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:

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.

The converse of the theorem is also true:
For any three positive numbers a, b, and c such that a2 + b2 = c2, there exists a triangle with sides a, b and c, and every such triangle has a right angle between the sides of lengths a and b. An alternative statement is:

For any triangle with sides a, b, c, if a2 + b2 = c2, then the angle between a and b measures 90°. This converse also appears in Euclid's Elements (Book I, Proposition 48): "If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right."

It can be proven using the law of cosines or as follows:
Let ABC be a triangle with side lengths a, b, and c, with a2 + b2 = c2. Construct a second triangle with sides of length a and b containing a right angle. By the Pythagorean Theorem, it follows that the hypotenuse of this triangle has length c = √a2 + b2, the same as the hypotenuse of the first triangle. Since both triangles' sides are the same lengths a, b and c, the triangles are congruent and must have the same angles. Therefore, the angle...

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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...

...The PythagoreanTheorem 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.
ThePythagoreanTheorem 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 PythagoreanTheorem is a statement about triangles containing a right angle. The PythagoreanTheorem 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...

...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 WIZARD OF OZ 2
The Wizard of Oz
Scarecrow’s Speech on PythagoreanTheorem
The Pythagoreantheorem 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 Pythagoreantheorem. 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 Pythagoreantheorem 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 Pythagoreantheorem is a statement about triangles containing a right angle. A right triangle is a triangle with a ninety-degree angle. With the Pythagoreantheorem, 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...

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PythagoreanTheorem
Pythagoras was born in Samos, Greece around 570 BCE. From there he emigrated to Croton, Italy where most of his most important ideas and theories would develop. Early on, Pythagoras founded a society of disciples where he introduced the idea eternal recurrence into Greek thought, and it was Pythagoras’ ambition to reveal the basis of divine order. This is how Pythagoras came to mathematics, and he saw math as a purifier of the soul, and saw numbers in everything. He was convinced that the divine principles of the universe can be expressed in terms of relationships of numbers.
Over 4000 years ago, the Babylonians 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. Pythagoras 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. This can be written in the form a^2 + b^2 = c^2, and today this is what is known as the PythagoreanTheorem.
The PythagoreanTheorem was one of the first times in human history that people could calculate a length or distance using only outside information. The train of thought used by Pythagoras was the first time the idea of a unset...

... 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...

...PythagoreanTheorem: Proof and Applications
Kamel Al-Khaled & Ameen Alawneh
Department of Mathematics and Statistics, Jordan University of Science and Technology
IRBID 22110, JORDAN
E-mail: kamel@just.edu.jo,
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Idea
Investigate the history of Pythagoras and the PythagoreanTheorem. Also, have the opportunity to
practice applying the PythagoreanTheorem to several problems. Students should analyze information on
the PythagoreanTheorem including not only the meaning and application of the theorem, but also the
proofs.
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1
Motivation
You’re locked out of your house and the only open window is on the second ﬂoor, 25 feet above the ground.
You need to borrow a ladder from one of your neighbors. There’s a bush along the edge of the house, so
you’ll have to place the ladder 10 feet from the house. What length of ladder do you need to reach the
window?
Figure 1: Ladder to reach the window
1
The Tasks:
1. Find out facts about Pythagoras.
2. Demonstrate a proof of the PythagoreanTheorem
3. Use the PythagoreanTheorem to solve problems
4. Create your own real world problem and challenge the class
2
2.1
Presentation:
General
Brief history: Pythagoras lived in the 500’s BC, and was one of the...