Detailed Analysis: Pythagorean Theorem, Methods, and Example Report

This report provides a detailed examination of the Pythagorean Theorem, a fundamental concept in Euclidean geometry. It begins with an introduction to the theorem, including its historical context and the relationship between the sides of a right triangle. The report then delves into the proof of the theorem, using geometric diagrams and algebraic methods to demonstrate its validity. Following the proof, the report explores various methods for generating Pythagorean triples, including the two-unit-fraction method, the two-fractions method, and the m, n formula. Finally, the report includes example problems to illustrate the application of the theorem. The report concludes with a summary of the key concepts and a list of references. This assignment aims to provide a comprehensive understanding of the theorem, its proof, and its practical applications.

Mathematical work
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Table of Contents
INTRODUCTION...........................................................................................................................3
EVALUATION OF PYTHAGORAS TRIPLE THEOREM...........................................................3
STATEMENT OF PYTHAGORAS THEOREM...........................................................................3
PROOF OF PYTHAGOREAN THEOREM...................................................................................4
METHODS RELATED TO GENERATING PYTHAGOREAN TRIPLES..................................7
EXAMPLE RELATED TO PYTHAGOREAN THEOREM..........................................................9
CONCLUSION..............................................................................................................................17
REFERENCES..............................................................................................................................18
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INTRODUCTION
According to detailed analysis, it can be said that Pythagorean Theorem is considered as
one of most critical aspect in the mathematics. It has been noticed that key fundamental relation
in context to euclidean geometry need to be consider as critical aspect. In addition to this, it can
be said that theorem is greatly dependent upon three values. The theorem can be written as an
equation relating the lengths of the sides a, b and c, often called the Pythagorean Triples
(Lipowsky, 2009). It has been noticed that Pythagoras theorem was initially taken into account in
for better understanding of diagonals. Along with this, the right angle aspects within triangle can
be explored in critical manner for sustainable development under the following theory.
EVALUATION OF PYTHAGORAS TRIPLE THEOREM
Pythagorean was a mathematician who was born in Greece in 570 BC. Theorem was
mainly designed in order to understand the concept of triangles. He was one of the first Greek
mathematical thinkers. The statement related to the following theorem was invented on a
Babylonian tablet circa 1900− 1600 B.C. In 5000 B.C, Professor R. Smullyan had revealed some
important some important things about Pythagorean Theorem in his book. Another Philosophical
Fantasies was also discussed about his experiment. In his geometry classes, he drew a right
triangle on the board with squares on the hypotenuse and legs. On the basis of this, he was
observed that the square on the hypotenuse had a larger than either of the other two squares.
STATEMENT OF PYTHAGORAS THEOREM
The famous theorem given by Pythagoras states the relationship between the three sides
of a right triangle. As per the theory, the sum of the squares of the two right angle sides will
always be the same as the square of the long side that is called the hypotenuse. In symbolic way,
it can be represent as A2+B2=C2
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It has been noticed that if triangle has one reflection which is counted at the scale of right
angle (90 degree angle). It means the triangle is indication the specific relationship between
lengths of all sides. By having improved focus on the theorem the measurement of three sides of
triangles can be accomplished in desired manner. As same if the longest side of triangle is
referred as hypothesis (Reddy, 2014). Mainly is it reflected as a h so that side can be recognised
effectively as compared to other two sides. Remaining two sides of right angle are considered as
side a and b. In order to have better understanding about the concept and effective application of
theorem the formula need to be covered. For example, a2 + b2 = h2.
PROOF OF PYTHAGOREAN THEOREM
As per the above figure, the area of the inner square is CxC or C2 and area of the outer
square is as follows:
The area of the outer square = The area of inner square + The sum of the areas of the four right
triangles around the inner square.
On the basis of this, the area of the outer square is
A2 + B2 + 2AB = C2 + 41/2 AB, or A2 + B2 = C2.
With an assistance of the formula the key values can be employed in desired manner so
that measurement of all sides can be taken into account. In addition to this, it can be said that the
square of the longest side will remain same as per reference to sum of square of other two sides
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(Serra, 2009). For example, 1+1=2 can be considered as an outcome of theorem. It means the
square of side a is 1 and side b square is also 1 then the overall outcome will be 2. It reflects that
the square of h will be 2. It has been noticed that such kind of theorem is mainly applied at the
situation where the right angles' triangle can be accomplished.
It has been witnessed that if all the angles of a triangle are less than 90° then the overall
circumstances created will be referred as h^< a^ + b^. Along with this, it also reflects that a^ and
b^ is greater than the h^. With an assistance of this, overall measurement of triangle can be taken
into account. In order to have better understanding about the subject the application of example
is need to be referred effectively (Okun, 2008). It has been noticed that equilateral triangle holds
the equal measurement of all sides and the degree of angle will be 60 degree. In this the
measurement can be 1 of all the sides. On the other side, if one of the angles associated with the
triangle is calculated at the scale of more than 90 degree. It means the calculation of h^ is greater
as compared to total of a^ and b^. According to the theorem the angles of all triangles cannot be
considered as equalised in general manner (Moutsios-Rentzos, Spyrou and Peteinara, 2014).
Moreover, sum of the longest side cannot be high as compared to sum of the other sides. It has
been noticed that sum of the longest side can be equal to the sum of the other sides.
Few of visual proofs are being present in respect to the Pythagoras triple theorem. It has
been noticed that both diagrams reflect the same size of square which is hold by the sides a and
b. It means the total also provides a same size of square. Along with this, it can be said that both
squares are also being contained of same four identical right angles' triangle which is highlighted
in the white section of the diagram (Kalanov, 2013). Along with this, the situation of side a, b
and c also need to be evaluate properly so that measurement can be taken into account
effectively. In addition to this, it can be stated that the left square also has two blue squares with
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areas a^ and b^ whereas the right hand one replaces them with one red square of area c^. It has
been noticed that it cannot be dependent over the lengths of a,b,c and the sides' dependency is
not associated the right angle triangle. When it is focused on the red box, it has been witnessed
that two blue squares are also equal. It means the measurement of a and b is equal to the h side so
that theorem can provide better understanding about the angle (Reid and Knipping, 2010). This
makes an effective visual aid by pushing the squares from their locations on the left to where
they are shown on the right. Don't turn them or flip them, just move them to their respective
corners. Moreover, the proper consideration of the squares is also significant according to
theorem because all sides of triangle is equal.
According to above diagram, it can be said that there are two triangles which are opposite
to each other. But when it is combined the square is being generated effectively. Square all sides
can also be measured at the scale of same so evaluation of sides is significant. In addition to this,
it can be said that largest square that split in the half with a help of the diagonals so that
measurement can be taken into account effectively (Kalanov, 2013). A base of the square is also
focused towards determination of the vertices so that internal square measurement can be taken
into account. Along with this, two flexible strings which are connected to the vertices of internal
square need to be analysed effectively. By having a proper comparison between all the sides of
two triangles the each string of triangle can be analysed in appropriate manner. It also reflects the
perpendicular in the square so that goals and objectives can be accomplished in desired manner.
It also reflects that the overall sum of the original internal square so that goals and objectives can
be accomplished in desired manner (Okun, 2008).
Along with this, the overall objective in context to the 3-4-5 triangle also needs to be
referred effectively in the theorem. It has been noticed that the perhaps the surprising the some
tight angles triangles the sides can be measured effectively (Moutsios-Rentzos, Spyrou and
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Peteinara, 2014). Along with this, the three sides of triangle understanding need to accomplished
according to lengths. In addition to this, it can be said that a=3, b=4 and h=5. It means the
triangle can be referred as 3-4-5 triangle. As per the consideration of the theorem, 3^+4^= 25 can
be measured. Square of 3 is 9 and the square of 4 is 16.
METHODS RELATED TO GENERATING PYTHAGOREAN TRIPLES
There are some methods of generating Pythagorean Triples which are as follows:
A simple two-unit-fraction method of generating PTs: It is a very simple and effective method
generating Pythagorean Triples. This is based on developing the sum of two units fractions of
consecutive numbers either they are even or odd in nature (Conte and Rosca, 2015).
Two consecutive odd unit fractions: Under this, two odd numbers are taking into the accounts
where the difference between them is 2 such as instance 3 and 5. By making their unit fraction
and adding them, the generated outcome will always be a sum of two sides of a primitive
Pythagorean Triples. Example related to the following one is as follows:
Odd A Next B = A+2 1/A+1/B Hyp
1 3 4/3 5
3 5 8/15 17
5 7 12/35 37
7 9 16/63 65
9 11 20/99 101
11 13 24/143 145
Two consecutive even unit fractions: In this, two even numbers consider where the difference
between them is 2 such as 2 and 4. At the time of calculating primitive Pythagorean triangle,
consider number convert into unit fraction and add them respectively. This concept can be
understood by an example which is as follows:
Even A Next B = A+2 1/A+1/B Hyp
2 4 3/4 5
4 6 5/12 13
6 8 7/24 25
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8 10 9/40 41
10 12 13/60 61
12 14 13/168 169
The Two-Fractions method of generating Pythagorean Triples: It is a very simple method to
generate various Pythagorean triangle. Under this, two numbers consider whose products are 2
and their fraction doing not have to be in their lowest form (Ger, 2013). After this, the fraction
value adds with each other and crosses multiply to transform them into whole numbers. Some
relevant examples of this method are as follows:
Factor one Factor two Pythagorean triangle
1 2 3 4 5
2/2 = 1 4/2 = 2 6 8 10
2/4 = 1/2 8/2 = 4 5 12 13
3/3 = 1 6/3 = 2 9 12 15
2/3 3 8 15 17
4/4 or 2/2 = 1 4/2 or 8/4 = 2 12 16 20
1/3 6 7 24 25
3/2 4/3 20 21 29
1/4 8 9 40 41
The m, n formula for generating Pythagorean Triples: With the help of two different positive
integer values, Pythagorean Triples can be generated. Once the finding of one triple, other ones
can be generated by just scaling up all the sides by the similar factors. When a Pythagorean triple
that is not a multiple of another one then it is called a primitive Pythagorean triple. For example,
3, 4, 5 and 5, 12, 13 are primitive Pythagorean triples but 6, 8, 10; 333,444,555 and 50,120,130
are not coming in this category because they are multiple of another Pythagorean triples. The
formula for generating this is as follows:
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(m2 – n2)2 + (2 m n) 2 = (m2 + n2)2
EXAMPLE RELATED TO PYTHAGOREAN THEOREM
Question 1. Use the Pythagorean Theorem to determine the length of X
8cm X= ?
6cm
Step 1:
Determining the legs and the hypotenuse of the right triangle in which value of legs has 6 and 8
and X is the hypotenuse
Step 2:
Substitute values into the formula
(a2+b2) = X
82+62 = X
X = 10cm
(Calculation also attached in Excel sheet)
Question 2: Use the Pythagorean Theorem to determine the length of Y.
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9cm 15cm
Y=?
Step 1:
Determining the legs and the hypotenuse of the right triangle in which value of legs has 9 and Y
and 15 is the hypotenuse
Step 2:
Substitute values into the formula
(a2+b2) = 15
92+Y2 = 15
92+Y2 = 152
Y2 = 152-92
Y = 152- 92
Y = 12cm
(Calculation also attached in Excel sheet)
Question 3: Is a triangle with sides’ lengths of 4 cm, 7 cm and 8 cm a right triangle? If it is
a right triangle, then the sum of the squares of the two smaller sides will equal the square
of the largest side.
By using Pythagorean Theorem, the calculation for the following question is as follows:
4cm 8cm
7 cm
42+72= 82
16+49 = 64
65cm ≠ 64cm
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(Calculation also attached in Excel sheet)
Apart from the above stated examples of Pythagorean Triples, some other examples can
be taken into the consideration which is as follows:
Real World Pythagorean Theorem Problems
Question 1: Sujena runs diagonally across a rectangular field that has a length of 36 meter and a
width of 18 meter. What is the length of the diagonal that Sujena runs? Round the answer nearest
whole number.
Solution: By understanding the above question, a relevant diagram for this is as follows:
On the basis of the above diagram, a rectangular field has a length of 36 meter and a
width of 18 meter. By applying Pythagorean Theorem, the length of the diagonal that Sujena run
is:
(BD)2 = (BC)2 + (CD)2
(BD)2 = (36)2 + (18)2
(BD)2 = 1620
BD = 40.24 = 40 meter
Therefore, the length of the diagonal that Sujena runs is 40 metres.
(Calculation also attached in Excel sheet)
Pythagorean Theorem Word Problems
Question 2: Jon leaves home to go school. He walks 9 blocks North and then 12 blocks west.
How far is he from the home?
Solution: From the given problem, the distance between home to school is the length of
hypotenuse. So, it can be assume that a = 9 blocks and b = 12 blocks and needed to determine the
value of c. By applying Pythagoras Theorem i.e. c2 = a2 + b2
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c2 = 92+122
c2 = 81+144
c2 = 225
c = 15
Therefore the distance from home to school is 15 block.
(Calculation also attached in Excel sheet)
Pythagorean Theorem Story Problems
Question 3: Peter has let out 50 metre of kite string when he observes that his kite is directly
above a point on the ground 30 meters away from him. Find out how high is the kite in the sky
from the ground?
Solution: By observing the question, it has determined that Peter has let out the string of kite
about to 50 meter and the distance between point A and him is 30 meter. The height of kite from
the ground point is as follows:
(50)2 = (30)2 + (h) 2
h2 = 2500-900
h2 = 1600
h = 40 meter \
From the above calculation, the height of Peter kite from ground has 40 meter.
(Calculation also attached in Excel sheet)
Question 4: A ladder has placed on the wall with the distance of 35 feet to the wall. Be
measuring the distance from top of the wall to the ground straight is 60 feet. Determine whether
the ladder reaches at top of the wall or not?
Solution:
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From the above diagram, it has found that the length of the ladder is 50 feet and the
distance from the wall to the bottom of the ladder is 35 feet. On the basis of this, by applying
Pythagorean Theorem on the case:
502 = 352+h2
2500 = 1225+h2
h2 = 1275
h = 35.70 feet
The ladder will never reach on the top of the wall. It will only reach at 35.70 feet. The
reason is the height of top of the building is from ground is 60 feet.
(Calculation also attached in Excel sheet)
Question 5: A person is walking 100 meter from X position situated in the north to B position in
east direction and then move to the west of Y to reach at final destination point Z. This position
is situating in the north of X with the distance of 60 meter. Determine the distance between X
and Y.
Solution:
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From the above figure, it assume that XY = xm and YZ = (100-x)m. In triangle XYZ, Z
is 90 degree angle. On this, by applying Pythagoras theorem
XY2 = YZ2 + XZ2
x2 = (100 – x)2 + 602
x2 = 10000 –200x + x2 + 36002
200x = 10000+3600
200x = 13600
x = 68 meter
Therefore, the distance between point X and Y is 65 meters.
Question 6: Determine the value of perimeter of a rectangle whose length is 80 meter and
diagonal is 100 meter
Solution: By using Pythagoras theorem the value of another leg of rectangle is as follows:
802+x2 = 1002
10000-6400 = x2
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x2 = 3600
x = 60 meter
Therefore, perimeter of the rectangle WXYZ = 2 (length + width)
= 2 (80+60) meter
= 2*140 meter
Therefore, perimeter of the rectangle WXYZ = 280 meter
(Calculation also attached in Excel sheet)
Question 7: The heights of two buildings are respectively 29 meter and 34 meter. If the distance
between both of the buildings is 12 meter and determine what will be distance between their
tops?
Solution:
From the above draw diagram, the height of two vertical buildings AB and CD are 34
meter and 29 meter. On the basis of the given question, it has determined that DE is
perpendicular on AB. So AE = AB-EB. Therefore,
AE = 34 meter – 29 meter
AE = 5 meter
Now, the figure shows AED is the right angle triangle and value of ED is equal to BC.
So, value of AD with the help of Pythagoras theorem is as follows:
AD2 = AE2 + ED2
AD2 = 52 + 122
AD2 = 25 + 144
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AD2 = 169
AD = √169
AD = 13 meter
(Calculation also attached in Excel sheet)
Question 8: Marry wants to plant a rose garden in her farmhouse. For this, she builds a
triangular flowerbed with the help of landscaping timbers. The values of two legs of that
triangular flowerbed have 8 feet and 6 feet. Determine how many feet of landscaping timers does
Marry need to buy for this.
Solution:
From the above figure, it has clearly shown that values of two legs of that triangular
flowerbed have 8 feet and 6 feet. By applying Pythagorean Theorem,
Y2 = 82+62
Y2 = 64+36
Y2 = 100
Y = 10 feet
Therefore, 10 feet of landscaping timers does Marry need to buy for build a triangular flowerbed.
(Calculation also attached in Excel sheet)
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CONCLUSION
From the above research, it can be concluded that Pythagorean Theorem has frequently
used in maths to make the calculation more advance and accurate. The application of the
following theory has included that calculating the distance between points on a plane,
transformed between polar and rectangular coordinates, figure out perimeters, surface areas and
volumes of various geometric shapes and calculating maxima and minima of perimeters, or
surface areas and volumes of various geometric shapes. In daily life, Pythagorean Theorem has
used in several modes such as find out the view of size if a TV that can be considered as a most
purchasing decision for the customers. Along with this, it has used in the construction industry to
determine the lay the foundation for the corners of a building.
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