Pythagorean Triples
To begin you must understand the Pythagoras theorem is an equation of a2 + b2 = c2. This simply means that the sum of the areas of the two squares formed along the two small sides of a right angled triangle equals the area of the square formed along the longest. Let a, b, and c be the three sides of a right angled triangle. To define, a right angled triangle is a triangle in which any one of the angles is equal to 90 degrees. The longest side of the right angled triangle is called the 'hypotenuse'. Once you have this basic understanding you can apply the understanding that if a, b, and c are positive integers, they are called Pythagorean Triples. Our textbook explains that “the numbers 3, 4, and 5 are called Pythagorean triples since 32 + 42 = 52” (Bluman, pg. 522) and that there is a set of formulas that will generate an infinite number of Pythagorean triples. Here are some examples:

| | |
3,4,5 Triangle 5,12,13 Triangle 9,40,41 Triangle| | | 32 + 42 = 52 52 + 122 = 132 92 + 402 = 412(Ganesh, 2010.)| | | The set of Pythagorean Triples is endless. Let n be any integer greater than 1 as in this example which is a set of Pythagorean triple, is true because: (3n)2 + (4n)2 = (5n)2 Some other Pythagorean Triples are (5,12,13); (7,24,25); (8,15,17); (9,40,41); (17,144,145); and (25,312,313).| | | To summarize, it is fairly easy to show that no matter what numbers you use for n you can use this method to show that there are infinitely many Pythagorean Triples. Since the Pythagorean Theorem states that every right triangle has side lengths satisfying the formula a2 + b2 = c2; the Pythagorean triples describe the three integer side lengths of a right triangle.

References:
Bluman, A. G. (2005). Mathematics in Our World (1st Ed.) Ashford University, Custom Edition. New York: McGraw-Hill. Ganesh, J. (2010). Pythagorean Triples...

...PythagoreanTriples
Ashley Walker
MAT126
Bridget Simmons
November 28, 2011
A Pythagoreantriple is a triple of positive integers a, b, and c such that a right triangle exists with legs a, b, and hypotenuse c (Bluman, 2005). A Pythagoreantriple is a triple of positive integers (a, b, c) where a2 + b2 = c2. A triple is simply a right triangle whose sides are positive integers. An easy way to generate Pythagoreantriples is to multiply any known Pythagoreantriple by an integer (any integer) (Vargas, 2008).
In project #4, pg. 522, (Mathematics in Our World) introduced some new information to add to my mathematics knowledge of numbers. The numbers 3, 4, and 5 are called Pythagoreantriples since 32 + 42 = 52. The numbers 5, 12, and 13 are also Pythagoreantriples since 52 + 122 = 132. Can you find any other Pythagoreantriples? Actually, there is a set of formulas that will generate an infinite number of Pythagoreantriples and write a brief report on the subject (Bluman, 2005).
When asked to find any other Pythagoreantriples, I found 5, choosing 2 integers, m and n, with m less than n (Manuel, 2010). Three formulas I choose to form the...

...Pythagorean Theorem
Diana Lorance
MAT126
Dan Urbanski
March 3, 2013
Pythagorean Theorem
In this paper we are going to look at a problem that can be seen in the “Projects” section on page 620 of the Math in our World text. The problem discusses Pythagoreantriples and asks if you can find more Pythagoreantriples than the two that are listed which are (3,4, and 5) and (5,12, and 13) (Bluman, 2012). The Pythagorean theorem states that for any right triangle, the sum of the squares of the length of the sides of the triangle is equal to the square of the length of the side opposite of the right angle (hypotenuse) and can be shown as a² + b² = c² (Bluman, 2012). We will be using a formula to find five more PythagoreanTriples and then verify each of them in the Pythagorean Theorem equation.
The formula that I have decided to illustrate is (2m)² + (m2 - 1)² = (m2 + 1)² where m is any natural number, this formula is attributed to Plato (c. 380 B. C.) (Edenfield, 1997). A natural number is any number starting from one that is not a fraction or negative (MathIsFun, 2011). The triples will be the square roots of each part of the equation. We will test this formula with the natural numbers of 5, 8, and 10. When we use 5 the formula looks like this: (2x5)² + (5² - 1)² = (5² + 1)², 10² + (25 – 1)² = (25 + 1)², 100 + 24² =...

...Anmol Mehrotra
Pythagoreantriples
Math Bonus
A
Pythagoreantriple
consists of three positive
integers
a
,
b
, and
c
, such
2
2
2
that
a
+
b
=
c
. Such a triple is commonly written (
a
,
b
,
c
), and a
wellknown example is (3, 4, 5). If (
a
,
b
,
c
) is a Pythagoreantriple, then so
is (
ka
,
kb
,
kc
) for any positive integer
k
. A
primitive Pythagoreantriple
is
one in which
a
,
b
and
c
are
coprime
. A right triangle whose sides form a
Pythagoreantriple is called a
Pythagorean triangle
.
The name is derived from the
Pythagorean theorem
, stating that every
right triangle
has side lengths satisfying the formula
a2
+
b2
=
c2
; thus,
Pythagoreantriples describe the three integer side lengths of a right
triangle. However, right triangles with noninteger sides do not form
Pythagoreantriples. For instance, the
triangle
with sides
a
=
b
= 1 and
c
=
√2 is right, but (1, 1, √2) is not a Pythagoreantriple because √2 is not an
integer. Moreover, 1 and √2 do not have an integer common multiple
because √2 is
irrational
.
Examples[...

...The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students.
The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea.
The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that:
"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."
Figure 1
According to the Pythagorean Theorem, the sum of the areas of the two red squares, squares A and...

...square root of their product. The harmonic mean of two numbers is the arithmetic mean of their reciprocals. Write a program that takes two floating-point numbers as inputs and displays these three means.
7. Write a C++ program to take a depth (in kilometers) inside the earth as input data; compute and display the temperature at that depth in degrees Celsius and Fahrenheit. The relevant formulas are:
Celsius = 10 x (depth) + 20 (Celsius temperature at depth in km)
Farhrenheit = 1.8 x (Celsius) + 32
8. The Pythagorean Theorem states that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. For example, if two sides of a right triangle have lengths 3 and 4, then the hypotenuse must have a length of 5. The integers 3, 4, and 5 together form a Pythagoreantriple. There is an infinite number of such triples. Given two positive integers, m and n, where m > n, a Pythagoreantriple can be generated by the following formulas:
Side1 = m2- n2
Side2 = 2mn
Hypotenuse = side12 + side22
Group 2
1. In a laboratory, the time of an experiment is measured in seconds. Write a C++ program to enter the time in seconds, convert and print out it as a number of hours, minutes and seconds. Use appropriate format for the output.
2. Write a C++ program to enter an integer number of 4...

...Algebra Week 5 Assignment: Pythagorean Quadratic
Algebra has been around for many years. Some people feel there is no use for algebra in the real world. There are those who feel it is a waste of time because only certain professions really use it. However, algebra is used in many ways which can relate to everyday situations. Algebra truly is a part of our everyday life and there really is no escaping it. This week, we are asked to solve a problem using thePythagorean Theorem. The Pythagorean Theorem is a relation in geometry among the three sides of a right triangle. The Theorem reveals that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This information can provide useful information and save lots of time and aggravation. People in the construction business find this information quite helpful because it allows them to build a sound structure. Architects and those in the engineering profession also need this information and apply it to their everyday tasks. All three of these situations really do occur daily for many people and it is extremely important to be able to figure these things out.
On page 371 of our text, Elementary and Intermediate Algebra, question #98 tells us, “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...

...PYTHAGOREAN THEOREM
More than 4000 years ago, the Babyloneans and the Chinese already knew that a triangle with the sides of 3, 4 and 5 must be a right triangle. They used this knowledge to construct right angles. By dividing a string into twelve equal pieces and then laying it into a triangle so that one side is three, the second side four and the last side five sections long, they could easily construct a right angle.
A Greek scholar named Pythagoras, who lived around 500 BC, was also fascinated by triangles with these special side ratios. He studied them a bit closer and found that the two shorter sides of the triangles squared and then added together, equal exactly the square of the longest side. And he proved that this doesn't only work for the special triangles, but for any right triangle. Today we would write it somehow like this: a2 + b2= c2. In the time of Pythagoras they didn't use letters yet to replace variables. (They weren't introduced until the 16th century by Vieta.) Instead they wrote down everything in words, like this: if you have a right triangle, the squares of the two sides adjacent to the right angle will always be equal to the square of the longest side.
We can't be sure if Pythagoras really was the first person to have found this relationship between the sides of right triangles, since no texts written by him were found. In fact, we can't even prove the guy lived. But the theorem a2 + b2= c2 got his name. Another Greek, Euclid,...

...In mathematics, the Pythagorean theorem — or Pythagoras' theorem — is a relation in Euclidean 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 Pythagorean theorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its discovery and proof,[2][3] although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they used it in a mathematical framework.[4][5]
The theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed,...