The Wizard of Oz
Scarecrow’s Speech on Pythagorean Theorem

The Pythagorean theorem is one of the earliest theorems known to ancient civilization. The well-known theorem is named after the Greek mathematician and philosopher, Pythagoras. In the Wizard of Oz, after the Scarecrow gets a brain, he states the Pythagorean theorem. However, he mistakenly says it applies to an isosceles triangle when it applies to a right triangle. He not only says the wrong triangle, he also gets the equation wrong. The Scarecrow says, “The sum of the square root of two sides of an isosceles triangle is equal to the square root of the third side.” The correct equation for the Pythagorean theorem is, “The sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.” The isosceles triangle is a triangle with at least two equal sides; it also has two equal angles. The Pythagorean theorem is a statement about triangles containing a right angle. A right triangle is a triangle with a ninety-degree angle. With the Pythagorean theorem, you take a triangle with a right angle and make a square on each of the three sides; the biggest square has the exact same area as the two other squares put together. A square root of a number is a value that can be multiplied by itself to give the original number. Here is an example of a square root; the square root of nine is three because when three is multiplied by itself you get nine. To square a number, you just THE WIZARD OF OZ3

multiply it by itself, as in the Pythagorean theorem. You can also square negative numbers, when you square a negative number you get a positive answer.
Although the Scarecrow got a brain from the wizard, he didn’t necessarily get the knowledge of having a brain. He messed up the Pythagorean theorem multiple times. He said that it had to do with square roots and isosceles triangles when the correct equation has to do with right...

...In mathematics, the Pythagoreantheorem — 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).
Thetheorem 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 Pythagoreantheorem 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...

...the 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 PythagoreanTheorem 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 Pythagorean triple. There is an infinite number of such triples. Given two positive integers, m and n, where m > n, a Pythagorean triple 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 digits and print it out...

... When using the PythagoreanTheorem, the hypotenuse or its length is often labeled with a lower case c. The legs (or their lengths) are often labeled a and b.
Either of the legs can be considered a base and the other leg would be considered the height (or altitude), because the right angle automatically makes them perpendicular. If the lengths of both the legs are known, then by setting one of these sides as the base ( b ) and the other as the height ( h ), the area of the right triangle is very easy to calculate using this formula:
(1/2)
This is intuitively logical because another congruent right triangle can be placed against it so that the hypotenuses are the same line segment, forming a rectangle with sides having length b and width h. The area of the rectangle is b × h, so either one of the congruent right triangles forming it has an area equal to half of that rectangle.
Right triangles can be neither equilateral, acute, nor obtuse triangles. Isosceles right triangles have two 45° angles as well as the 90° angle. All isosceles right triangles are similar since corresponding angles in isosceles right triangles are equal. If another triangle can be divided into two right triangles, then the area of the triangle may be able to be determined from the sum of the two constituent right triangles. Also the Pythagoreantheorem can be used for non right triangles. a2+b2=c2-2c
The side lengths of a...

...Pythagorean Triples
Tammie Strohl
MAT 126
David Gualco
November 9, 2009
Pythagorean Triples
PythagoreanTheorem states 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.
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If a, b, and c are positive integers, they are together called Pythagorean Triples.
The smallest such Pythagorean Triple is 3, 4 and 5. It can be seen that 32 + 42 = 52 (9+16=25).
Here are some examples:
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Endless
The set of Pythagorean Triples is endless.
It is easy to prove this with the help of the first Pythagorean triple, (3, 4, and 5):
Let n be any integer greater than 1: 3n, 4n and 5n would also be a set of Pythagorean triple. This is true because:
(3n)2 + (4n)2 = (5n)2
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So, you can make infinite triples just using the (3,4,5) triple.
Euclid's Proof that there are Infinitely Many Pythagorean Triples
However, Euclid used a different reasoning to prove the set of Pythagorean triples is unending.
The proof was based on the fact that the difference of the squares of any two consecutive numbers is always an odd number.
For example, 22 - 12 = 4-1 = 3, 152 - 142 = 225-196 = 29.
And also every odd number can be expressed as a difference of the squares of two consecutive numbers. Have a look at this table...

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PythagoreanTheorem
In mathematics, the Pythagoreantheorem 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 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.
These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and about lengths. Tobias Dantzig refers to these as areal and metric interpretations.[2][3] Some proofs of the theorem are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in geometry and the other in algebra, a connection made clear originally byDescartes in his work La Géométrie, and extending today into other branches of mathematics.[4]
The Pythagoreantheorem has been modified to apply outside its original domain. A number...

...PythagoreanTheorem
Diana Lorance
MAT126
Dan Urbanski
March 3, 2013
PythagoreanTheorem
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 Pythagorean triples and asks if you can find more Pythagorean triples than the two that are listed which are (3,4, and 5) and (5,12, and 13) (Bluman, 2012). The Pythagoreantheorem 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 Pythagorean Triples and then verify each of them in the PythagoreanTheorem 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 =...

...PYTHAGOREANTHEOREM
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...

...PythagoreanTheorem:
Some False Proofs
Even smart people make mistakes. Some mistakes are getting published and thus live for posterity to learn from. I'll list below some fallacious proofs of the Pythagoreantheorem that I came across. Some times the errors are subtle and involve circular reasoning or fact misinterpretation. On occasion, a glaring error is committed in logic and leaves one wondering how it could have avoided being noticed by the authors and editors.
Proof 1
One such error appears in the proof X of the collection by B. F. Yanney and J. A. Calderhead (Am Math Monthly, v.3, n. 6/7 (1896), 169-171.)
Suppose the theorem true. Then AB² = AC² + BC², BC² = CD² + BD², and AC² = AD² + CD². Combining the three we get
AB² = AD² + 2CD² + BD².
But CD² = AD·BD. Therefore,
AB² = AD² + 2AD·BD + BD².
From which
AB = AD + BD,
which is true. The supposition is true.
Critique
By the same token, assume 1 = 2. Then, by symmetry, 2 = 1. By Euclid's Second Common Notion, we may add the the two identities side by side: 3 = 3. Which is true, but does not make the assumption(1 = 2) even one bit less false.
As we know, falsity implies anything, truth in particular.
Proof 2
This proof is by E. S. Loomis (Am Math Monthly, v. 8, n. 11 (1901), 233.)
Let ABC be a right triangle whose sides are tangent to the circle O. Since CD = CF, BE = BF, and AE = AD = r = radius of circle, it is easily shown...