TABLE OF CONTENT TITLE | PAGE | TABLE OF CONTENT | 1 | APPRECIATION | 2 | OBJECTIVE | 3 | METHOD INVESTIGATION | 4 | PART I | 5-7 | PART II | 8-10 | PART III | 11-13 | PART IV | 14-16 | REFLECTION | 17-18 | CONCLUSION | 19 |

APPRECIATION
First of all, I would like to say thank you for giving me the power to complete this project work. Not forgotten my parents for providing me any material to complete this project and their support which are the most needed for this project. Internet, books, computer and the list goes on. They also supported me and encouraged me to complete this project so that I will not procrastinatein doing it. For their strong support, I would like to express my gratitude to my beloved parents. They also helped me to find the mark to complete this project. They have always been at my side and I hope that they will still be there in the future. Besides, I would like to thank my Additional Mathematics teacher, Pn. Phong Bee Bee as she had gave us some important guidance and commitment during this project work. She has been a very supportive figure throughout the project. Then I would also want to thank my teacher for guiding me and my friends throughout this project. We had faced some difficulties in doing this task, but she taught us patiently until we knew what to do. She had tried to teach us until we understand what we supposed to do with the project work. I also want to thank my friends for assisting me to complete the project. They have done a great job at collecting the materials and sharing information needed to complete the project. Without them this project would never have its conclusion. Last but not least, my friends who were doing this project with me and share our ideas. They were helpful that when we combined and discussed together. We had this task done. Last but not least, I would also like to thank all the classmates for sharing their information related to this project. Not forgetting to all the people who were involved in this project. I thank you all.

OBJECTIVE
The aims of carrying out this project work are enable students to : a. Apply mathematical knowledge to everyday situation and appreciate the importance and the beauty of mathematics in everyday lives.; b. Improve problem solving skills, thinking skills, reasoning and mathematical communication; c. Develop positive attitude and personalities and intrinsic mathematical values suck as accuracy, confidence and systematic reasoning; d. Stimulate learning environment that enhances effective learning, inquiry-based and teamwork ; e. Develop mathematical knowledge in a way which increases students’ interest and confidence; f. Apply and adapt a variety problem solving strategies to solve problems; g. Promote effective mathematical communication; h. Express mathematical ideas precisely

METHOD INVESTIGATION
In solving and finishing this project work done, some methods are used:

1.Lesson Session
The lesson session in the class help me in solve problem by using heuristics what I learnt in the class.
2.Communication
Discussion with teacher and friends to help in solving problem. The information from this discussion used as a reference material to success this project.

3. Reference
Some of the information from various reference material help me to find the method to solve the problem. For this Additional Mathematics project, I can get the reference from library, internet, my fellow friends, teachers and my parents.

PART I
HISTORY ABOUT HERON
Hero (or Heron) of Alexandria (c. AD 10–70) was an ancient Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. He is considered the greatest experimenter of antiquity and his work is representative of the Hellenistic scientific tradition. Hero published a well-recognised description of a steam-powered device called an aeolipile (hence sometimes called a "Hero engine"). Among his most famous inventions was a wind-wheel, constituting the earliest instance of wind harnessing on land. He is said to have been a follower of the Atomists. Some of his ideas were derived from the works of Ctesibius.Much of Hero's original writings and designs have been lost, but some of his works were preserved in Arab manuscripts. It is almost certain that Hero taught at the Musaeum which included the famous Library of Alexandria, because most of his writings appear as lecture notes for courses in mathematics, mechanics, physics and pneumatics. Although the field was not formalized until the 20th century, it is thought that the work of Hero, his automated devices in particular, represents some of the first formal research into cybernetics.
Hero described the construction of the aeolipile (a version of which is known as Hero's engine) which was a rocket-like reaction engine and the first-recorded steam engine (although Vitruvius mentioned the aeolipile in De Architectura some 100 years earlier than Hero). It was created almost two millennia before the industrial revolution. Another engine used air from a closed chamber heated by an altar fire to displace water from a sealed vessel; the water was collected and its weight, pulling on a rope, opened temple doors. Some historians have conflated the two inventions to assert that the aeolipile was capable of useful work. The first vending machine was also one of his constructions, when a coin was introduced via a slot on the top of the machine; a set amount of holy water was dispensed. This was included in his list of inventions in his book, "Mechanics and Optics". When the coin was deposited, it fell upon a pan attached to a lever. The lever opened up a valve which let some water flow out. The pan continued to tilt with the weight of the coin until it fell off, at which point a counter-weight would snap the lever back up and turn off the valve.
Next,,Hero also invented many mechanisms for the Greek theatre, including an entirely mechanical play almost ten minutes in length, powered by a binary-like system of ropes, knots, and simple machines operated by a rotating cylindrical cogwheel. The sound of thunder was produced by the mechanically-timed dropping of metal balls onto a hidden drum. In optics, Hero formulated the Principle of the Shortest Path of Light: If a ray of light propagates from point A to point B within the same medium, the path-length followed is the shortest possible. It was nearly 1000 years later that Alhacen expanded the principle to both reflection and refraction, and the principle was later stated in this form by Pierre de Fermat in 1662; the most modern form is that the path is at an extremum.
Besides, Heron described a method of iteratively computing the square root. Today, though, his name is most closely associated with Heron's Formula for finding the area of a triangle from its side lengths.The imaginary number, or imaginary unit, is also noted to have been first observed by Hero while calculating the volume of a pyramidal frustum.
Heron’s formula can be used to determine the area of a triangle if the lengths of three sides of the triangle are known if S represents the semi-perimeter of a triangle ABC, that is S = 12(a+b+c), then the Heron’s formula, also known as Heron’s formula, gives the area of the triangle ∆ = SS-aS-b(S-c) x ( c-x )
N
h

The formula can be derived as follows
Diagram shows a triangle ABC where the height CN is h. By representing the area of ∆ = 12ch -
If AN = x and BN = (c-x),
Using Pythagoras’ theorem for triangles ANC and BNC,
We obtain : x2+h2=b2 - (c-x)2+ h2 = a2
Equation - x2 -(c-x)2=b2-a2 x2-c2-2cx+x2= b2-a2
-c2+2cx=b2-a2
b2+c2-a2=2cx -
Equation × c2 c2x2+ c2h2 = c2x2 c2h2 =c2b2- c2x2 =(cb-cx)(cb+cx)
4c2h2 =(2cb-2cx)(2cb+2cx)-

Sub into
4c2h2=2cb-b2+c2-a2[2cb+b2+c2-a2]
= [2cb –b2-c2+a2][2cb+b2+c2-a2] = [-(b2-2cb+c2) +a2] [b2+2cb+c2-a2] = [-(b-c)2+a2] [(b+c)2-a2] = [a2-(b-c)2] [(b+c)2-a2] = [a-(b-c)] [ a+(b-c)] [(b+c)-a] [(b+c)+a] = [a-b+c] [a+b-c] [b+c-a] [ b+c+a]

As the semi-perimeter s of the triangle is given by S = 12 (a+b+c), we have 2s = a+b+c, with (a+c)-b = (2s-b)-b = 2s-2b –
(a+b)-c = (2s-c)-c = 2s-2c –
(b+c)-a = (2s-a)-a = 2s-2a –
Sub , , into
4c2h2 = (2s-2b)(2s-2c)(2s-2a)(2s) = 16(s-a)(s-b)(s-c)(s)
14c2h2 = s(s-a)(s-b)(s-c)
12ch =ss-as-b(s-c), ch > 0
Area of triangle ABC = ss-as-b(s-c)
3cm
B
6cm
Part II
4cm
C
9cm
4cm
8cm
P
Q
R
A

Method 1
Area of triangle ABC= A□ –Ap-AQ-AR =9(8)-12(6)(8)-12 (3)(4)-12(9)(4) =72 – 24 – 6 – 18 =24 units2
Method 2
By Pythagoras’ theorem,
AC= 92+42 =97 units
AB=82+62=10 units
BC= 32+42 =5 units
By Heron formula,
S=12(97 +10+5)
=12(15+97)
Area of A = S(s-a)(s-b)(s-c) =12(15+97 )(1215+97 -10)(12(15+97)-5

=576 =24 units2

A
Method 3
10cm
5cm
C
B

97

By cosine rule, c² = a² + b² - 2abcosC (97 )2 = 52 + 102 – 2(5)(10)cosC cosC = 25 +100 - 97100 = 0.28 C = 73◦44΄ Method 4 Let A = (0,0) , B = (9,4) , C = (6,8) Area of ∆ABC = 1200946800 = 1204+98+60-09-46-8(0) = 1272-24 = 1248 = 24 unit2 5
B
Method 5 73°44'
97
C

10

A

From cosine rule, C = 73°44΄ sin73°44΄ = h5 h = 5(0.96) = 4.8cm Area of Δ ABC = 12x h x AC = 12(4.8)(10) = 24 unit2

PART III V

10

12
D
A
C
B

V 10

h

m
N
M

12
6
A B i) Length of VM = 102+62 = 136 m Area of ΔVBA = 12 (h)(12) = 12 (136)(12) = 69.9714m2 Since4 of the triangle have the same area,
∴ Total surface area of the surface of the pyramid
=4(69.9714) + 12(12)
=423.886 cm2

ii)

12m
12m
10m

C
BG

A
D

V

DB = AD2+AB2 = 122+122 = 288 m VB = VD2+DB2 = 102+288 = 388m VA = VD2+DA2 = 102+122 = 244m By Heron’s formula, S = 12 (VA + VB + AB) = 12(244 + 388 + 12) = 23.659 m Area of triangle VAB = SS-VAS-VB(S-AB) =23.65923.659-24423.659-388(23.659-12) = 8784 = 93.723m2 ∴ Area of VAB = Area of VBC = 93.723 cm2 Area of VDA = area of VDC = 1210(12) = 60m2 ∴ Total surface area = 2(93.723) + 2(60) + 12(12) = 451.446m2 Type I is more economically to build in terms of cost due to has smaller surface area

Part IV a) C a a

A 80-2a B Perimeter = 80 S = 12(80) = 40 Area of ∆, A = Ss-as-b(s-c) = 4040-a40-a(40-80-2a) = 4040-a2(2a-40) = 40(40-a)2a-40 b) When triangle can be formed A>0 4040-a2a-40 >0
(40-a)2a-40 >0
40-a(2a-40)12>0
∴(2a-40)12is always positive
40-a>0
a<40

(2a-40)12>0 2a-40>0 a>20

20 40
∴to get of a=20<a<40

c)to get maximum area ,by differentration method. dAda=0 (0)(40-a)(2a-40)+40(-1)2a-40)+40(40-a)122a-40-122=0
402a-40=4040-a2a-40
402a-40=4040-a 3a=80 a=803
∴maximum area A=4040-8032803-40 =307.92m2

Method 2 :By Heron’s formula,the max area will be the smallest number substrated from the
Semiperimeter.Hence, the side length are 803m will has the greatest area

B

803803

A C

h=8032-(403)2 =2.094cm
∴A=12hAC
=1223.094803 =307.92cm

REFLECTION
In the making of this project, I have spent countless hours doing this project. I realized that this subject is a compulsory to me. Without it, I can’t fulfill my big dreams and wishes…I used to hate Additional Mathematics…It always makes me wonder why this subject is so difficult…I always tried to love every part of it…It always an absolute obstacle for me…Throughout day and night…I sacrificed my precious time to have fun…From…Monday, Tuesday, Wednesday, Thursday ,Friday and even the weekend that I always looking forward to…
While conducting the project, I have learnt a lot of thing such that patience, careful and cooperation. While I conducting this project, a lot of information that I found.I have learnt how geometrics 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 up 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 two weeks 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.
Furthermore, I learnt the value of cooperation. I cooperated with my friends to complete this project. Without cooperation, it is not possible to complete this project. Besides, I learned how to cooperate with friends. My friends and I discussed about the project and we shared ideas amongst ourselves. This discussion has made me more confident when doing something. I also learned to be disciplined in terms of punctuality, doing project and not least to avoid plagiarism. In a nutshell, I have learned something about the future and how serious can it be if we make a wrong decision. Moreover, work that we done must be original and not totally derived from other people. These two images have expressed my opinions from this project.
In the process of conducting this project, I have learnt that perseverance pays off, especially when you obtain a just reward for all your hard work. For me, succeeding in completing this project work has been reward enough. I faced some problems while conducting this project, but I never give up. My perseverance finally pays off when I successfully completed this project. Besides, I learnt to be confident. Although this project seemed to be challenging, I was confident enough to do it. I realized that confidence is very important when we are carrying out a task. In addition, I learnt the value of responsibility. I was responsible when t conducted this project by obeying the rules and making sure that every part of the project is complete and correct. Moreover, I learnt to be precise. When I am making calculations and looking for information for this project, I did carefully and precisely. I made sure that all the calculations and information are correct, relevant and do not contain mistakes. Furthermore, I learnt to be appreciative. My parents, teachers and friends gave me some help when I was conducting this project. I appreciated their help very much.
I also learnt that mathematics is used everywhere in daily life, from the most simple things like doing geometrics. Besides that, I have learned many moral values that I practice. This project work had taught me to be more confident when doing something especially the homework given by the teacher. I also learned to be a more disciplined student who is punctual and independent.
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.

CONCLUSION After doing research, answering question and some problem solving, I saw that the usage of geometric is important in daily life. Geometry is widely used in the field of civil engineering. Heron formula is even a crucial formula in constructing building with pyramid shape. Anyway, geometric is a daily life necessity. Without it, constructing of building, development of city cannot be conducted. Therefore, we should be thankful of the people who contribute in the idea of geometry. Geometry tend to be studied in the first five years of school as an introduction to basic mathematics. This is due to geometry is very important that we used to apply it in the future of our life. Geometry deals with calculating coordinates, areas and gradient of lines and different types of shape. Coordinates are the points of a particular place and they also can be joined together to form particular shape. Meanwhile, gradients are the steepness of a line. Coordinates, areas and gradients tend to be used widely in the field of construction. They are a must mastered mathematics topics for every single civil engineer. Building is not as easy as we think. Civil engineer need to put their heart and soul to calculate all the angles, gradients, coordinates and areas of the building accurately. A tiny mistake would sometimes put the building at the risk to collapse. Geometry is also important in the development of town, city or a country. The area of the town or city should be measured accurately for the development in order not to cause wastage land. In conclusion, geometry is a daily life necessity. Without it, man’s creativity will be limited. Therefore, we should be immensely thankful to the people who contribute in the idea of geometry. In the end, I can say that SMJK Chio Min School is new-built.