Essay On Assessing Maths Assignment

Essay on Assessing Maths Assignment
Access Diploma in Adult Learning
Assessing Maths Assignment
Landscaping a Garden
I've been asked me to cost his landscaping project for him using the prices quoted by a local
supplier, and to give him a full breakdown of the calculations required and how I arrived at the final
cost.
Plan
I plan to do this firstly by breaking up the garden plan into 5 sections.
1. Decking and border.
2. Flowerbed and crazy paving
3. Fish pond, safety fence, bridge and rail
4. Perimeter fence
5. Grass.
Decking and Border
The decking area consists of two right angle triangle. The two edges around the decking are equal in
length. I need to work out the length of the edges and the area of the decking, how much materials
required and cost.
In ... Show more content on Helpwriting.net ...
the flowerbed using the equation, this will give me the service area of the flower bed
Area +
I then will work out the area of the larger semi–circle marked D using the above equation and
subtract the area of the smaller circle (flowerbed). This will give me the area of the crazy paving
I will then work out how much crazy paving required / m².
I will then work out the cost of the paving @ £3.50 + VAT per m²
I will work out how many bulbs required for the area in m² for the flower bed, and the cost at £6.40
per m².
Fish pond, safety fence, bridge and rail
The fish pond has a depth of 75cm enclosed by a safety fence which has a 1m wide bridge over it in
the shape of a quadrant. The bridge is fitted with a handrail on both sides.
Firstly I need to decide what length the sides of the pond are going to be. (Pond marked E)
To work out the amount of safety fence required, I will work out the perimeter of the square fish
pond subtracting 2m (1m for each side of the bridge at 1m each side).

Perimeter = 4 x sides – 2(1m)
I will need to work out how many meters of safety fencing/ m required and then cost it at £8.70 per
m
To work out the quadrant shape bridge marked F. As a quadrant is quarter of a circle I can work out
the length of the outside edge of the bridge by using the circle theorem. I will calculate the
circumference using the radius and dividing by 4.
Equation to find Quadrant
Circumference =
When will then cost the
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Pythagoras Accomplishments
Did you know that Pythagorean Theorem was actually created well before our generation's time
period? Yes, it seems that is pretty basic knowledge, but did you know it was actually created in
B.C. and instead of A.D.? The brilliant mind to thank for such an achievement is the man who goes
by the name Pythagoras. However the Pythagorean Theorem isn't his only accomplishment, he was
so advanced for his time, that he had many achievements. But to the best way to examine these
accomplishments is to start at the beginning. In around 569 B.C on a small island of Samos,
Pythagoras was born. His mother, named Pythais, was a native to the island Samos and his father,
named Mnesarchus, was a traveling merchant. Due to his father being a traveling merchant,
Pythagoras spent his most of his childhood traveling with his father, thus causing young Pythagoras
to become more intelligent and worldly compared to other youth at the time. Along the travels he
picked up reading and reciting poems created by Greek Poet, Homer. Though there were two ...
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Many of the theorems we have today our proven fact, while back in Pythagoras' day math was more
critical thinking rather than problem solving. One example of Pythagoras' critical thinking comes in
the form of a simple triangle which he called Tetractys. A tetractys, or also known as tetrad, a
triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in
each row, which is the geometrical representation of the fourth triangular number. This later became
the mystic symbol of the Pythagoreans. Speaking of triangle's Pythagoras' most famous contribution
to the world of mathematics, The Pythagorean Theorem was proved during this time period.
Pythagoras did not create the Theorem, the creation is credited to the Babylonians and Indians,
however Pythagoras was the first man to successfully prove the Theorem. The iconic equation looks
like

Fast Class Practice
Joe receives a weekly salary of $350 plus a 10% commission on all sales. Last week, Joe had a total
of $1,000 in sales. How much total money did Joe make last week? | | $250 | | | $300 | | | $450 | | |
$900 |
A chef uses one–half of his spices when cooking breakfast. Two–thirds of the spices he uses on
breakfast are used when making omelets. What proportion of his total spices are used on making
omelets? | | | | | | | | | | | |
The perimeter of a square is 8 inches. What is the area of the square if each side is a whole number?
| | 32 | | | 8 | | | 4 | | | 2 |
Jordan is one year younger than two–thirds the age of Jack. How old is Jordan if Jack is 30 years
old? | | 17 | | | 19 ... Show more content on Helpwriting.net ...
How many pennies does she have? | | 98 | | | 0.98 | | | 48 | | | 0.48 |
Sylvia makes chocolate truffles to give to her family. She keeps one–tenth of them for herself. If she
made 10 dozen truffles, how many did she keep? | | 1 | | | 12 | | | 1.2 dozen | | | 2 dozen |
After Tuesday's shipment was unloaded, a manager found that 7, or 28 percent, of the items in the
shipment were either dented or scratched. How many items were received on Tuesday? | | 4 | | | 21 | |
| 25 | | | 196 |
The fraction is equal to what percent? | | 10% | | | 20% | | | 25% | | | 40% |
A blouse is on sale for 35% off, and the sale price is marked as $19.50. What was the original price
of the blouse? | | $26.32 | | | $30.00 | | | $54.50 | | | $55.71 |
A right triangle has hypotenuse 20. One leg has length 16. What is the length of the other leg? | | 12 |
| | | | | 6 | | | 36 |
A right triangle has an area of 54. If the base is 9, what is the perimeter? | | 27 | | | 36 | | | | | | |
The area of a right triangle is 2. The length of the base is the same as the length of the height. Find
this length. | | 2 | | | 4 | | | | | | |
In a right triangle, the base is of the height. The area is 150. What is the length of the hypotenuse? | |
| | | | | | 25 | | | 20 |
Math Practice Set #1: Answers

Pythagoras Research Paper
Pythagoras was one of many math Mathematicians and a Greek Philosopher . He was born 570 BCE
Samos, Ionia and died 500–490 BCE Metapontum, Lucanium. He also was the first philosopher
ever. Pythagoras came up with the Pythagorean Theorem. The Pythagorean Theorem is a among all
3 sides of the triangle. There's also a formula that goes along with Pythagorean Theorem, the
formula is (a2 + b2 = c2). We still use Pythagorean Theorem til this day. That formula only applies
to right triangles. The Pythagorean theorem has shocked people for nearly 4,000 years. There are
now almost 367 different ways to do it. Pythagoras left Samos and went to Italy to continue with the
Theorem. One of Pythagoras famous quotes was "As soon as laws are necessary for

Trigonometry: Right Triangle Trigonometry
Right Triangle Trigonometry is the study of triangle measurements. When the Egyptians first used a
sundial around 1500 B.C., they were using trigonometry (Burrill, Gail p 376). Trigonometric ratio is
a ratio of the lengths of two sides of a right triangle. The three common ratios. They are sin, cosine,
and tangent. They have abbreviations are sin, cos, and tan. The triangle shown to the left is a right
triangle. Each side has a number because that is its side length. As we see there is a number 5. This
side is called the hypotenuse. The hypotenuse is always the side the is directly across from the 90o
angle. There is a number 3. This side is called adjacent. And there is also a number 4. This side is
called opposite. There is an acronym ... Show more content on Helpwriting.net ...
Give the lengths to the nearest tenth.There are many ways to get started, but today we are going to
start off with the side XW. We first need to find out what a is. You can use a calculator. We know
that a is opposite of angle Y. So now if you see we are able to use the tangent function because we
have the opposite side of 65 degrees and the adjacent side of 65 degrees which is 5. So Tan 65o=
a/5. Now we can solve for a. So we are going to multiply both sides of the equation by 5. And we
are left with a= 5 x tan 650. Now we can get our calculators out and figure out what this is to the
nearest tenth. Now the number you will get is 10.7225346025. But it says to round to the nearest
tenth. So the answer for a is approximately equal to 10.7. So we now know that a has an
approximate length of 10.7. Now its time to tackle the side YW. We can go two ways, we can use
Trigonometric Function, or we can use the Pythagorean Theorem. I am going to use the
Trigonometric Functions. So Cosine deals with the Adjacent over the Hypotenuse. To continue we
are going to take cos 65o= 5/b. and then we multiply both sides by b, and you are left with b x cos
65o = 5. And then to solve for b you are going to divide both sides by cos 65o which is equal to
b=5/cos 65o. And now you ender that into your Calculator and you end up with b is approximately
11.8. We are almost done salving this right triangle. The next thing we do is we need to find the
measure of

Questions On The Pythagorean Theorem Essay
Tasfia Haque Math Research Paper
December 8, 2015 Mr. Rubinstein Period 6 The Pythagorean theorem is a theorem that states that
the sum of the squares of two legs of a right triangle, a and b, is equal to the square of the
hypotenuse, c. This can be written and shown as the equation, a2+b2=c2. Because a2+b2=c2, we
can solve for the sides of the legs of the right triangles, in terms of this formula of the Pythagorean
theorem.
C=√(A2+B2)
A=√(C2–B2)
B=√(C2–A2)
This diagram represents the Pythagorean theorem as well. Because the squares of each side of the
right triangle are used in the theorem, this can be shown as an extension to each side of the triangle,
where there are three squares and one side of each square is apart of the triangle. The sum of the
areas of the two squares attached to the legs of the triangle is equal to the area of the square attached
to the hypotenuse. This is because to find the area of the square, you square its side, which also
happens to be the side of the triangle. This perfectly shows and represents the Pythagorean theorem
because the sides have to be squared and added to create the equation, a2+b2=c2. This can also be
shown through other extensions and shapes that are attached to each side of the triangle, not just
with squares. For example, the sum of the area of circles or hexagons attached to the two legs of the
right triangle is equal to the area of the circle or hexagon attached to the hypotenuse. This is

6.03 Calorimetry Honors Lab
I notice that angle H is in the opposite of the right angle and therefore angle H has a value of 90
degrees. This will mean that the sum of the angles that are listed as (8m – 18) and (5p + 2) will add
up to 90. And since the angle (7m + 3) is opposite to the (5p + 2) angle, they're equal. Therefore (8m
– 18) + (7m +3) = 90. (8m – 18) + (7m +3) = 90 8m – 18 + 7m + 3 = 90 15m – 15 = 90 15m = 105
m = 7 Since m = 7, I immediately know that (8m – 18) = 38, and (7m + 3) = 52. And because of the
opposite angles, I also know that (5p + 2) = 52, so p = 10, and (11t – 17) = 38, so t = 5 The line CE
has been divided into 2 equal half's by point D, so 5a + 12 = 9a – 12 Solving for a, gets 5a + 12 = 9a
– 12 12 = 4a – 12 24 = 4a 6 = a So a = 6 The two angles off of point E are marked as congruent, so
... Show more content on Helpwriting.net ...
s = 2 t = 5 a = 6 m = 7 p = 10 And upon doing so I see the word "stamp" as the answer to the riddle
"What sits in a corner but travels around the world?" 1. I noticed that segment BHF and CDE are
congruent, another segment I see is AHE and GHC are vertical. The diagrams show BHF and CDE
are congruent because they are parallel, segment AHE and GHC connect and make vertical angles.
2. A=6 work shown above 3. I can be sure that I solve the equation correctly because I know line BF
is 180 and I have the measurement of <AHB and <GHF and the work are shown above how I got
the answer for A. 4. S=2 4a. <HEF and <HED are congruent angles. 4b. the equation that correctly
relates m<HEF and m<HED is 21s+6+48=90. 4c. The solution of the equation from part b is 5.
<HEF and <HED are adjacent complementary because they both are sharing point H and E and
when those two angles are connecting the make 90

Right Triangle Research Paper
There are six trigonometric ratios that one must know in order to find any angle in a right triangle.
There are various of ways to remember these trigonometric ratios, but the most common way is
through SOHCAOTOA. By having this clear in your memory, it will allow one to remember at least
the three basic trigonometric ratios: Sine (sin.), Cosine (cos.), and Tangent (tan.). Before one learns
about how SOHCAOTOA is split up, we must learn about the angles in a right triangle. First off, the
hypotenuse is the longest line in a triangle, then in order to find the adjacent and opposite, one must
locate where the angle. Upon locating the angle, we can conclude that the opposite is further away
from the angle, whereas the adjacent is the closer one ... Show more content on Helpwriting.net ...
In short, these three trig. ratios are the reciprocal of sine (sin.), cosine (cos.), and tangent (tan.).
Although many may assume that cosecant is the reciprocal of cosine, it is actually that of sine,
which means that cosecant is hypotenuse over opposite. Thereafter, that leaves us with secant, which
is in fact the reciprocal of cosine, demonstrating that it is hypotenuse over adjacent. Nonetheless,
cotangent is the last trig. ratio, meaning that it is the reciprocal of tangent, being rather adjacent over
opposite. Now, we know the formulas for these trigonometric functions being:
Csc=HypotenuseOpposite, Sec=HypotenuseAdjacent, and Cot=AdjacentOpposite . For example, the
triangle on the next page is a 7–24–25 right triangle, and we must determine the six trigonometric
ratios for angle C of the right triangle. Based off this information, we can determine that the adjacent
is 7, with the opposite being 24 and hypotenuse as 25. In order to find the sine of angle C, we must
write out Sin C =OppositeHypotenuse and plug in the numbers to equal Sin C=2425. Next, in order
to find the cosine of angle C, by writing out Cos C=AdjacentHypotenuse and plugging in the
adjacent and hypotenuse number to equal Cos C=725. After solving for the sine and cosine of the
right triangle, we must find the tangent being Tan C=OppositeAdjacent.

A Brief Biography Of Pythagoras
A Brief Biography of Pythagoras Pythagoras was a Greek religious leader who made huge
developments in math. He is described as the first pure mathematician. Pythagoras is a very
important figure in the development of mathematics and he made many contributions to the math we
use today. Pythagoras was born c.570 BCE in Samos, Ionia, Greece and died on c.495 BCE on
Matapontion. There is not a lot of information about his childhood. In 532 BCE, Pythagoras
emigrated to Italy, where he attended a scientific school. One of the teachers who had a big
influence in Pythagoras life was Pherekydes. As well as Thales, a philosophe. Thales introduced him
to math ideas and suggested to go to Egypt to learn more about every subject. So he took ... Show
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That at its deepest level, reality is mathematical in nature, 2. That philosophy can be used for
spiritual purification, 3. That the soul can rise to union with the divine, 4. That certain symbols have
a mystical significance, and 5. That all brothers of the order should observe strict loyalty and
secrecy. In this society were accepted both man and women. Brumbaugh says that it hard for us
today, familiar as we are with pure mathematical abstraction and with the mental act of
generalization, to appreciate the originality of this Pythagorean contribution because we do not
know in what sense Pythagorean and mathematikoi were studying mathematics. Regarding to this
we do not know what work is his anf what is his followers because there wasn't much writing. Even
though from the Pythagoras' actual work nothing is known, Braumbaugh writes that he studied
proportion of numbers, odd and even numbers, perfect numbers, etc. and he states: "Each number
had its own personality – masculine or feminine, perfect or incomplete, beautiful or ugly. This
feeling modern mathematics has deliberately eliminated, but we still find overtones if it in fiction
and poetry. Ten was the very best number: it contained in itself the first four integers – one, two,
three, and four [1 + 2 + 3 + 4 = 10] – and these written in dot notation formed a perfect triangle."
And today we know Pythagoras for his famous Pythagorean

Special Right Triangle Report
To identify a Special Right Triangle. The angle measure must be known first. If the given angle
measure is 45°–45°–90° Special Right Triangle. Also if the given angle measure is a 30° or a 60°.
Then the triangle is a 30°–60–°–90° Special Right Triangle In a 45°–45°–90° triangle. There are two
legs and a hypotenuse. The hypotenuse is opposite the 90° angle. The legs are opposite the 45°
angles. Since the two legs are of equal length. When given the length of one. The other leg always
have the same length. To find the hypotenuse when given the length of a leg. Multiply the leg length
by the square root of two. And to find the length of the legs when given the length of the
hypotenuse. Divide the length of the hypotenuse by the

How Did Egypt Use Triangles
The Greeks and the Egyptians used triangles as early as 3500 BCE. They used these triangles as
rules of thumb. They could apply these rules to specific applications. For example, the Egyptians
knew that the 3:4:5 ratio was a right triangle. They could derive this because for them to create a
right triangle the Egyptian land surveyors used a rope divided into twelve equal parts, creating a
triangle with three pieces on one side, four pieces on the second side, and five pieces on the last
side. The right angle was found where the three–unit side came together with the four–unit side.
This was a very efficient way to create right triangles. It's a mystery as to how the Egyptians came
up with this, but this was later used by Pythagoras (c.571 –

Freen Friend Number Two Circle
In my opinion friend number one is correct. Friend one is saying the circles are concentric, meaning
every ray you draw from the center of the circles will cut the big circle once and the little circle
once. This makes a correspondence between the points of the one circle and the points of the other
circle. So they have the same amount of points. If you look below this, I have taken the circle and
drew a few lines on it. Showing you how each ray, or line, cuts the big circle one and the little circle
once. Looking at the picture you see that each ray cuts each, big and little, circle once. Making
friend number one's statement correct, because they do have the same amount of point on the grid.
Friend number two's statement was correct to a certain point. They had said, "If you draw a
horizontal line through the center. At first, the line cuts both circles in two places, but as move the
line up or down you will see line will cut the big circle twice, but only just touch the smaller circle
at one point. And if you get farther away than that, the line will touch the bigger circle (still twice!),
but not touch the little circle at all. All points on the smaller circle are related to some point on the
bigger one, but some points on the bigger one are not related to point on the smaller one. So the
larger circle has more points." ... Show more content on Helpwriting.net ...
No matter how far up or down they move (still touching the small circle) the line it is going to cut
the small circle two times. If they would have said "At first the line cuts both circles in two places,
but as move the line up or down and the farther away you get from the small circle, the line will
touch the bigger circle (still twice!), but not touch the little circle at all." Then they would have also
been correct. Looking at this circle below I have shown you how it makes two cuts on each circle,
expect when the line is not even touching the small

Rubaiyat Of Omar Khayyam : Poem : Mathematics And Mathematics
As a mathematician, he is most notable for his work on the classification and solution of cubic
equations, where he provided geometric solutions by the intersection of conics.[5][6] As an
astronomer, he composed a calendar which proved to be a more accurate computation of time than
that proposed five centuries later by Pope Gregory XIII.[7]:659[8] Omar was born in Nishapur, in
northeastern Iran. He spent most of his life near the court of the Karakhanid and Seljuq rulers in the
period which witnessed the First Crusade. There is a tradition of attributing poetry to Omar
Khayyam, written in the form of quatrains (rubāʿiyāt ‫)رباعیات‬. This poetry became widely known to
the English–reading world due to the translation by Edward FitzGerald ... Show more content on
Helpwriting.net ...
In 1076 Khayyam was invited to Isfahan by the vizier and political figure Nizam al–Mulk to take
advantage of the libraries and centers in learning there. His years in Isfahan were productive. It was
at this time that he began to study the work of Greek mathematicians Euclid and Apollonius much
more closely. But after the death of Malik–Shah and his vizier (presumably by the Assassins' sect),
Omar had fallen from favour at court, and as a result, he soon set out on his pilgrimage to Mecca. A
possible ulterior motive for his pilgrimage reported by Al–Qifti, is that he was attacked by the clergy
for his apparent skepticism. So he decided to perform his pilgrimage as a way of demonstrating his
faith and freeing himself from all suspicion of unorthodoxy.[4]:29 He was then invited by the new
Sultan Sanjar to Marv, possibly to work as a court astrologer.[1] He was later allowed to return to
Nishapur owing to his declining health. Upon his return, he seemed to have lived the life of a
recluse.[13]:99 Khayyam died in 1131, and is buried in the Khayyam Garden. Mathematics[edit]
"Cubic equation and intersection of conic sections" the first page of two–chaptered manuscript kept
in Tehran University. Khayyam was famous during his life as a

Pythagoras was a Greek mathematician known for formulating the Pythagorean Theorem. He was
born in 570 BC on the islands of Samos and passed away 495 BC at around the age of 75 in
Metapontum. He was once a philosopher who taught that numbers were the essence of all things and
was described as the first pure mathematician. People describe him as an extremely important
person in mathematical history and yet not many people know much about him. There is little
reliable records about his life and accomplishments. He linked numbers with virtues, colors, music
and other qualities. He also believed that the human soul is immortal and he believed that after death
human soul moves into another living being. Pythagoras created and organized a group ... Show
He also believed that that the sun, moon, and other planets had their own movements. His beliefs led
to the Copernican theory of the universe. The principles of the Pythagorean Theorem had already
been known by the Egyptians before Pythagoras formulated it. No one today is sure how Pythagoras
himself proved the Pythagorean Theorem because he never allowed anyone to record his teachings
in writing. Most likely, like most ancient proofs of the Pythagorean Theorem, it was geometrical in
nature. The Pythagoreans knew that any triangle whose sides were in the ratio 3:4:5 was a right–
angled triangle. But, they had a desire to find mathematical of all things led them to prove the
geometric theorem. Although the Egyptians were the first to discover the theorem, Pythagoras was
the first to prove it.
Pythagoras Theorem was a demonstration that the combined areas of squares with side length of a
and b. These side lengths will equal the area of a square with sides of length c, where a, b, and c
represent the lengths of the two sides and hypotenuse of a right triangle. However, the Pythagoreans
did not consider the square on the hypotenuse to be that number c multiplied by itself c^2. Instead, it
was conceptualized as a geometrical square c constructed on the side of the

William Dunham 's Journey Through Genius
The author of Journey through Genius, William Dunham, begins this chapter by depicting how
mathematics was spurred and developed in early civilizations. Dunham focuses primarily on the
works' and achievements' of early Egypt, Mesopotamia, and Greece in this section. These ancient
societies, as they developed, produced mathematicians such as; Thales, Pythagoras, and
Hippocrates, who turned a basic human intuition for space and quantity into applicable everyday
mathematics. The primary influences driving the development of early mathematics were the issues
of growing civilizations, most notably counting commodities, taxation, and the division of land
equally, rather than a pure desire for understanding that is seen in mathematics today. These
influences culminated in the development of early arithmetic and geometry. The first civilization
that is discussed is early Egypt. Records have been found referring to mathematics done in Egypt
showing a rudimentary understanding of the Pythagorean Theorem as it pertains to the construction
of triangles with whole number sides before the creation of the Pythagorean Theorem. An example
lies in an ancient Egyptian construction of a rope with knots forming 12 evenly spaced segments
along the length of the rope. They knew if a triangle having sides of 5, 4, and 3 segments was
formed it would form a triangle containing a right angle. However, it is important to state that
evidence has not been found that ancient Egyptians understood exactly

Math 221 Week 5 Assignment Essay
Buried Treasure
Ashford University MAT 221
Buried Treasure
For this week's Assignment we are given a word problem involving buried treasure and the use of
the Pythagorean Theorem. We will use many different ways to attempt to factor down the three
quadratic expressions which is in this problem. The problem is as, ""Ahmed has half of a treasure
map, which indicates that the treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa
has the other half of the map. Her half indicates that to find the treasure, one must get to Castle
Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their
information, then they can find x and save a lot of digging. ... Show more content on
Helpwriting.net ...
Running with this information can now write out the equation AB2 + BC2 = AC2. One important
thing is that we must note that AB is equal to "X" and the line segment of BC is equal to that of
2x+4, and that AC will be equal to that of 2x+6. So we will now input this information to create (x)2
+ (2x + 4)2 = (2x + 6)2 and begin factoring each term into two sections. These two sections will be
as x*x + (2x + 4)(2x + 4) = (2x + 6)(2x + 6). x times x is x2. An important tool to use now would be
the FOIL method, so we will take (2x + 4)(2x + 4) and create 4x2 + 16x + 16. Right off the bat we
notice that we have like terms. So we will add x2 to 4x2 to get 5x2. This will create 5x2 + 16x + 16
= 4x2 + 24x+ 36. Now we will use the subtraction property to get 5x2 – 4x2 + 16x – 24x + 16 – 36
= 0, however we still have like terms, so because 5x2 is a like term with –4x2 we will add them
together to get x2. We will also combine 16x and –24x and also 16 and –36 which are also like
terms and create –8x and –20, our equation should now look like x2 – 8x –– 20 = 0.We will now
factor the equation from left to right, first factoring x2 which has 1 coefficient so the fact will be 1
and –1. The other term will be 20 which have no coefficient so we will do 5x4 and then 4 still can be
divided so 2x2. This will create 20=225.
We will now take a look using the Prime Factorization formula which will aid us in finding the
number

Similarity and Congruence
Similarity and congruence Two triangles are said to be similar if every angle of one triangle has the
same measure as the corresponding angle in the other triangle. The corresponding sides of similar
triangles have lengths that are in the same proportion, and this property is also sufficient to establish
similarity. A few basic theorems about similar triangles: * If two corresponding internal angles of
two triangles have the same measure, the triangles are similar. * If two corresponding sides of two
triangles are in proportion, and their included angles have the same measure, then the triangles are
similar. (The included angle for any two sides of a polygon is the internal angle between those two
sides.) * If three ... Show more content on Helpwriting.net ...
Theorem 5–9. SSS Similarity Theorem If all three pairs of corresponding sides of two triangles are
proportional, then the two triangles are similar. Transversal In geometry, a transversal is a line that
passes through two lines in the same plane at different points. When the lines are parallel, as is often
the case, a transversal produces several congruent and several supplementary angles. When three
lines in general position that form a triangle are cut by a transversal, the lengths of the six resulting
segments satisfy Menelaus' theorem. The Triangle Inequality Theorem states that the sum of any 2
sides of a triangle must be greater than the measure of the third side. In mathematics, the
Pythagorean theorem – or Pythagoras' theorem – is a relation inEuclidean geometry among the three
sides of a right triangle (right–angled triangle). In terms of areas, it states: In any right–angled
triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is
equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a
right angle). The theorem can be written as an equation relating the lengths of the sides a, b and c,
often called the Pythagorean equation:[1] where c represents the length of the hypotenuse, and a and
b represent the lengths of the other two sides. The

Pythagoras’ Impact in Geometry
The most common thing people associate the mathematician Pythagoras with is the Pythagorean
Theorem that describes the relationship of the the sides of a right triangle, which is a^2 + b^2 = c^2.
Some know him as the first pure mathematician. (Mastin, 2010) His teachings come before other
famous philosophers and thinkers, such as Plato and Aristotle. Who is Pythagoras and how did he
impact the mathematical world of geometry? In order to answer the previous question, there must be
an understanding of who he was, what his teachings were, and how his teachings are applied today.
Pythagoras was born in Samos, Greece in 569 B.C. His marital status is unknown. He was well
educated and could recite Homer. He was not only interested in mathematics, but also philosophy,
Astronomy, and music. He was taken prisoner and sent to Babylon, where Magoi priests taught him
arithmetic, music, and mathematical sciences. When he was set free, he created a school called the
Semicircle. He left the school because the leaders of Samos wanted him to be a politician. He then
made a school where his followers, the Pythagoreans, were taught, had lived, and had worked. The
followers were men and women also known as mathematikoi. The theories and ideas in mathematics
that his followers made were accredited to him. They were also a very secretive group, so not much
is known how they came up with their ideas and philosophies. (Douglas, 2005) He died in
approximately 500 B.C. and his followers, "continued

How Did Pythagoras Influence The Ancient World
Pythagoras was from c. 580 B.C.– c. 500 B.C., known as Pythagoras of Samos. He was a famous
Greek philosopher, religious leader, and mathematician best known for his work on geometry is the
Pythagorean theorem, a^2+b^2=c^2. He and his followers created a secret society which was known
as the Pythagoreans. They explored, discovered, and proved a lot of the principles, ideas, and
concepts of mathematics. Pythagoras was an excellent man who made many important discoveries
in different areas of mathematics and science. Those discoveries influenced the ancient world as
well as today. And, people continue to apply them today.
In Mathematics, the Pythagorean believed that "numbers were like gods, pure and free from material
change ... They believed ... Show more content on Helpwriting.net ...
It states that "for any right triangle, the square of the hypotenuse length c is equal to the sum of the
squares on the two shorter leg lengths a and b – which is written as a^2+b^2=c^2" (Pickover, 2009,
p. 40). An ancient Chinese text on mathematics, the Chou Pei also gives an evidence that the
Chinese knew about the Pythagorean theorem many years before Pythagoras or one of his followers
(Joseph, 2000, p. 180). The Pythagorean discovered and first proved it, so that the theorem is named
after Pythagoras.
Pythagoras' Proof:
Let a and b be the legs, and c be the hypotenuse of the given right triangle. Each big square below
has a+b as its side. The left big square is separated into six pieces. They are the two squares one
with a as its side and one with b as its side, and four right triangles congruent to the given triangle.
The right big square is separated into five pieces. They are the square with the hypotenuse, c as its
side and four right triangles congruent to the given triangle. Then we write out the area for each big
square. Since they have congruent sides, they areas will be congruent (Head, 1997, p.

Three Basic Trig Functions Of Trigonometry
SIX BASIC TRIG FUNCTIONS
Trigonometry, stemming from the greek words trigonon and metron , is the branch of mathematics
in which sides and angles within in a triangle are examined in relation to one another. A right
triangle has six total functions used in correlation to its angles, represented by the greek letter theta
(θ). The primary operations are sine (sin), cosine (cos) and tangent (tan) which serve also as the
reciprocal of cosecant (csc), secant (sec), and cotangent (cot) in that order. All six have
abbreviations shown in parenthesis. Sine is the trigonometric function that is equal to the ratio of the
side opposite a given angle (θ) to the hypotenuse. Cosine is the ratio of the side adjacent to θ to the
hypotenuse. Tangent is the ratio of the opposite side from the given angle to the adjacent side. These
functions are used in order to calculate unknown angles or distances from some known and/or
measured aspects in a geometric figure.
BRIEF HISTORY Trigonometry began as a method born from necessity to model the motions of
galactic objects in mathematical astronomy. In order to predict their movements with geometric
accuracy Hipparchus began the development of mathematical tools which would allow astronomers
to convert arc lengths to distance measurements. Many contribute Hipparchus as the father of
trigonometry. He was the first to construct a table of values of a trigonometric function and created
the notion that all triangles as being inscribed into a circle

Simple Math Working Models
You Can prove that radius to the point of contact of a tangent is perpendicular by
Take two Iron Rings with a radius of a pin that you have like a stitching needle.. The iron rings
should be at a thickness of 1 cm... Join the two rings by superimposing but not exactly, without gap.
just leave a gap between the two circumference of the circle such that there is parallel gap
throughout the two rings then join them at two points{one at any where and another at straight opp
to the other} using m~seal. no find the center by drawing a circle with same radius with compass.
(no need to measure the radius just estimate the radius by having the ring)then keep the rings on it
and then take a wire make it as the diameter with the contact points as ... Show more content on
Helpwriting.net ...
To place a number of marks or stations in any given direction the horizontal plates are clamped
together, the telescope pointed in the desired direction and then moved vertically and focused on
each mark as required.
Although the theodolite looks like a difficult instrument to use, its basic concept is very simple. The
surveyor would begin by picking a distinct point in the distance. After centering the theodolite over
the primary point of interest, the surveyor would use the eye piece to align the sight axis with the
point of interest in the distance. Next the surveyor would zero the horizontal and vertical axes
graduated circles. Finally the surveyor would move the sight axis using the eye piece to the final
point of interest and determine the horizontal and vertical angles between the points. With this
known angle and the triangulation principle, the distance between each of the three points may be
determined.
See, told you that if you understand it is quite easy to use. One major difference between the
clinometers and theodolite is that theodolite is used in professional using ( NOT THIS ONE ::: THIS
ONE IS HOMEMADE). THEY USE THE DIGITAL ONES |
U can go for anything.
1) Make use of thin sticks or Straws. Have papers cutting fix between two sticks to show angle.
Decorate it according to yourself.
2) since you are in 10(C.B.S.E) your study of Trigonometry is concerned to 2–D only . even then
you can try 3–d. Following is a description:

Summary Of Dualism By Antoine Arnauld
Antoine Arnauld objection aims to refute a part of Descartes's point of view on dualism. Arnauld
aims a specific claim to disprove Descartes, which is by challenging him with an example of the
pythagorean theorem. Descartes theory of the mind describes the theory of the mind and body being
separate from each other. The soul is symbolizes the mind and Descartes explains the body as just
existing without even knowing the body's exist, Descartes does not think body and soul correlate
with each other. Arnauld's conclusion of the objection is explaining the mind and body through the
example of the pythagorean theorem. Arnauld says, "Suppose someone knows for certain that the
angle in a semi–circle is a right angle, and hence that the triangle formed

Bloodstain Pattern Analysis Lab
Bloodstain pattern analysis (BPA) is the interpretation of bloodstains at a crime scene in order to
recreate the actions that caused the bloodshed. Analysts examine the size, shape, distribution, and
location of the bloodstains to draw about what happened. BPA uses principles of biology (behavior
of blood), physics (cohesion, capillary action and velocity) and mathematics (geometry, distance,
and angle) to assist investigators in answering questions like:
From where did the blood originate from?
What was the cause of the bloodshed? (weapons used)
From what direction was the victim wounded?
What were the positions of the victim/ attacker during the bloodshed?
Did anyone/anything move around after the bloodshed?
How many possible ... Show more content on Helpwriting.net ...
Blood drops are falling freely and only moving because of the pull of gravity on it. At low
velocities, larger bloodstains are produced. Usually, low velocity bloodstains are a result of blood
dripping from a victim. Dripping blood often falls at a 90 angle. This causes a circular bloodstain to
form on the surface that hits it lands upon. The droplets range from 4mm–10mm. However, this
usually happens if the victim is not moving and completely still. If droplets are falling from a
moving object or person, (walking or running) they fall to the ground at an angle (see angle of
impact) and the direction of the movement can be established. Spines often appear on bloodstains
left behind and can be caused by drops repeatedly landing in the same place, by the distance the
drop falls (larger spines indicate larger distance), or by the surface upon which the blood lands
(rough surfaces cause more spines to form). Low velocity blood may also be found in the trail of a
person who is bleeding and larger pools of blood may indicate where the person

Blood Spatter Analysis In Criminal Justice
In my essay, I'm going to look into how blood spatter analysis is used in processing a crime scene
and how useful it is in finding and convicting a suspect. I will look into how important blood spatter
analysis is and what it can show within a crime scene. I believe that first I should give a clear
definition of blood spatter analysis. It is the examination of shapes, distribution patterns of
bloodstains in order to show an interpretation of the physical events that created them. First I'm
going to look at how blood spatter analysis is used in processing a crime scene. There are many
different points to blood spatter analysis such as; The interpretation of blood patterns found on items
of clothing, this interpretation can help determine things like; the position of the victim, so whether
they were standing, ... Show more content on Helpwriting.net ...
It can tell the investigators with hand was holding the gun, but also reconstruct the position of the
gun when it was fired. This again could help determine the how a crime was committed, but also
with what weapon and by whom relating to angle of the gun. There was a study conducted by
(S.N.Kunz et al 2013). Were looked into an apparent suicide, which had unusual characteristics for a
suicide, which included an unusually blood spatter pattern on the man's hands, it was determined by
studying the crime scene photos and also the angle of the wound that the man did commit suicide,
however, the unusual pattern was caused by an abnormal position of the gun and supporting hand.
This can show that blood spatter analysis was needed to help determine the event. However it
should be taken into account that this was the findings of a few analysts and when it comes to blood
spatter analysis it is based on the analyst onion and interpretation of the

David Hume Summary To Human Understanding
An Enquiry to Human Understanding based of David Hume. David Hume tries to explain that our
awareness of causation is a product of experience. He focuses on the notion that knowledge comes
from experience. He attempts to explain the original impressions involved in causation. He begins
by distinguishing impressions and ideas. According to Hume, impressions are invoked in our senses,
emotions and anything else that is of mental phenomena. Ideas are the memories or random thoughts
that our minds connect to our impressions. There are three ways, according to Hume, that our minds
come up with ideas. These include resemblance, contiguity and cause and effect. Matters of fact and
ideas are further distinguished by Hume. He explains that the association of ideas cannot be denied.
He uses mathematical and geometrical as examples of the relationship and association of ideas. The
square of the two sides is equal to the square of the hypotenuse. This shows the relationship of the
sides to the hypotenuse. These theorem do not require observation. There can be a contrary of ideas.
This may also be its point of connection considered a blend of causation and resemblance, according
to Hume. Matters of fact are observable. According to Hume, fact is known by observation. It is
based on the relationship of cause and effect. For instance, a man is believed to ... Show more
content on Helpwriting.net ...
It explains importance and usefulness of experience. Customs spare us the ignorant we would be
engulfed in. they explain past events. Experience goes beyond memory and senses, and it explains
matters of fact. If one finds remains of the building in the desert, they will have concluded that there
was civilization in that part, but experience restricts that conclusion. The person will then try to find
evidence to support this theory. Hume then concluded that, all matters of facts can only be inferred
through a memory, senses or some connection between objects, like flame and

Pythagoras of Samos, son of Mnesarchus and Pythias born 570 BCE in Samos Ionia was a Greek
Philosopher, Mathematician and the founder of Pythagorean brotherhood. Although religious in
nature , the Pythagorean brotherhood formulated principles that influenced the thought of Plato and
Aristotle and contributed to the development of mathematics and Western rational philosophy.
Pythagoras was classified as the first "true" mathematician. The contributions he made were
noticeably important to the people today. But he also remained a controversial figure. Some
historians say that Pythagoras was married to a woman named Theano and had a daughter Damo,
and a son named Telauge. Pythagoras as a teacher and taught Empedocles. Others say that Theano
was ... Show more content on Helpwriting.net ...
Mathematical procedures, operations, or properties. source www.dictionary.com Pythagorean
theorem: The Pythagorean theorem is a theorem that shows a special relationship between the sides
of a right triangle. Source http://virtualnerd.com The basic concept of the pythagorean theorem is,
finding the missing hypotenuse or figuring out if the triangle is right or not. Both consist of the
equation a^2 + b^2= c^2. someone had to try and bring him down. Pythagoras believed that
numbers did not have numbers in between them but the two numbers had a relationship between
them a, proportion. Also that each number was its own being. Pythagoras was challenged by
Pathosis who told him off. Later he was killed by Pythagoras. The problem was if you have a
triangle with two sides the same the outcome would not be a rational number. He tried everything
but it would not settle. The fact that he was being proven wrong made him furious. It also lead to the
death of Pathosis. But the death of the Pathosis did not stop the word from getting out. It almost
works, but you start dividing this square evenly to fill up the two equal other squares, and you've got
this one odd one

Pythagoras Essay
Pythagoras
My name is Pythagoras of Samos. I believe I should win the fabulous two–week cruise on the
incomparable Argo because I dedicated my life to educating and caring for the future generations. I
risked my life to share my knowledge with anyone who wanted to learn. I was born on the island of
Samos, but lived most of my life in Crotona, Italy. When I was a young man, I traveled to many
different places to observe the different lifestyles and cultures. Some of the countries I visited were
India, Egypt, and Persia (Bulfinch). After viewing many different aspects of life I developed my
philosophies and beliefs. My most important philosophy is that almost everything in life can be
associated in some way with numbers . ... Show more content on Helpwriting.net ...
I am most famous for discovering the Pythagorean Theorem, which solves the length of the
hypotenuse of a right triangle. Use the equation a² + b² = c², where "a" and "b" are the two sides
forming the right angle to solve "c" which is the hypotenuse (Bruce E. Meserve 46). If I could meet
a Greek god or goddess I would undoubtedly choose Apollo. Apollo has many great characteristics
and I think we could be good friends if we talked with each other. One reason why I want to meet
him is because I am interested in seeing what he looks like. In books they say he is the most
beautiful god represented by the color gold. He also has a golden chariot with golden horses which I
want to ride. Another reason is that he is the god of things that I'm interested in like music,
mathematics, and medicine (Bernard Evslin 37). He is also a very thoughtful and kindhearted god.
The main reason I want to meet him is that he owns an oracle that can tell the future and even
though I have clairvoyant powers at this moment, I want to ask him if he will let me borrow his
oracle just in case if I lose my powers (Ellen Switzer 26). I asked the Oracle of Mother Earth to tell
me how modern Greece was dealing with its current political problems. She said that Greece's prime
minister was trying to resolve the Kosovo problem by talking with both political forces. The prime
minister felt that fighting and war was not the acceptable solution . He wants the

How Did Pythagoras Contribute To Modern Algebra
Pythagoras was one of the greatest mathematicians of all–time, developing some key points to
modern algebra, and his life story starts in Greece. In 569 BCE, Pythagoras was born on the island
of Samos: an island of Greece, closer to Persia than Greece itself. Pythagoras spent most of his days
in his home–town, but from time to time he traveled with his father, a merchant who roamed many
lands. Growing up, he was enticed by Homer's poetry, until great philosophers became of greater
interest. A man by the name of Thales created a strong impression on Pythagoras, exposing him to
the world of mathematics and astronomy. Hearing so much about this fellow, in 535 BCE he
journeyed across the Mediterranean Sea, and south to Egypt. When he arrived there, ... Show more
When he was given freedom, he formed a school in Samos, called "The Semicircle," but soon left to
travel to Italy. He traveled to southern Italy, to the town of Croton, where he founded a religion
based school. He also developed a small group of his top followers called the Mathematikoi. In this
school, Pythagoras made great discoveries. Some achievements of Pythagoras include, classifying
numbers into even and odd, classifying perfect numbers, and classifying triangle numbers. His
biggest discovery is most definitely the Pythagorean Theorem. This property's equation states that
a^2+ b^2 = c^2, with the variables a, and b acting as the two legs of a right triangle, and c acting as
the hypotenuse. The Pythagorean Theorem was the start of basic trigonometry, and geometry. When
you hear the phrase Pythagorean Theorem, most people revert to saying that Pythagoras invented it.
Well... no. Basically, Pythagoras heard the idea proposed in Babylon, so he stole it for himself and
refined it a bit. Even so, the little bit of refining he did was something that the Persians probably
couldn't have done. Pythagoras was proud of his achievement, but then one of his own students
turned on

Pythagoras was an Ionian Greek philosopher, mathematician, and putative founder of the
Pythagoreanism movement. He was born in Samos, Greece and died in Metapontum. His parents
were Mnesarchus and Pythais. Some say that Pythagoras had three daughters and one son. Others
say he never got married and had children. He is referred to as the first pure mathematician. He was
a well educated man and was interested in mathematics, philosophy, astronomy, and music. It is said
that he was influenced by Pherekydes (philosophy), Thales (mathematics and astronomy) and
Anaximander (philosophy, geometry). They don't know very much about his mathematical
achievements. They don't have any of his writing's like they do other philosophers. Pythagoras
settled

How Does The Pythagorean Theorem Work?
a2+b2=c2 is the famous theorem that Pythagoras discovered and named, calling it the Pythagorean
Theorem. This theorem applies to the right triangle stating, that by adding the length of both legs
squared you can then find the squared length of the hypotenuse. This theorem is set up in way that if
you know two of the variables, whether it is a leg(b or a) and the hypotenuse (c) or both legs (a and
b), you will always be able to find the third measurement. However, why does this theorem work?
Why does a2+b2=c2? That is the question that is asked hundreds of times by thousands of people.
The answer to it is not a complicated one, the reasoning behind that is because there are at least 367
Pythagorean Theorem proofs out there (Source four). They ... Show more content on
Helpwriting.net ...
Step Seven: You then solve the equation. c2=(a–b)2+2ab =a2–2ab+b2+2ab = a2+b2
Proof Four
Step One: Create four copies of a right triangle
(Note: Each has an area ofab/2)
Step Two: Combine the the triangle so that it forms a square with the side of (a+b) and a hole with
the side of c.
Step Three: Then calculate the area of the big square in two ways by creating the equation;
(a+b)2=4ab/2 +c2. (When simplified you will get the required identity.)
Step Four: We then use the identity created in the third proof and add it to the identity we created in
step three.
Step Five: Add the two identities together. c2=(a–b)2+4ab/2 c2=(a+b)2–4ab/2
2c2=2a2+2b2
c2=a2+b2
Overall there are hundreds of ways to prove the Pythagorean Theorem. As stated before these proofs
range from effortless geometric and algebraic proofs to intricate and complex trigonometric and
calculus proofs. The proofs shown above are predominantly straightforward and easy to understand
and do. Once again, these proofs are only a small portion of other proofs out there in the world.
However, it will never matter what proof you choose to prove the Pythagorean Theorem, because no
matter what they will all have the same end

Compass And Straightedge
Compass and Straightedge: Basic Constructions and Limitations In Euclid's Elements, Book 1, the
very first proposition states, "To construct an equilateral triangle on a given finite straight–line."
(Heiberg, Fitzpatrick, Euclid, pg 8) This proposition is saying that it is possible to construct an
equilateral triangle from a given segment. Euclid was able to perform this construction with just a
straightedge and compass. As The Elements was published in 300 BC (Heiberg, Fitzpatrick, Euclid,
pg 4) only the most basic tools had been invented. The straightedge Euclid used the Euclidean ruler,
used only the draw the straight line through any two points (Martin, pg 6). There were no markings
on the straightedge, unlike modern rulers. The compass ... Show more content on Helpwriting.net ...
With modern tools such as a protractor and marked ruler, this question is easy to do. However, it is
more difficult with only a straightedge and compass. For thousands of years people tried to solve
this problem and either ended up with an inexact construction or they had to use other tools than the
Euclidean tools. Finally in 1837 Pierre Laurent Wantzel, a French mathematicians, solved the
problem (Major, Jost, pg 81). It turns out that in general it is impossible to trisect any angle, but
there are cases, for example 90 degrees and 180 degrees, that can be trisected with a straightedge
and compass (Courant, Robbins, pg 147). The degrees that can be trisected come from the n–gons
that are constructible from a compass and straightedge. An example of this is an equilateral triangle.
The angles are 60 degrees, which is a third of 180 degrees. But in general, most angles cannot be
trisected. To prove that in general it is impossible to trisect an angle, it suffices to show that one
angle cannot be trisected. For example take an angle of 60 degrees. Let the angle be denoted by ϴ
and cosϴ = y. Need to find x = cos(ϴ/3). Cosine of ϴ/3 is connected to cosϴ=y=4〖cos〗^3(ϴ/3) –
3cos(ϴ/3) Thus trisecting an angle with cosϴ = y comes to 〖4z〗^3–3z–y=0. Then take ϴ =60
degrees, and y = cos(60) = 1/2. Thus 〖8z〗^3–6z= 0. There is no rational number that satisfies z,
thus the general holds, and it is impossible to trisect an angle (Courant, Robbins, pg

Scientific Notation Essay
Scientific Notation
Essential question: How can you use scientific notation to express numbers of different quantities?
Scientific notation is a method of writing or displaying numbers in terms of a decimal number
between 1 and 10, which is multiplied by a power of 10
Ex. Step 1 – 3 105 First, you would solve for the exponents Step 2 – 3 100000 Then, you would
multiply the factors that are left Step 3 – 300000 Lastly, you get your answer
Essential Answer: Scientific Notation can be used to express different quantities by reducing a large
or small number into a short mathematical sentence.
Integers
Essential Question: Why are using integers easier than a ... Show more content on Helpwriting.net ...
How to find a 3D figures Volume
Essential Question: What figures does volume apply to?
Volume is the amount of 3–dimensional space an object occupies. In the equations for volume, B
means area of the bases and h means distance between the bases. Remember to always add the
cubed sign at the end of your answer ( # unit3).
Ex. Find the volume of a figure that is 5 in. long, 7 in. wide, and 8 in. tall. Step 1– V = Bh First, find
the equation for the figure.
Step 2– V = (35)8 Then, find the base and substitute.
Step 3– V = 280 in3 Lastly, solve for V.
Essential Answer: It applies to only 3D figures.
How to find a 3D figures Surface Area
Essential Question: How is finding surface area different than finding volume?
Surface area is the total area of the surface of a three–dimensional object. For surface area, the
equation is SA = Ph + 2B. P stands for the perimeter of the base. The h stands for the distance
between the bases. B stands for area of the bases.
Ex. Find the volume of a box that is 2 feet long, 4 feet wide, and 3 feet

Kennedy Middle School Garden Case Study
The Kennedy Middle School fifth–grade class is growing a school garden. The students are very
excited to grow a garden, but animals have been eating the goods, so a fence needs to be put in. The
total cost to fence in the garden will be $159. The class calculated this by finding the perimeter (in
feet) and multiplying it by the cost per foot of fence, $1.50. The Pythagorean Theorem a2 +b2=c2
helped find the hypotenuses of the two right triangles that the garden can be broken into, which
helped the find the perimeter. The class first wrote the Pythagorean Theorem, which is a2 +b2=c2, to
find the hypotenuses of the triangles. The legs of the triangles were represented by a and b and the
hypotenuse was represented by c. After writing the equation,

Advertisement Analysis : Advertising Visual Analysis
ARTH 125 Advertisement Visual Analysis Jiayi Song "If only I could reach..." Ostensibly selling
super glue, this advertisement features a photograph of a muscular male toy with one broken leg. It
lies in the center of the photograph with the left broken leg of the figure, a skateboard and a super
glue are scattered around the figure. What is most striking in the image, however, is the pose of the
figure, his naked upper body with his left arm stretching out to get the super glue, and his desperate
facial expression. One can tell that the gender of the figure is male from his naked upper body and
the way he wears nothing but a shorts and a pair of shoes. The first reason he appears desperate is
because of his facial expression. One can see his mouth open as if he is saying "help" or "ouch." The
skin texture on his forehead also shows little wrinkles to demonstrate his craving for the super glue.
The second reason is his body language. His body lines are well defined and there are veins bulging
not only on his inner side of both his arms but also on his legs. He seems to be exerting his strength
by closing his right hand into a fist and stretching his left arm straight towards the glue. His upper
and lower body as well show his usage of force by bending a little bit away from the ground.
Moreover, his head also bends in the direction of the super glue. His awkward posture is reminiscent
of Renaissance paintings depicting Jesus's uncomfortable body posture on subjects such as

Case Study : ' Mqs61qj Midyear Project '
MQS61QJ Midyear Project
1. A calculator gives us rational approximations of irrational numbers, so we can "see" that
2*sqrt(3)> π. Explain how a circle inscribed in a regular hexagon allows us to "see" that 2*sqrt(3)>
π without any decimal approximations. A circle inscribed in a regular hexagon allowed us to "see"
that 2√3> π by simplifying the inequality that the area of the right triangle CHG was greater than the
area of the sector enclosed within triangle CHG. The reason why the area of triangle CHG was
greater than the area of the sector was because in the diagram, it was noticeable that triangle CHG
overlapped a greater amount of area than the sector. This was why a certain amount of "space" was
observed within the enclosure of ... Show more content on Helpwriting.net ...
As referred to the previous expression, rx/4> (πr^2)/12, (x√3)/2 can be substituted in for r, which
further simplified it to (x√3)/8> (πr^2)/12. I didn't substitute (x√3)/2 in for r on the other side of the
inequality, because I felt that it would be much easier to first multiply both sides by 24, which is the
LCM of both 8 and 12. As a result, the substitution of (x√3)/2 for r transformed the inequality to
3x^2 √3> (3x^2 π)/2. However, the inequality had to be multiplied by 2 and then divided by 3x^2 on
both sides before simplifying it to 2√3> π. Therefore, the inequality 2√3> π, derived from the area
of the triangle CHG and the sector, indicates how we can see that a circle inscribed in a regular
hexagon validates this specific inequality.
2. A cafeteria serves three meals a day and offers one fruit selection at each meal. The manager can
select from five different categories of fruit: fresh (raw), fresh (cooked), dried, canned, or juice.
How many different fruit menus (arrangements) can be offered on a single day if the same category
can be offered at most once each day and cannot be repeated at the same meal two days in a row?
Based on the scenario that five different categories of fruit can offered in a single day within 3
meals, but the same category can be offered at most once each day, I incorporated the formula for
permutation without repetition. The formula for permutation without reputation was n!/(n–r)!, in
which n stood

Why Living Organisms Respond To Their Environment
In our experiment our team answered the question, "Why do living organism respond to their
environment factors?" using pill bugs or Armadillidiidae. Our purpose to answer the question is to
understand the characteristics of a living organism. Specifically, the characteristic of how Living
Things Respond to Their Environment. Our hypotenuses that where that if the pill bug is exposed to
water, then the pill bug will be close to water; If the pill exposed to heat, then the pill bugs will stay
a distance and our third hypotenuse is if the pill bugs is exposed to a stable environment exposed to
nothing, then it will search for a suitable environment. The purpose of using pill bugs is to have a
creature that can be manipulated and record data easily.

Arcsine Equation Lab
Purpose
The purpose of the experiment is to find the existing relationship between the angle of inclination of
a straight ramp, and the acceleration of a ball on the ramp.
Background The angle of inclination, is the angle between the floor, and the elevated ramp. There is
no direct way to measure the angle. The simplest way to obtain the angle of inclination is through
trigonometry; specifically, inverse trigonometric functions. You can find the angle using arcsine.
Arcsine is the inverse sine function, the formula is: 〖Θ=sin〗^(–1) (x). In which, theta represents
the angle of inclination in radians, and x is the sine of the angle. The ratio of sine is:
sin=opposite/hypotenuse. Trigonometry can be applied because the elevation of the ramp ... Show
The hypothesis states the relationship between the angle of inclination and acceleration is linear. The
hypothesis is incorrect. The results indicate the accepted relationship between the variables is sine.
The theoretical function is y=9.8sin⁡
(x). Where the 9.8 m/s2 is the force of gravity. However, a
different function is produced from the lab data. The experimental function is y=3.849
sin⁡
(3.015x+5.071)+3.644. When looking at the graph, the acceleration of the experimental function
is too small compared to the theoretical function. The deviation of the experimental function from
the theoretical function is due to procedural error. One main error that is not addressed in the data
analysis is the reaction time of the experimenters when manually recording the time. The reaction
time is the time it takes for the experimenter to respond to the visual stimuli of the ball then pressing
start or stop on the stopwatch. The reaction time causes the time recorded to be greater than it
actually is. Supposedly, if the time it takes for the ball to travel the 2.2860–meter ramp at a certain
angle, took less time than recorded, because of the elimination of reaction time, that would alter the
average velocity. Average velocity is displacement divided by time. If the average time was smaller,
it would result in a greater average velocity. Then, the faster average velocity is used to find the final
velocity by multiplying it by two. That

The Pythagorean Theorem Was Discovered And First Proven By...
Background:
The Pythagorean Theorem was discovered and first proven by the Greek mathematician,
Pythagoras. The Pythagorean Theorem states that the sum of the squares of the two legs of a right
triangle equals the square of the hypotenuse of the triangle. In simpler words, when looking at the
right triangle below, a²+b²=c². This major discovery in the history of mathematics lead to the
accomplishments of many other basic things we do in life. The Pythagorean Theorem does not just
stop at the famous equation of a²+b²=c², but it has many other aspects. Whole numbers that can fit
into this theorem are commonly referred to as Pythagorean Triples. The Pythagorean Theorem
includes numerous amounts of poofs, from the basic proof to the ... Show more content on
Helpwriting.net ...
Once the concept of the Pythagorean Triples was discovered, mathematicians were eager to validate
the concept. Euclid created a widely accepted formula in deriving a Pythagorean Triple, which can
be proved with the help of the unit circle and algebra.
Euclid's Formula and its Proof
Euclid's formula for a Pythagorean triple is: a = 2mn, b = m^2 – n^2, c = m^2 + n^2
The variables: m and n: stand for positive, rational integers. It is important to remember that m has
to be greater than n, otherwise the statement valuing variable b on a Pythagorean triangle would be
false because b cannot
This can be understood in terms of the rational points on the unit circle. A unit circle is a circle with
the radius of 1.
We can draw a right triangle: a and b are the legs c is the hypotenuse
In the next part, we must use the trigonometric identity of sin2+cos2=1. For representational
purposes, we must look towards the following diagram for how the triangle would be represented in
this proof.
The next step in formulating this next equation would require us to find the sine and cosine of
triangleABC. The sine formula is opposite/hypotenuse, which would equate to a/c. Additionally, the
formula for cosine is adjacent/hypotenuse, equating to b/c. Knowing the trigonometric identity, we
can derive the new eqution:
For the next step, we will need to know the equation of a [unit] circle: x2 + y2 = 1
We know that

Exploring The Role Of Hipparchus In Greek Mythology
Hipparchus was born in 190 B.C. in Nicaea, Bithynia, and died in 120 B.C. Due to this , there is not
a lot information on his early life. He approximately started working as an astronomer in his 30's.
Hipparchus was best known to be an astronomer, aside from that he was also a Greek mathematician
and geographer. Most of the work that he did is now lost , but "Only one work by Hipparchus has
survived, namely Commentary on Aratus and Eudoxus and this is certainly not one of his major
works. It is however important in that it gives us the only source of Hipparchus's own writings".(1)
Nevertheless, most of his research and calculations are found in 'Almagest' by Claudius Ptolemy,
which is how mathematicians can see how he came across the beginnings of trigonometry.
First of all , one of Hipparchus' greatest contributions to mathematics was the creation of the
trigonometric table , which allows to tabulate the values of some or all six of the trigonometric
functions for various angles. Additionally, he was also known for discovering the ... Show more
All of the information regarding mathematics that he already knew was based on Babylonian and
Egyptians. In order to discover the precession of equinoxes, Hipparchus had to first find a way to
figure out the angles and measurements of the moon and sun. Therefore, he created a table of
chords, which later became a trigonometric table, which could be seen one of the first
calculators.When creating the table ,"He considered every triangle as being inscribed in a circle, so
that each side became a chord" , he then originated dividing a circle into 360 degrees, same as
Hypsicle's had done with the ecliptic , and he was the first

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