of (polygnos/polugnos, 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

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:

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

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

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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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.

pm

qm

100m

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

By using cosine rule and method 1 before (degree) 0 38.2145 44.1837 49.5826 55.7711 57.6881 58.9924 Area(m2) 0 2598.15 3057.92 3464.10 3968.63 4130.67 4242.64

p(m) 50 60 65 70 80 85 90

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95 99 100

105 101 100 Table 1

59.7510 59.9901 60

4308.42 4329.26 4330.13

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.

ym Area= xy m2 1400 2600 3600 4400 5000 5400 5600 5625

x 10 20 30 40 50 60 70 75

The maximum area is 5625 m2

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B. Regular Pentagon

=

41.2915 m

C. A Semicircle 2r +r = 300 r= Area = ( ) = 5247.55 m2

D. A Circle

r=

16

Area = (

) (

) = 7161.97 m2

E. Regular Hexagon

6a = 300 a = 50 Area = x 50 x 50 x 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 +r = 300 r= Area = ( ) = 5247.55 m2

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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. 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 -

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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, 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.

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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 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.

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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.

Dherbs
Natural care solution

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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, mans 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 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.

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