Add Maths Project Work

Add Maths Project Work
PROJECT WORK FOR ADDITIONAL MATHEMATICS 2012 POLYGONS IN OUR LIFE Name:
Class: Teacher: I/C number: CONTENT |No. |Title |Pages | |1 |Objectives | | |2 |Introduction | | |3
|Part 1 | | |4 |Part 2 ... Show more content on Helpwriting.net ...
Some other generalizations of polygons are described below. HISTORY OF POLYGONS 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.[citation needed] Non–convex polygons in general were not
systematically studied until the 14th century by Thomas Bredwardine.[20] In 1952, Shephard
generalized the idea of polygons to the complex plane, where each real dimension is accompanied
by an imaginary one, to create complex polygons.[21] [pic] The historical image of polygons 1969
SOME PICTURES OF POLYGONS METHODS OF FINDING AREA OF TRIANGLE Method 1:
h b If you know base (b) and height (h) of the triangle, the following formula can be applied. Area =
½ x b x h Method 2: a c b If you know three sides (a, b and c) of the triangle, Heron's Method can be
applied. s = (a+b+c) / 2 Area =( s (s–a) (s–b) (s–c) Method 3: a ( b If you know two
... Get more on HelpWriting.net ...

The Ottoman Hexagonal Tile With Floral Pattern
The Ottoman Hexagonal Tile with Floral Pattern is located in the Art from Islamic Lands gallery, in
room 2550, on the second floor of the Arthur M. Sackler Museum on the Harvard University
Campus. The object number is 1960.102. It is located with a series of Ottoman tiles on the
easternmost wall of the one–room gallery. The tile, created between 1520 and 1540, is from İznik,
Turkey. It is composed of fritware ceramic painted underglaze. The base of the tile is a white
underglaze glaze, and all decoration is in various shades of blue underglaze. The shape is a hexagon
with a circular, symmetrical floral pattern derived from a central focal point. This tile had
meticulous work put into its design, showing the concern of its artist to emulate cultural tradition
and destiny on a single ceramic hexagon.
Fritware is a form of pottery where frit is added to the clay in order to lower the necessary fusion
temperature. This version of ceramics was popular in İznik, Turkey, the leading producer of
decorative tiles during the Ottoman Empire. Underglaze is a technique in which decoration is
painted to the piece before it is glazed. However, underglaze uses certain pigments derived from
oxides that fuse with the glaze, thus restricting the color palette. The original color found to work by
the Chinese was cobalt blue, and this is present on the Ottoman tile. In addition, the İznik factories
used turquoise, purple, green and red in later years. Because this tile is from the earlier part

Eudoxus' Contribution to Calculus
Eudoxus was a notable mathematician and astronomer of ancient times, particularly 408 – 355 BC.
He lived in Greece and studied under Plato, one of the most notable philosophers ever. In Calculus,
Eudoxus is known for advancing Antiphon's ideas on the method of exhaustion. The method of
exhaustion is very important to calculus because one of the fundamental themes of calculus is
sending variables (or whatever it happens to be) to infinity, which is a branch of the method of
exhaustion. This is known as taking the limit. Eudoxus used the method of exhaustion to calculate
volume and area. One example of this is his work with the area of a circle. At the time they hadn't
established this yet. If you didn't know the formula for finding the area of a circle, you would need
to approximate it, just like how we have learned to approximate the area under a curve with Rieman
Sums. First, you could inscribe a circle in a triangle, and use the area of the triangle to approximate
the area of the circle. But that would not give you a very accurate answer, so next you would draw a
square around your circle. Still, this is not very accurate, so you would keep adding to the number of
sides of your polygon until you approached infinity, giving you the most accurate answer possible
without the formula. This is how Eudoxus would have figured out the area of a circle – with limits.
It seems thus that Eudoxus was the first person to develop the definition of a limit. This definition
has

Differences Between Formal Grammar And Functional Grammar
3.2 Development of Grammar
Language is always in flux and dynamic, so is grammar, an important component of language.
During the passage of time, there were many aspects in human society that have experienced
changes, which influenced language. Therefore, the development of grammar is also significant.
There were different frameworks and types of grammar being developed by degrees. In this section,
two distinctive types: formal grammar and functional grammar will be introduced. 3.2.1 Formal
Grammar
According to Khatim (2013), grammar is established when a significant number of people use the
language in the same structure, or particular ways, and rules for grammar would be established
gradually. There should be formal rules for grammar so as to act as a guidance for language usage.
As pointed out by Khatim ... Show more content on Helpwriting.net ...
McLaughlin points out that a grammar lesson with exactly the same contents can actually be a kind
of formal grammar teacher or functional grammar teaching, depending on the purpose of the teacher
that rather the teacher want to give mental training to the students, or want them to really apply what
they have leant to their speeches. As Coffin, Donohue and North (2013) mentioned, even the two
kinds of grammar could be presented in completely different ways, connections between them could
still be

Designing A Integrated Approach For Search And Rescue...
Keywords: Mobile Robot, Explore Unknown Map with Obstacles, Gas Detection, V– rep, Khepera
III, Albers Algorithm.
Abstract: Nowadays, using robots instead of humans in risk operations is an interesting point in the
field of robotics. In this paper we introduce an integrated approach for search and rescue operation
of detecting gas sources in large areas; e.g. houses, factories, and labs. This aims at saving people's
lives at bottlenecks resulting from gas leak and ignition of fires. Experiment is performed using
master Khepera robot supplied with gas sensor circuit to search map, in addition to two slave
Khepera supplied with gripper to take the best path to clinch the victim.
Moreover, this paper proposes an improvement of Albers exploration algorithm to reduce the time
required to explore unknown map with different polygon obstacles. The proposed approach aims to
minimizing the overall exploration time, making it possible to localize fire sources in an efficient
way, as demonstrated in
Vrep simulation as well as real world experiments with Khepera III robot. A comparison among
different algorithms has showed the effectiveness of the proposed one where the percentage of
performance speedup is about 30% to 57% depending on size of the map and number of obstacles.
1 INTRODUCTION
Autonomous Robots are meant to perform searching operations inside buildings, mines and caves ,
which are considered extremely serious. These robots perform such tasks in complex environments
which

Five Platonic Solids
The Five Platonic Solids The five platonic solids are the tetrahedron, cube, octahedron,
dodecahedron, and the icosahedron. They are named for the Greek philosopher Plato. Plato wrote
about them in The Timaeus (c.360 B.C.) in which he paired each of the four classical elements,
earth, air, water, and fire with a regular solid. Earth was paired with the cube, air with the
octahedron, water with the icosahedron, and fire with the tetrahedron. The fifth Platonic solid, the
dodecahedron, Plato says that, "...the god used for arranging the constellations on the whole
heaven". Aristotle later added a fifth element, the ether and postulated that the heavens were made
of this element, but he had no interest in matching ... Show more content on Helpwriting.net ...
Some people credit Pythagoras with their discovery. Others say he may have been only familiar with
them and that the octahedron and icosahedron belong to Theaetetus. Theaetetus was the first to give
a mathematical description of all five and could have been responsible for the first piece of proof
that no other regular polyhedra are out there. A net is the shape that is made by unfolding a 3D
figure. A net is made up of all the faces of a figure. For a solid to be considered a platonic it needs
each face to be a regular polygon and each vertex must come together because if only two came
together the figure would collapse on itself. The sum of each interior angle of the faces meeting at a
vertex must be less than 360 degrees. A polyhedra solid must have all flat faces, whilst a non–
polyhedra solid has a least one of its surfaces that is not flat. Regular means all angles are of equal
measure, all faces are congruent or equal in every way, and all edges are of equal length. 3D means
that shape has width, depth and height. A polygon is a closed shape in a plane figure with at least
five straight edges. A dual is a Platonic Solid that fits inside another Platonic Solid and connects to
the midpoint of each

Koch Snowflake Investigation Angus Dally
Koch Snowflake Investigation Angus Dally
Background:
In 1904, Helge von Koch identified a fractal that appeared to model the snowflake. The fractal was
built by starting with an equilateral triangle and removing the inner third of each side, building
another equilateral triangle where the side was removed, and then repeating the process indefinitely.
The process is pictured below, showing the original triangle at stage zero, and the resulting figures
after one, two and three iterations.
Method:
Let Nn=the number of sides, Ln=the length of a single side, Pn= the length of the perimeter and An=
the area of the snowflake, all for the nth stage. Using an initial side length of 1, create a table that
shows the exact values of N_n,〖 ... Show more content on Helpwriting.net ...
Consequently, the area of each of these triangle is 1/9 of the original triangles.
Therefore the total area added is:
Area added=3×1/9
=1/3
Therefore, to find the area of snowflake n=1, these two area ratios must be multiplied by the area of
the original triangle at n=0.
A_1=√3/4 (1+1/3)
A_1=√3/3
For snowflake n=2, twelve additional triangles are added onto the initial triangle, with side lengths
of 1/9. Consequently, the area of each of these triangle is 1/81 of the original triangles.
Therefore the total area added is:
Area added=12×1/81
=12/81
=4/27
Therefore, to find the area of snowflake n=2, this area ratio, as well as the previous two, must be
multiplied by the area of the original triangle at n=0.
A_2=√3/4 (1+1/3+4/27)
A_2=(10√3)/27
At n=3, following the previous pattern:
48 triangles with side length 1/81, and area 1/729 are added.
Area added=48×1/729
=16/243

Therefore, to find the area of snowflake n=3, this area ratio, as well as the previous three, must be
multiplied by the area of the original triangle at n=0.
A_3=√3/4 (1+1/3+4/27+16/243)
A_3=(94√3)/243
A_n Data Table
Iteration An n=0 √3/4 n=1 √3/3 n=2 (10√3)/27 n=3 (94√3)/243
The area ratios being added in each iteration yields a pattern:
1/3,4/27,16/243...= 4^((1)–1)/3^(2(1)–1)

Character Analysis Of The Hanging Garden By Toshiaki Toyoda
Based on the homonymous book by Mitsuyo Kakuta, Hanging Garden is Toshiaki Toyoda's best
movie to date.
The Kyobashis appear to be a regular family of four members: Eriko, the mother, retains a part time
employment in a restaurant, otherwise busying herself with housekeeping; Takashi, the father, is a
regular salary man; the two kids, Mana and Ko, 16 and 14 years old respectively, are students. What
makes the family special is that they have agreed, following an initiative of Etsuko's, not to have any
secrets from and always be sincere with each other. Presumably, this results into constant awkward
situations.
However, as the script deepens on the characters, the facade of regularity shutters. Takashi, who has
not had intercourse with his wife for five years, retains two extramarital relationships, with whom he
often engages in sadomasochistic acts, one of which is his spouse's colleague. The other one, at
some point, is hired by his son to tutor him, thus her regular visit at their home. Ko, clearly has
sexual inclinations towards her and even asks her to have intercourse with him. Mana, who rarely
attends school, spends her time shopping at the department store, getting photographed for adult
magazines and even having sex with strangers. ... Show more content on Helpwriting.net ...
Ryuhei, the father is a successful senior member of a company with a more than adequate income.
At some point though, he is dismissed thus resulting in the egress of the family's issues. Shamed by
his dismissal, he keeps it a secret, by hypocritically continuing his everyday routine. What he
actually does though, is that he goes to the employment agency in the morning and spends the
remainder of the day roaming the streets, until his usual time of getting home arrives. At some point,
he meets Kurosu, another individual like him, who explains that there are a lot of men in their
situation and eves shows him some tricks to better conceal the very

Science Fair Question : Can You Accurately Measure Pi...
Question
My testable science fair question is: Can you accurately measure pi through scientific
experiments? I will conduct four separate scientific experiments throughout my project. My
hypotheses for
Hypothesis these experiments will vary greatly. My first experiment will be finding several large
circles in our environment, then pacing off the circumstance and the diameter. I will then calculate
pi by dividing the measured circumference by the measured diameter. My hypothesis for this
method is that it won 't be very accurate and only round pi to a whole number (3) because of the
inaccuracies of heel to toe measuring. In my second experiment, I will throw cylinders on the floor
marked with tape strips. This method uses the random location of the cylinders and calculates pi by
seeing how many cylinders cross over lines. This experiment is a variation of Buffon 's needle, one
of the oldest problems in the field of geometry. I expect this method to calculate pi between a range
of 3.1 to 3.2. because although this experiment is supported scientifically, it 's still a game of chance.
In the third experiment, I will program a computer to repeat my throwing cylinders experiment
repeatedly to get more accurate results. I believe that this experiment will be just as accurate as my
second experiment because I expect after 200 repeats the results won 't get much more accurate. My
fourth experiment will be using regular polygons (triangles, squares, and octagons)

Evidence And Assessment Of Student Learning
Evidence and Assessment of Student Learning How will you know whether students are making
progress toward your learning goals for each of the following types of performance: exceeds
expectations, meets expectations, and below expectations. (Be sure to include both content and
language, assessed either separately or together.) Students will be meeting expectations throughout
the lesson if they are correctly using the vocabulary with the aid of the word bank or the use of the
textbook while explaining their answers and reasoning. Students will be exceeding expectations if
they are interacting with their fellow classmates and are using the lesson's vocabulary without any
aid while explaining their mathematical reasoning. Students will be ... Show more content on
Helpwriting.net ...
Students exceed expectation if they are able to use several of the lesson's proper term when
explaining their answers. Students who are at the expanding level will meet expectation if they
recognize the proper terms of the lesson in questions or problems. Students will not meet
expectation if they do not incorporate the proper term in their responses, this indicate a degree of
confusion. Students will exceed expectation if they incorporate the term of this lesson in their
explanation when they are expressing verbally. Student feedback How will your provide students
with feedback? (Include all types of feedback – electronic, peer, teacher, answer key, etc.; include
not only correction but other) Teacher and peers will give the feedback throughout the lesson. The
peer–peer feedback will be allowed during the lesson; teacher's feedback will be given immediately.
How will you support students to meet their goals? (Most of the Lesson Plan goes here.) How will
you get the lesson started? What questions, texts, inquiry modeling and/or other techniques will you
use to engage students? ➢ Opening/Hook Students will be introduced to this new unit by watching
two brain pop videos; the first one regarding geometry, and the second one regarding angles. ➢
Prior Knowledge Activation Subsequent from watching the video the teacher will have a discussion
with students in order to access their schema

Contrasting Theories, Explanations And Policies
5.1. Intellectual Skills
These skills include critical, analytical, problem–solving and synthesising skills which include the
assimilation of new knowledge, development of critical analysis and the application of knowledge in
wider contexts (Transkills.admin.cam.ac.uk, 2017).
5.1.1. Assessing the merits of contrasting theories, explanations and policies
Health and safety policies are implemented in every part of the workplace in the United Kingdom.
The merits of following health and safety policies are that the risk of injury are greatly reduced
using good practice after identifying potential risks in the workplace. Although the risks of injury in
a GIS lab are low, sitting for long periods of time behind a computer screen has been proven to be
adverse to many aspects of health including cardiovascular disease, diabetes and muscular skeletal
issues (Dunstan et al., 2012). Regular breaks were encouraged and walks around the local area were
commonplace to stretch and exercise during the workday.
The main structure of the business utilises the Feed In Tariff (FIT) (Ofgem.gov.uk, 2017) for their
solar PV systems and the Renewable Heat Incentive (RHI) for their biomass heating systems
(Ofgem.gov.uk, 2017). These policies allow Eden ... Show more content on Helpwriting.net ...
Although Google Earth Pro included features which were useful, other applications or websites were
needed to determine other required data. An alternative application known as Marble (Nienhüser,
2017) has many of the same features as Google Earth Pro (Google Earth Pro, 2017) but it has the
option of ad–ons such as yearly sunshine, temperature and precipitation data which would need to
be searched elsewhere if Google Earth Pro were used. Although this software may have been a
useful tool to use, as the process outlined by Scott was tried and tested the method shown was

Language
Assignment 2: Figurative Language versus Literal Language
The lack of exposure to non literal forms of language makes it difficult to engage in productive
thinking. Having the capacity to understand figurative language increases our ability to
communicate with each other. By increasing our word bank we expand our knowledge base and
increase our thinking capacity. Below are a list of ten words with their meaning, definitions,
examples and appropriate circumstances in which to use them.
1. Describe the meaning and function of each term.
1. Idiom is a language, dialect or speaking style peculiar to a people.
2. Analogy is a similarity between two like subjects on which a comparison can be based.
3. Metaphor is a figure of speech in ... Show more content on Helpwriting.net ...
A good example is from Wordsworth's "I Wandered Lonely as a Cloud":
A host of golden daffodils; Beside the lake, beneath the trees, Fluttering and dancing in the breeze.
He doesn't say "many" or "a lot of" daffodils, he uses the word "host." That means a huge number of
daffodils. Later, he personifies the daffodils, and personification will be covered later on.
Another example is from "The Eagle" by Tennyson,
"He clasps the crag with crooked hands."
The hard consonant sounds add even more to the imagery here.
Simile
A simile compares two things using the words "like" and "as." Examples include: * busy as a bee *
clean as a whistle * brave as a lion * stand out like a sore thumb * as easy as shooting fish in a barrel
* as dry as a bone * as funny as a barrel of monkeys * they fought like cats and dogs * like watching
grass grow
Metaphor
When you use a metaphor, you make a statement that doesn't make sense literally, like "time is a
thief." It only makes sense when the similarities between the two things become apparent or
someone understands the connection.
Examples include: * the world is my oyster * you are a couch potato * time is money * he has a
heart of stone * America is a melting pot * you are my sunshine
Alliteration
Alliteration is the easiest of the examples of figurative language to spot. It is a repetition of the first
consonant sounds in several words. Some good examples

Tiling Research Paper
The topic of my essay is tiling. I chose this because I really enjoyed geometry when I took it, and I
wanted to gain a more in–depth knowledge of tessellation. Tiling as an art form also fascinates me,
so when I found M.C. Escher's work during my research for this topic I knew this one would be
right for me. Learning how drastically a curved image represented on a non–curved medium is
changed never seemed as real to me in sixth–grade geography as it did when learning that the image
below is comprised completely of regular pentagons.
In this exploration, I would like to learn more about how shapes tessellate. I will learn which shapes,
regular and irregular, tile on a Euclidean plane, and how tessellations work on a hyperbolic plane. I
will ... Show more content on Helpwriting.net ...
All quadrilaterals do, too, for the same reasons triangles do. Since triangles are essentially half of a
quadrilateral, the angles of a quadrilateral always add up to 360. This means that no matter what sort
you choose– rhombus, trapezoid, or just a run of the mill square– it will tessellate.
When it comes to the tessellation of pentagons, however, things are not so cut and dry. A regular
pentagon will not tessellate, as each of its angles is 108, and 108 x 5=540, meaning five pentagons
would not be able to join at one point. Four pentagons would also be too large to perfectly meet at a
vertex, as 108 x 4=432, and three pentagons would not be large enough, as 108 x 3=324; close, but
no cigar.
However, there are fifteen types of irregular pentagons that have been found to tile the plane. These
classes of pentagons were discovered at different times, and below is a table containing the formula
for some of them. ∠A–E denote an angle, and just simply A–E denotes a side. The formulas have
different equalities, with certain angles equalling each other so as to properly make angle amounts
that factor into the 360 needed for tiling, and with different sides being equal or proportional to each
other so as to facilitate the shape being right for

Solid Mensuration
CPR
(MATH13– B10)
Members: C06 Wrenbria Ngo C07 Julie – Ann Parañal C08 Dani Patalinghog C09 Marino Penuliar
C10 Michael Sadsad
CPR
(MATH13– B10)
Members: C06 Wrenbria Ngo C07 Julie – Ann Parañal C08 Dani Patalinghog C09 Marino Penuliar
C10 Michael Sadsad
Prof. Charity Hope Gayatin
Prof. Charity Hope Gayatin
Homework 1.1
#15. Find the sides of each of the two polygons if the total number of sides of the polygons is 13,
and the sum of the number of diagonals of the polygons is 25.
Assume: ... Show more content on Helpwriting.net ...
Find the lengths of the three sides if the area of the triangle is 576cm2 .
Soltion : c/17=9/10=b/9
A= ss–as–b(s–c) s= a+b+c2 s= 10a+9a+17a20 b= 9(40)10 s= 95a b=36
576= 18a250; c= 17(40)10
9a2= 14400 c= 68 a2= 1600 a=40 Answer: 40cm, 36cm, 68cm
#15. Given triangle ABC whose sides are AB=15in., AC=25 in., and BC= 30in. From a point D on
side AB, a line DE is drawn to a point E on side AC such that angle ADE is equal to angle ABC. If
the perimeter of triangle ADE is 28 in., find the lengths of the line segments BD and CE. Given: ? A
D 30in 15in
B E C
Required: BD =? ; CE =?
Solution: For BD P ADEP ABC = ADAB P ADE=28in Answer: The length of segments BD and CE
is 9in and 10in AD = 15in( 28in)70in P ABC=70in AD = 6in P ADEP ABC= AEAC BD = 9in AE =
25in (28in)70in AE = 10 in

#17. What is the sum of the areas of the two triangles formed in number 16? Given: 3

The Modeling Technique That You Were Assigned For Your Own...
Final Exam
1. Describe the modeling technique that you were assigned in your own words.
The modeling technique that I was assigned Extended Backus Naur Form (EBNF). In software
engineering, expanded Backus–Naur frame (EBNF) is a group of metasyntax documentations, any
of which can be utilized to express a setting free sentence structure. EBNF is utilized to make a
formal depiction of a formal language which can be a PC programming language. They are
augmentations of the fundamental Backus–Naur frame (BNF) metasyntax documentation. EBNF is
a code that communicates the sentence structure of a formal language. An EBNF comprises of
terminal images and non–terminal creation rules which are the limitations administering how
terminal images can be joined into a legitimate arrangement. Cases of terminal images incorporate
alphanumeric characters, accentuation marks, and whitespace characters. BNF is the first, most
straightforward, for the most part utilized as a part of scholarly papers of hypothetical setting, for
conveying to people, instead of being utilized as a part of compiler or parser. EBNF implies
Extended BNF. There 's not one single EBNF, but rather numerous.
Reusing existing syntax learning dwelling in gauges, speciﬁcations and manuals for programming
dialects, confronts a few difficulties. A standout amongst the most signiﬁcant of them is the differing
qualities of syntactic documentations: without loss of all inclusive statement, we can express that
each and

Comparison Of Soccer Balls
Background
Although a soccer ball may look like a sphere is not a real sphere. When you look closely at the
actual ball you will see that it is polyhedron made of different shapes. A polyhedron is a three
dimensional object composed of flat polygon sides typically connected at the edges (Wolfram 1999).
The most common shape of modern soccer balls is called a truncated icosahedron. Icosahedron is a
shape made up of 20 triangles. To create a more round object the icosahedron is truncated. Here,
truncated applies to cutting off the edges which makes the icosahedron look more round
("Icosahedron," 2014). The truncated icosahedron has 32 total faces including 12 regular pentagons
and 20 regular hexagons with 90 edges and 60 vertices ("Truncated ... Show more content on
Helpwriting.net ...
Check that the lengths of all the lines is 5 cm and that the angles are all 108º. Carefully cut out the
regular pentagon.
12) Now repeat the previous steps 11 more times to create the 12 regular pentagons. Always check
that the compass and lines are 5 cm. B) Create a regular hexagon
To draw a regular hexagon we followed the instructions provided online (Hartley, 2013).
1) Set the radius of the compass to 5 cm. It is critical that it stays at this length. Throughout the
process check that the length stays at 5 cm.
2) Draw a line 5 cm long and label the points A and B (.
3) Draw 2 small arcs centered at point A and point B towards the center of the line AB. Label the
point where the two arcs cross as point 1. This is the center of the regular hexagon and the vertices
of the hexagon will be on a circle centered at this point.
4) Initially draw a full circle centered at point 1. In future steps you only need to draw arcs near
where the vertices occur.
5) Using point B as the center, draw a short arc that crosses the circle. Label the point of intersection
point

Archimedes Of Syracuse And The Current State Of Computing
As long as the field of mathematics has existed, people have been searching for shortcuts to
eliminate the monotony and difficulty of calculating figures accurately. As a result, human beings
began to develop new technologies to simplify this process. In ancient history, the abacus was a
useful device in calculating simple numbers requiring addition and subtraction. In the seventeenth
century, the first mechanical calculators were able to perform multiplication and division through
repetitions of addition and subtraction. Calculators were then programmed in order to multiply and
divide automatically. From these early devices emerged the first computers and calculators, which
were originally intended to calculate figures. Now, modern computers are expected to perform a
variety of functions, outside of calculations, quickly and effectively. However, back in the time of
abacuses, an ancient Greek mathematician was discovering the formulas and primitive mechanical
devices that have evolved into the current state of computing technology. Archimedes of Syracuse
was born in 287 BC in Syracuse, Sicily. Not much is known about his early life or his parents, but it
is believed he studied alongside his mathematical contemporaries Conon of Samos and Eratosthenes
of Cyrene in Egypt. In 212 BC, he was killed by the invading Roman soldiers when he refused to
comply while solving geometrical problems in the dirt. Although he died prematurely, Archimedes
made a significant contribution to

Regular Polygon
Polygon
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For other uses, see Polygon (disambiguation).
Some polygons of different kinds
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 ... Show more content on
Helpwriting.net ...
The polygon is also equilateral. * Tangential: all sides are tangent to an inscribed circle. * Regular:
A polygon is regular if it is both cyclic and equilateral. A non–convex regular polygon is called a
regular star polygon.
Miscellaneous
* Rectilinear: a polygon whose sides meet at right angles, i.e., all its interior angles are 90 or 270
degrees. * Monotone with respect to a given line L, if every line orthogonal to L intersects the
polygon not more than twice.
Properties
Euclidean geometry is assumed throughout.
Angles
Any polygon, regular or irregular, self–intersecting or simple, has as many corners as it has sides.
Each corner has several angles. The two most important ones are: * Interior angle – The sum of the
interior angles of a simple n–gon is (n − 2)π radians or (n − 2)180 degrees. This is because any
simple n–gon can be considered to be made up of (n − 2) triangles, each of which has an angle sum
of π radians or 180 degrees. The measure of any interior angle of a convex regular n–gon is radians
or degrees. The interior angles of regular star polygons were first studied by Poinsot, in the same
paper in which he describes the four regular star polyhedra. * Exterior angle – Tracing around a
convex n–gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way
around the polygon makes one full turn, so the sum of the exterior angles must be

Solid Mensuration
CPR (MATH13– B10) Members: C06 Wrenbria Ngo C07 Julie – Ann Parañal C08 Dani Patalinghog
C09 Marino Penuliar C10 Michael Sadsad CPR (MATH13– B10) Members: C06 Wrenbria Ngo
C07 Julie – Ann Parañal C08 Dani Patalinghog C09 Marino Penuliar C10 Michael Sadsad Prof.
Charity Hope Gayatin Prof. Charity Hope Gayatin Homework 1.1 #15. Find the sides of each of the
two polygons if the total number of sides of the polygons is 13, and the sum of the number of
diagonals of the polygons is 25. Assume: 8 and 5 D1= n2(n–3) ... Show more content on
Helpwriting.net ...
d1=13m Given: Answer: d2=24m d2= ? Required: d2= ? Solution: A = 12 d1d2 2(156) = 13 d2 156
m2=12(13)(d2) d2=24m #20. A piece of wire is shaped to enclose an equilateral triangle in which
the area is 163cm2 . It is then reshaped to enclose a rectangle whose length is 9cm. Find the area of
the rectangle. A1= ? A1= ? Given: A1=163cm2 9cm Required: A =? Solution: A=b2 sin θ 323= x23
A1A2=X1X22 163 = X2 sin 60° x2=32 163A2= 4292 A = 70.15cm2 163= X232 x=42
A2=81(163)32 2(163) = x23 #23. A quadrilateral contains two equal sides measuring 12cm each and
an included right angle. If the measure of the third side is 8cm and the angle opposite the right angle
is 120°, find the measure of the fourth side and the area of quadrilateral. Given: x=? A= (30.55–
12)230.55–830.55–29.10–(29.10)(12)28cos2 12cm A= 153.46cm2 8cm 12cm Required: Fourth side
= x =? Solution: For d1 For θ For

Khepera Mobile Robots For Search And Rescue Operation
Khepera Mobile Robots for Search and Rescue Operation
Keywords: Mobile Robot, Explore Unknown Map with Obstacles, Gas Detection, V– rep, Khepera
III, Albers Algorithm.
Abstract: Nowadays, using robots instead of humans in risk operations is an interesting point in the
field of robotics. In this paper we introduce an integrated approach for search and rescue operation
of detecting gas sources in large areas; e.g. houses, factories, and labs. This aims at saving people's
lives at bottlenecks resulting from gas leak and ignition of fires. Experiment is performed using
master Khepera robot supplied with gas sensor circuit to search map, in addition to two slave
Khepera supplied with gripper to take the best path to clinch the victim.
Moreover, this paper proposes an improvement of Albers exploration algorithm to reduce the time
required to explore unknown map with different polygon obstacles. The proposed approach aims to
minimizing the overall exploration time, making it possible to localize fire sources in an efficient
way, as demonstrated in
Vrep simulation as well as real world experiments with Khepera III robot. A comparison among
different algorithms has showed the effectiveness of the proposed one where the percentage of
performance speedup is about 30% to 57% depending on size of the map and number of obstacles.
1 INTRODUCTION
Autonomous Robots are meant to perform searching operations inside buildings, mines and caves ,
which are considered extremely serious. These

William Jones and Pi Essay
William Jones is a famous mathematician the created, and was the first to use, pi. William was born
on a farm in Anglesey, then later moved to Llanbabo on Anglesey, then moved again after the death
of William's father. He attended a charity school at Llanfechell. There his mathematical talents were
spotted by the local landowner who arranged for him to be given a job in London. His job was in a
merchant's counting house. This job had Jones serving at sea on a voyage to the West Indies. He
taught mathematics and navigation on board ships between 1695 and 1702. He was serving on a
navy vessel which. Navigation was a topic which greatly interested Jones and his first published
work was "A New Compendium of the Whole Art of Navigation" It was ... Show more content on
Helpwriting.net ...
Jones was also friends with the Parker family, and had done business with them at their castle at
Shirburn. It was greatly helped that he had a good friendship with Philip Yorke and George Parker,
especially after he lost all his money when the bank that he kept his money had been run down. His
two former students were later men of great influence and were able to help Jones with obtaining
many different types of jobs with various positions, and he was paid well. The first man to really
make an impact in the calculation of pi was the Greek, Archimedes of Syracuse. Where two people
by then name of Antiphon and Bryson left off with their inscribed and circumscribed polygons,
Archimedes took up the challenge. However, he used a slightly different method than they used.
Archimedes focused on the polygons' perimeters as opposed to their areas, so that he approximated
the circle's circumference instead of the area. He started with an inscribed and a circumscribed
hexagon then doubled the sides four times to finish with two 96–sided polygons. Archimedes
approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular
polygons: the polygon inscribed within the circle and the polygon within which the circle was
circumscribed. Since the actual area of the circle lies between the areas of the inscribed and
circumscribed polygons, the areas of the polygons gave upper and lower

Different Types Of Polygons By Nancy Tobin Capture The...
This book introduces different types of polygons through objects we find around us everyday. Shape
Up! uses cheese slices, pretzel sticks, a slice of bread, a pencil, and more to introduce various
polygons, flat shapes with varying numbers of straight sides. Children can learn simple geometry
with this playful but informative math concept book that takes the fear out of math and puts the fun
back in. There 's a wisecracking cartoon kid who accompanies readers throughout and makes
silliness and math seem to go together like pretzels and salt. It 's a tremendous learning package that
will be as pleasing to math teachers as it is to the kids who will probably discover this book on their
own. The illustrations by Nancy Tobin capture the reader's eye with her colorful and aesthetically
appealing pictures.
I believe this book will be very affective because it gets children involved with geometry first hand
and lets them find shapes, not just in food, but where ever they go. Using healthy foods to learn
geometry will also develop an interest in trying new things as well. Learning shapes through an
interactive activity will help them develop a relationship with finding shapes and problem solving
without even realizing it! Developing a fun activity with a lesson engages students and captures
their full attention.
1st Day– Read the book aloud and distinguishes the shapes that were discussed in the book.
2nd Day– Students will pair up and develop a scavenger hunt list of the shapes

Celta
Assignment 2 – Focus on the learner LEARNER'S PROFILE Sonia Meirelles is a 27–year old
Brazilian student, in the Intermediate English class, who has been studying English for 13 months
(since August, 2011). Sonia has a degree in Biology and used to work as a biology teacher in a
regular school, however she is currently unemployed. She then decided to start taking English
lessons, so she could have more chances when trying to find a job. She enjoys the English classes as
she believes it is not only about learning a language but also about learning a different culture. After
observing the student and applying questionnaires, I could determine Sonia has high visual and
interpersonal intelligence. During classes, Sonia ... Show more content on Helpwriting.net ...
Whenever she receives an email from any of the teachers she usually replies it and asks questions
related to what she could not understand (e.g.: Teacher, what does Rgds mean?) Although she is very
motivated, during the lessons she prefers to be nominated for activities. Whenever talking to the
teacher, she often asks for feedback and after making a mistake, she tries to say it again until she
feels she understood it. Specific assessment of learner's language – Strengths, weaknesses and
Analysis of Students errors Grammar: Sonia is very confident when using both present simple
(using the auxiliary verb DO in interrogative sentences and DON'T in negative sentences, placing
adverbs and conjugating the verbs with He, she and It correctly) and past simple tenses (using the
auxiliary verb DID in interrogative sentences and DIDN't in negative sentences –and the verbs in the
base form – and being aware of regular and irregular verbs). e.g.: When do you have holidays? What
does regard mean? I usually go to the beach. Jennifer... when she has holidays, she usually visits her
parents. However, she still omitts articles in her speech (e.g: I don't have boss. / I'm Biologist), and
uses " have" instead of "There is" (e.g.: Have a Science Museum in the city.) , both mistakes are
made due to L1 interference. In Portuguese, as in the

Regular Expression Of A Web App
Regular Expression To FSM– Web App
3.1 Algorithm
3.1.1 Regular Expression Parsing
I built my own parser which will suit the recursive nature of my regular expression to Finite State
Machine Algorithm. To parse the regular expression, first, the expression is converted from infix to
prefix form. To convert from prefix to infix form, the string is read from the back and on meeting an
operator, it is pushed to an operator stack if top of operator stack has lower precedence than the read
operator else pop it to append to front of output string. If an operand is met, simply append it to
beginning of the output string.
After converting the expression to prefix, second step is to construct expression tree where each
inner node including root node has operator as its value and the leaf nodes are operands. The entire
infix string is read from the end. If the character is an operand it is pushed to the operand stack else
if an operator is read, operands are popped from the operand stack and assigned as left and right
child of node with having the operator as the node value and pushed again into the stack. Since the
backward construction algorithm is recursive in nature, such a syntax tree makes it more suitable for
the algorithm where expression will be taken as input.
3.1.2 Regular Expression To FSM– Backward Construction
After converting input expression to prefix and constructing expression tree, next steps is to
implement the Backward Construction algorithm for

Unit 64 Case Study
There is a standard 8–by–8 checkerboard made up of 64 squares. These squares can be made of
other squares within the checkerboard. How many squares of various sizes are there? Also, how
many squares can be in any–sized checkerboard (e.g. 9–by–9 checkerboard). There are many steps
to figure out the problem. First, I figured out how many squares there were in the checkerboard
squares which was 64.(Green should explain how you got 64.) To find 64 I simply looked at the top
of the square and added the squares up to get 8, and I did the same exact thing to the side of the
square and got 8 then I multiplied them to get 64 squares.. Then, I counted the squares along the side
and the top and bottom to find one size.(Did you use paperrater.com? You

Evaluation Of The Elementary School
This lesson was taught November 12, 2015 at Rosa Taylor Elementary School in Miss. Baggott's
fifth grade class. That week the students went to see Miller Middle School November 10th and were
out of school for Veteran's Day on November 11th. This week in math the students were learning
about polygons and their characteristics. Triangles have a first and last name describing the sides
and types of angles. There are different types of quadrilaterals and other polygons with five for more
signs. For the summative assessment given to the students by the cooperative teacher, the students
had to identify the shapes by their name, number of sides, number of angles, and (for triangles) their
"first and last names." My lesson was titled "review stations" where the class was broken up into
groups of three or four. There were six stations that included: polygon collage, the shape game,
foldable focus group, triangle board game, tangrams, and computer games–provided by the teacher.
Each station had a red folder that consisted of everything the student needed for that particular
station. The folder had written directions either on the sheet of paper or on the folder itself. For the
mini–lesson, we collectively wrote a chart in their interactive journals for notes. The chart consisted
of five columns, including: prefix, name, number of sides, number of angles, and shape. Their
assessment for Friday asked questions about number of sides and number of angles a particular
polygon had. Each

Taking a Look at Tessellations
Most people recognize the artistry of walls in ancient palaces, of mosaic pictures, and even of
honeycombs. Likewise, the artistry and intricacies of M.C. Escher's drawings astound most people.
When we look at these objects and artwork we recognize the shapes within them; we see squares,
hexagons and other shapes without giving them much thought. We might not even know that these
patterns of shapes have a name, and we certainly do not think of mathematics when we see them.
But, in fact, these patterns – or tessellations – are part of the field of geometry.
When a space is covered with a pattern of flat shapes with no overlaps or gaps it is known as a
tessellation or a tilling. Tessellations have been around for many centuries and in many different
cultures and are still prevalent today. In Latin the word tesserae means small stone cube they were
used to make up tessellata– the mosaic pictures forming floors and tiling in Roman buildings.
Making a repeating pattern with a regular polygon creates regular tessellations. Triangles, squares
and hexagons are the only three shapes that can make a regular tessellation. In order for a
tessellation to be regular the pattern is identical at each vertex. A tessellation created with two or
more regular polygons is known as a semi–regular tessellation. Just like in a regular tessellation in a
semi–regular tessellation the pattern at each vertex is the same. The third type of tessellation is a
demi regular tessellation however

How Did Gauss Contribute To Math
Carl Friedrich Gauss, from his youth, was destined to be a great mathematician. By the time Gauss
turned three, he had already taught himself to read and write. Additionally, Gauss often told
acquaintances and friends that before he learned to speak, he learned to make mental calculations.
Throughout his lifetime, Gauss made discoveries which would benefit many fields within
mathematics. Gauss contributed greatly to the fields of arithmetic, statistics, geometry, algebra, and
astronomy. The origin of this famed mathematician began in Brunswick, Germany on April 30th in
the year of 1777. Gauss had humble beginnings, and lived in a small house with his parents. Gauss's
parents were both peasant–laborers, which was expressed by the local dialect they spoke (West 15–
16). Gauss's father, Gebhard Dietrich Gauss, worked in various areas; he worked as a "brick layer,
gardener, canal–tender, street butcher, and accountant for funeral society" (West 16). Gauss's
mother, when she was Dorothea Benze, worked for seven years as a maid before she married
Gebhard. Gauss was Dorothea's only child, but he had a half–brother, Johann George Heinrich, who
was the son of Gauss's father from an earlier marriage (West 16). Gauss was an intelligent child, and
at the age of three, he corrected an ... Show more content on Helpwriting.net ...
The idea is despite how meticulously an item is measured in order to receive the correct
measurement, one is likely to observe different lengths each time something is measured. If one
takes a measurement x amount of times, the lengths will likely be similar in value, but they will not
all be equal. Therefore, the "true" value is unknown. Gauss developed a formula in which all the
desired measurements are taken and the optimal "true" value can be calculated. Today, statisticians
use the Method of Least Squares when they need to measure errors; the formula allows them to
calculate answers (West

Alphabet of Lines: Geometric Construction
In antiquity, geometric constructions of figures and lengths were restricted to the use of only a
straightedge and compass (or in Plato's case, a compass only; a technique now called a Mascheroni
construction). Although the term "ruler" is sometimes used instead of "straightedge," the Greek
prescription prohibited markings that could be used to make measurements. Furthermore, the
"compass" could not even be used to mark off distances by setting it and then "walking" it along, so
the compass had to be considered to automatically collapse when not in the process of drawing a
circle. Because of the prominent place Greek geometric constructions held in Euclid's Elements,
these constructions are sometimes also known as Euclidean ... Show more content on
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It is possible to construct rational numbers and Euclidean numbers using a straightedge and compass
construction. In general, the term for a number that can be constructed using a compass and
straightedge is a constructible number. Some irrational numbers, but no transcendental numbers, can
be constructed. It turns out that all constructions possible with a compass and straightedge can be
done with a compass alone, as long as a line is considered constructed when its two endpoints are
located. The reverse is also true, since Jacob Steiner showed that all constructions possible with
straightedge and compass can be done using only a straightedge, as long as a fixed circle and its
center (or two intersecting circles without their centers, or three nonintersecting circles) have been
drawn beforehand. Such a construction is known as a Steiner construction. Geometrography is a
quantitative measure of the simplicity of a geometric construction. It reduces geometric
constructions to five types of operations, and seeks to reduce the total number of operations (called
the "simplicity") needed to effect a geometric construction. Dixon (1991, pp. 34–51) gives
approximate constructions for some figures (the heptagon and nonagon) and lengths (pi) which
cannot be rigorously constructed. Ramanujan (1913–1914) and Olds (1963) give geometric
constructions for . Gardner (1966, pp. 92–93) gives a geometric construction for Kochanski's
approximate construction for

Formal Instruction Of Grammar In School
Topic: Argue whether formal instruction of grammar in school is helpful and enlightening for
someone who wants to become a good writer. Today, formal grammar is essential in our academic
careers, requiring us to write concise and professional assignments in order to exercise creativity
and leave impressions on instructors. Although, the effectiveness of the explicit instruction of formal
grammar is still under speculation; even individuals from academic and research backgrounds are
skeptic about whether explicit formal grammar instruction is effective at all. Although, it sounds
only nonsensical to even consider abolishing this instructional method, for why should learning
grammar be based on one's own knowledge of it? Why should the student's initial experience with
grammar be deemed as more important than proper instruction? The explicit instruction of formal
grammar in society is necessary, for the consequences of removing this practice can prevent core
grammar from being taught in the first place; also, even if other methods would prove more
effective, applying this would be too much of a risk, and ultimately keep students from discovering
talents, and make it more difficult for foreign students to learn English as a second language. The
debate about grammar instruction does not entail whether grammar should be taught, but rather how
it should be taught. Formal grammar instruction, also known as traditional grammar instruction, is a
teaching method where students

Importance Of Learning A Foreign Language Essay
Learning a foreign language involves developing new skills and going through different stages. The
four skills you need to develop are listening, speaking, reading, and writing. You need to be able to
understand when someone speaks the language you are learning. In addition to that, you need to be
able to express yourself in that language. Most of the time, the written language is more complex
than the spoken language. You want to be able to understand a text you read. Moreover, you want to
be able to express your ideas in writing, with the right words and correct grammar. Acquiring these
four skills requires various study patterns.
Building Vocabulary
First of all, you need to build vocabulary. Building vocabulary involves learning the meaning of the
words, their spelling, and pronunciation. Unlike in English, each word has a gender in many
languages like French, Spanish, and German. You need to memorize the gender of a word in order to
use it correctly. You also need to learn the conjugations of verbs and adjectives. That is where you
cross the border between the vocabulary and grammar.
Studying Grammar
In order to learn a foreign language, you need to study its grammar, the structural rules of a
language. Most of the time, grammar is a complicated subject to understand, to learn, and to use
correctly. To make things even more complicated, you will need to memorize the exceptions to the
grammar rules you learn.
Reading
If you have sufficient vocabulary and a good

The Thought Fox Figurative Language
"Figurative language is by no means just ornamental, but an important part of guiding cognitive
construal."(Dancygier, 2014 p196) Attention to figurative language when analysing a text is critical
as making judgement on a text can be undetermined when taking in the literal sense, perhaps the text
when taken literally means nothing or has a shallow and uninteresting meaning but when figurative
language is taken into consideration a deeper new meaning can be taken from the text. The aim of
figurative language first and foremost is to force the reader to imagine what it is the writer is trying
to express and to explain the concept in an interesting way. This language is not supposed to be
taken literally and through comparisons to another concept, a deeper undertone is revealed to the
reader. Two of the poems in which make use of figurative language are "Mary's Song" by Sylvia
Plath and Ted Hughes "The Thought Fox" In Plath's "Mary Song" she uses metaphors to portray a
deeper message to the reader. There are three main metaphors all overlapping within Plath's poetry.
This is the goal of figurative language within the text. It is not supposed to be taken for its ... Show
more content on Helpwriting.net ...
The poem can not be taken for its literal mean but instead, it compels the reader to delve deeper into
the immediate. The main connotation is the Fox. Foxes are known worldwide as sly, they do not
attack suddenly but instead, they plan and sneak up slowly on their prey. Hughes is perhaps, using
this image to get the reader to imagine this is how inspiration comes to a writer. As the fox sneaks
into the room so too will inspiration to write.The fox inches in "that now And again now, and now,
and now." The fox moves slowly, carefully, incrementally in a series of steps. It inches in carefully.
Hughes exploits this aspect of a foxes characteristics to establish this deep

The Genius of M.C. Escher Essays
The Genius of M.C. Escher
Mathematics is the central ingredient in many artworks. While notions of infinity and parallel lines
brought "perspective" to the artistic realm in creating realistic representations of depth and
dimension, mathematics has influenced art in a more definite way – by actually becoming art. The
introduction of fractal geometry and tessellations as creative works spawned the creation of new and
innovative genres of art, which can be exemplified through the works of M.C Escher. Escher's
pieces are among the most recognized works of art today. While visually stimulating and deeply
meaningful, his art reflects many ideas of mathematics through geometry, symmetry, and patterns.
Maurits Cornelius ... Show more content on Helpwriting.net ...
He once said, "Although I am absolute innocent of training or knowledge in the exact sciences, I
often seen to have more in common with mathematicians than my fellow artist" (Totally Tessellated:
Escher Biography & Timeline, 1998).
Finally, in 1930, Escher received widespread acclaim for his lithograph entitled Castrovolva. He
continued to incorporate geometry and patterns in his pieces, and found that his work began to be
displayed in science museums rather than art galleries. From 1951–1954 Escher completed some
400 works, by this time a prominent figure in the world of art, the majority of which included such
mathematical principles as polyhedra, infinity, knots, and tessellations.
A polygon is a closed figure bounded by three or more straight line segments. A polyhedron is a
geometric entity composed of polygons connected at their edges to enclose space. Among the most
popular piece that exemplifies the use of polyhedra is entitled Gravity (Figure 1:
http://library.advanced.org/16661/escher/trends/1/html) In this Escher work, dinosaur–like creatures
emerge from a series of pyramids fit together like a star.
Along with polyhedrons, Escher also incorporated the idea of infinity into numerous pieces.
Fascinated by the concept of bounding infinity, that is representing infinity in an enclosed plane,
Escher attempted to demonstrate this in his work. He once

Advantages And Disadvantages Of Handcrafted Rules And...
NER has reached maturity for many languages (e.g. English and French) but Arabic NER still attract
researchers for more enhancements for the task.
In literature, NER approaches have been classified into two main streams: handcrafted rules and
machine–learning (ML) based. ML itself is classified into supervised, semi supervised and
unsupervised ML.
The approaches that have been encountered for Arabic NER are reviewed below.
2.1 Handcrafted Rule Based NER
The Rule based approach, also known as knowledge engineering approach, relies on grammar rules
coming from the linguistic knowledge and heuristic rules to identify names, such rules are
implemented as regular expressions or finite state transduction based grammar for pattern matching.
In ... Show more content on Helpwriting.net ...
Zaghouani et al. (2010) adapted the Europ Media Monitor (EMM) platform by introducing three
components, i.e. preprocessing, lookup full names and local grammar to recognize unknown names,
the main difference between this approach and others that they used language–independent rules
along with the language dependent ones. The evaluation was made over a corpus compiled and
annotated from two newspapers ( Assabah and Alanwar newspapers14) . The overall F–measure
achieved was 74.95%.
RENAR is the same system but had been renamed after two years of introducing the initial
approach,it was evaluated over the ANERcorp dataset. The aim of this evaluation is to compare the
performance of this method with other machine learning based studies (such as Benajiba and Rosso
(2007)). RENAR outperforms other systems for LOC NEs, scoring an F–measure of

What Students Should Know?
What Students Should Know According to the learning progressions report, coming into third grade,
students know how to analyze, compare, and classify two–dimensional shapes by their properties.
When students do this, they relate and combine these classifications that they have made (The
Common Core Standards Writing Team, p. 13). Because the students have built a firm foundation of
several shape categories, these categories can be the "raw material" for thinking about the
relationships between classes. Students have learned that they can form larger, superordinate,
categories, such as the class of all shapes with four sides, or quadrilaterals, and recognize that it
includes other categories, such as squares, rectangles, rhombuses, ... Show more content on
Helpwriting.net ...
15). In fourth grade, students learn to represent angles that occur in various contexts as two rays,
explicitly including the reference line, which is a horizontal or vertical line when considering slope
or a "line of sight" in turn contexts. They understand the size of the angle as a rotation of a ray on
the reference line to a line depicting slope or as the "line of sight" in computer environments (The
Common Core Standards Writing Team, p. 15). In fourth grade students also might explore line
segments, lengths, perpendicularity, and parallelism on different types of grids, such as rectangular
and triangular (isometric) grids. Finally, students also learn how to reason about these above
mentioned concepts (The Common Core Standards Writing Team, p. 16). According to the learning
progressions report, coming into fifth grade, students should know how to represent angles that
occur in various contexts. They should also already understand the size of the angle as a rotation of a
ray. Finally, they should have developed explicit awareness of and vocabulary for many concepts
they have been developing, including points, lines, line segments, rays, angles (right, acute, obtuse),
and perpendicular and parallel lines (The Common Core Standards Writing Team, p. 17). In fifth
grade, students develop competencies in shape composition and decomposition, and especially the
special case of spatial structuring of rectangular

Discussion Questions On Formal Language
ENG1502
59579064
Assignment 3 [837921]
Question 1
1.1) One expects to find formal language in any place of work and authority such as businesses,
court, parliament, newspapers etc. Any profession which is associated with a high level of education
and order. Formal language is also known to reflect on the speaker's social class. It shows what type
of school you attended, how you were raised and your heritage. Informal language is more
commonly heard in a social environment. For example when you are at a braai or attending a family
function. The people present usually know each other so the atmosphere is more relaxed and there is
not an essential requirement for formality. Informal language is usually used on social media
platforms or instant messaging.
1.2) If you were to address your colleagues, you would need to do it in a formal manner. You should
gather as many facts as possible and also listing as many possible solutions as you can find. Facts
and solutions help in strengthening your argument. This information would be presented to the
board to try and convince them to agree with your point of view. Whereas in a social environment
with your friends informal language could be used. You could discuss teenage pregnancy and give
your personal thoughts and opinions on the topic. Your thought would not need to be supported by
facts as it could be opinion based.
1.3) Whenever you are addressing someone professional, you are required and expected to use
formal language.

Questions and Answers Regarding Equilateral Triangles
Since the centroid of a triangle lies along each median, 2/3 of the distance from the vertex to the
modpoint of the opposite side, we have:
■(t=2/3∙√3/2∙c=c/√3;@u=2/3∙√3/2∙b=b/√3.)
And in this case (1) becomes (2):
〖3s〗^2=b^2+c^2–2∙b∙c∙cos(A+60°).
Expanding the cosine of the sum, we have (3): cos(A+60°)=cos(A)/2–(√3∙sin(A))/2. Substituting (3)
into (2) yields to (4):
〖3s〗^2=b^2+c^2–b∙c∙cos(A)+√3∙b∙c∙sin(A).
Now we will apply the Law of Cosines to ∆ABC (5): a^2=b^2+c^2–2∙b∙c∙cos(A). And recall (6):
2∙S_∆ABC=b∙c∙sin(A).
Substituting (5) and (6) into (4) gives us (7):
3s^2=(a^2+b^2+c^2)/2+2∙√3∙S_∆ABC
As (7) is symmetrical in a,b and c, it follows that the triangle connecting the three centroids is
equilateral:
IH=GH=GI=s=√((a^2+b^2+c^2)/2+2∙√3∙S_∆ABC )/√3=√((a^2+b^2+c^2)/6+(2∙S_∆ABC)/√3).
The Napoleon's triangle is equilateral. Proved.
Answer: Proved.
Rotating by the point:
Fix the point I as the center of rotation and rotate the entire figure by 120°, and superimpose the
rotated copy on the original figure.
Uder rotation the ∆CAF maps to itself (C maps to A, A maps to F, F maps to C and I maps to itself).
Points BB, DD, EE, GG and HH are the images of the points B, D, E, G and H, respectively.
Because of this he triangle ∆A.EE.BB= ∆BCE.
Connect D to EE and G to HH. By the rigidity of the rotation, ∆GHI=∆GG.HH.I. In particular
GH=GG.HH.
Consider the six triangles that coverge on point A. Triangles ∆ABD, ∆ACF and ∆A.EE.BB are
equilateral.
The angles of triangles sum to 180° and

Application Of Computational Geometry On View
Application of Computational Geometry on View
Planning
Name: Pravakar Roy
Student ID:4927267
Graduate Student
Department of Computer Science
University of Minnesota, Twin Cities
April 27, 2015
Abstract
View planning is a crucial part of building vision system for autonomous robots or critical coverage
problems. In computational geometry the problem of covering/guarding a region is known as the art
gallery theorem. The version where static guards are re– placed by mobile guards is known as the
watchman route theorem. Both these classic theorem adopts the straight line notion of visibility
which is impractical for real world sensors. In this work we review the existing techniques for
solving the classical ver– sion of these problems and also discuss the modi cations needed to
handle real world sensors. We also discuss some recent methods that takes physical visibility
constraints into account.
1 Introduction
Producing autonomous robots that can operate without any sort of human intervention is one of the
ultimate goals in robotics [42]. Such robots should be capable of exploring their environment,
interpret the surroundings and act accordingly. Vision is clearly one of the most critical capability
such a robot must have to act autonomously. When exploring an unknown environment, a robot is
rst required to extract information about the surroundings.
Following this exploration, it should be able to carry out speci c task such as nding potentially
hazardous materials in the

Imagery In 'The Man I Killed And Ambush'
In the short stories "The Man I Killed" and "Ambush," one of Tim O'Brien's purposes is to describe
how society wrongfully portrays soldiers gain a sense of pride and victory when they take lives of
other human beings instead of the guilt–driven battle they have to deal with for the rest of their
lives. O'Brien tries to disprove this theory and instead show they are actually stuck with this tragedy
for the rest of their lives as they lose their innocence and sense of humanity. O'Brien shows this
through the use of imagery to portray and help develop this concept/theme. O'Brien describes
through vivid imagery and details in, "The Man I Killed," the dead man's eye, "His one eye shut, his
other eye was a star–shaped hole" (118). Here O'Brien uses this imagery to symbolize the star shape
of the dead man's eye as a sign of hope as a shooting star, yet, he ties this beautiful image with
death, to show that his hope/future has betrayed him. O'Brien purposefully places this star–shaped
wound on the soldier's eye, for it is with the eyes that both the dead man and O'Brien gaze upon the
stars in the sky. As if he was gazing more upon the stars, upon his future, which in this case his
future comes to an end with O'Brien's fatal doing. O'Brien's innocence has left him as he has become
in a sense "dirty" after taking this man's life. So, in this case, O'Brien has not taken this killing with
pride and victory, but with sadness and guilt. This goes back to his purpose to show that soldiers

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