1. The Theorem Of Pythagoras Brief description and proof of the Pythagorean theorem by dissection, based onsquares of sum and difference. http://www-spof.gsfc.nasa.gov/stargaze/Spyth.htm

Site Map (M-6) The Theorem of Pythagoras Pythagoras of Samos was a Greek philosopher who lived around 530 BC, mostly in the Greek colony of Crotona in southern Italy. According to tradition he was the first to prove the assertion (theorem) which today bears his name: If a triangle has sides of length ( a,b,c ), with sides ( a,b ) enclosing an angle of 90 degrees ("right angle"), then a + b = c A right angle can be defined here as the angle formed when two straight lines cross each other in such a way that all 4 angles produced are equal. The theorem also works the other way around: if the lengths of the three sides ( a,b,c ) of a triangle satisfy the above relation, then the angle between sides a and b must be of 90 degrees. For instance, a triangle with sides a b c = 5 (inches, feet, meterswhatever) is right-angled, because a + b = 9 + 16 = 25 = c Ancient Egyptian builders may have known the (3,4,5) triangle and used it (with measured rods or strings) to construct right angles; even today builders may still nail together boards of those lengths to help align a corner. Many proofs exist and the easiest ones are probably the ones based on algebra, using the elementary identities discussed in the preceding section, namely

2. Theorem Of Pythagoras Gives a geometric proof of the rule for measuring the sides of a right triangle. http://www.alphaquark.com/Traduction/Pythagore.htm

Pythagore Demonstration of the theorem homepage Source of this page Author : Thibaut Bernard Number of visitor Update: Monday 1th August 2005. Alphaquark author's Note : This page is a translation of with the help of Altavista translation I hope this translation is good, but if there are any errors, you can write me If this translation is successful, perhaps I will try to translate another document of Alphaquark Construction of the geometrical figure which will be used for the demonstration Let us take a rectangle of width A and height B. This rectangle which we make swivel of 90 o in the following way : For each rectangle, let us divide into two in the following way : Let us make swivel of 90 o the right-angled triangles in the following way yellow and purple : We thus find ourselves with four right-angled triangles. We note that one finds oneself with a square inside another. Demonstration Notation Let us take again our last diagram to indicate each sides by the following letters: One has four right-angled triangles of which : the side opposed by a

3. Pythagorean Problem A method of disproving the theorem of pythagoras is presented. The author is adamant that this is intended only as a puzzle to find the mistake in the arguments, and not as a serious proposal. http://www.geocities.com/ResearchTriangle/System/8956/problems/pyth.htm

PLEASE NOTE: The following work is presented as a mathematical puzzle. It is NOT a valid proof, but serves to illustrate the problems that can arise if one is not familiar with postulates and conditions of various theorems. Read it and try to find the problem, but PLEASE do not preach to the world that Pythagoras' Theorem is false. A Disproof of Pythagoras' Theorem The Theorem of Pythagoras In a right triangle, the sum of the squares of the lengths of the two side sides is equal to the square of the hypotenuse. a + b = c DISPROOF: Start by defining a coordinate system with a along the x-axis and b along the y-axis. Let y = f(x) define the hypotenuse. Furthermore define a sequence of functions f n n n (x) converges uniformly to f(x). Clearly the length of the path defined by f (x) is a+b (or length a depending upon exactly how defines the path). Similarly, for any value of n the length of the path defined by f n (x) is also a+b. Since the functions f n (x) converge uniformly to f(x) the length of the path defined by f(x) is a+b.

4. The Pythagorean Theorem Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. Pythagorean Theorem -- From MathWorld Dixon, R. The theorem of pythagoras. §4.1 in Mathographics. Talbot,RF Generalizations of Pythagoras Theorem in n Dimensions. Math. http://mathworld.wolfram.com/PythagoreanTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Geometry Plane Geometry Triangles ... Triangle Properties Pythagorean Theorem For a right triangle with legs and and hypotenuse Many different proofs exist for this most fundamental of all geometric theorems. The theorem can also be generalized from a plane triangle to a trirectangular tetrahedron , in which case it is known as de Gua's theorem . The various proofs of the Pythagorean theorem all seem to require application of some version or consequence of the parallel postulate : proofs by dissection rely on the complementarity of the acute angles of the right triangle, proofs by shearing rely on explicit constructions of parallelograms, proofs by similarity require the existence of non-congruent similar triangles, and so on (S. Brodie). Based on this observation, S. Brodie has shown that the parallel postulate is equivalent to the Pythagorean theorem.

6. Pythagoras' Theorem - By Seth Y-Maxwell Pythagoras was a great Mathematician who was the first to create the music scale NEW ! Pythagorean Theorem Proof Calculator Pythagorean http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. An Interactive Proof Of Pythagoras' Theorem An Interactive Proof of Pythagoras' theorem This Java applet was written by Jim Morey. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

8. Pythagorean Theorem move Drag the central red point. Reference "Pythagoras Theorem" OYA, Shinichi, 1975, Tokai University Press. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

9. Pythagoras [Internet Encyclopedia Of Philosophy] Pythagoras (fl. 530 BCE.) problem at this date was the duplication of the square, a problem which gave rise to the theorem of the square on the http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

10. Pythagoras' Playground (by The Event Inventor) site, we will use the amazing properties of the Pythagorean Theorem to explore the world around us. 2500 years ago, Pythagoras of Samos and his http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

11. NOVA Online The Proof Pythagorean Puzzle Theorem Demonstrate the Pythagorean Theorem. Think of each side of a right triangle as also being a side of a square that's attached to the triangle. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

12. Pythagorean Theorem The Pythagorean Theorem. Pythagoras, for whom the famous theorem is named, lived during the 6th century B.C. on the island of Samos in the Aegean http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

13. Coordinates--lesson Plan #9 The theorem of pythagoras, for instance, was proved by a method using not numbersbut triangles, rectangles and squares. Even today students in schools http://www-spof.gsfc.nasa.gov/stargaze/Lcelcoor.htm

Lesson Plan #9 http://www.phy6.org/Stargaze/Lcelcoor.htm (5c) Coordinates Part of a high school course on astronomy, Newtonian mechanics and spaceflight by David P. Stern This lesson plan supplements: Scelcoor.htm , on the web http://www.phy6.org/stargaze/Scelcoor.htm home page and index: on disk Sintro.htm , on the web http://www.phy6.org/stargaze/Sintro.htm Goals : The student will Learn to use cartesian coordinates (x,y) for defining the position of a point in 2 dimensions. Learn to use cartesian coordinates (x,y,z) in 3-dimensional space. [Optional: learn to appreciate there exist two ways of defining the z axis, and which of them is used.] Become familiar with some of the tools and terms used by surveyors: theodolite, azimuth, elevation, zenith, [nadir]. Optional items below are for students familiar with trigonometry and with the theorem of Pythagoras (areas of math also covered by web files linked to httm://www.phy6.org/stargaze/Smath Terms: Cartesian coordinates, axes, origin (of coordinates) [polar coordinates] Theodolite, azimuth, elevation, zenith, [nadir]. Stories and extras: Guiding questions and additional tidbits (Suggested answers in parentheses, brackets for comments by the teacher or "optional")

14. Project MATHEMATICS!Theorem Of Pythagoras The theorem of pythagoras http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

15. Theorem Of Pythagoras Proof of Pythagoras Theorem. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

16. Project MATHEMATICS!--Theorem Of Pythagoras Discovering the theorem of pythagoras. 3. Geometric interpretation. 4. Pythagoras.5. Applying the theorem of pythagoras. 6. Pythagorean triples http://www.projectmathematics.com/pythag.htm

The Theorem of Pythagoras Video Segments Three questions from real life Discovering the Theorem of Pythagoras Geometric interpretation Pythagoras Applying the Theorem of Pythagoras Pythagorean triples The Chinese proof Euclid's elements Euclid's proof A dissection proof Euclid's Book VI, Proposition 31 The Pythagorean Theorem in 3D Contents The program begins with three real-life situations that lead to the same mathematical problem: Find the length of one side of a right triangle if the lengths of the other two sides are known. The problem is solved by a simple computer-animated derivation of the Pythagorean theorem (based on similar triangles): In any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. The algebraic formula a + b = c is interpreted geometrically in terms of areas of squares, and is then used to solve the three real-life problems posed earlier. Historical context is provided through stills showing Babylonian clay tablets and various editions of Euclid's Elements . Several different computer-animated proofs of the Pythagorean theorem are presented, and the theorem is extended to 3-space.

17. The Theorem Of Pythagoras The theorem of pythagoras is one of the earliest and most important results theorem of pythagoras. Given any right angle triangle, if one forms a square http://www.math.uwaterloo.ca/navigation/ideas/grains/pythagoras.shtml

UWdir Math Google Search Math Index Search Skip to the content of the web site. Math Home ... UW Home Pythagoras (fl. 500 BCE) The theorem of Pythagoras is one of the earliest and most important results in the history of mathematics. It has immense practical value and led to the discovery of irrational numbers - a right triangle with unit sides leads via Pythagoras to the square root of 2! For further history: St. Andrews' history of Mathematics site Theorem of Pythagoras Given any right angle triangle, if one forms a square on each side of that triangle then the area of the largest square (that of the hypoteneuse) is equal to the sum of the areas of the two smaller squares (those which are formed on the sides about the right or 90 degree angle). Proof of the theorem is demonstrated through the following Quicktime animation. Use the controls to animate the movie. Note that the area of a given colour remains the same in the animation - no matter how the shape of the figure changes! Notes on the demonstration: Textual details of the proof are intentionally absent from the movie. This encourages the student to work through why this is in fact a proof and how they might produce a formal proof based on the demonstration.

18. The Theorem Of Pythagoras ... Key To Proof The key to the proof of the theorem of pythagoras. The theorem states that thearea of the large white square (square of the hypoteneuse) in the following http://www.math.uwaterloo.ca/navigation/ideas/grains/pythagoras-key.shtml

UWdir Math Google Search Math Index Search Skip to the content of the web site. Math Home ... UW Home The key to the proof of the theorem of Pythagoras The theorem states that the area of the large white square (square of the hypoteneuse) in the following diagram is equal to the sum of the areas of the two squares (of the two sides) in the next diagram It's important to notice that the orange triangle in both pictures is the same one, just rotated to better show the squares (and to match the animation). The proof begins by changing the solid white square to a blue one outlined in white. Then the same orange triangle is placed at each side of the square. The inside edges of the four triangles form the hypoteneus square. The outside edges of these four triangles form a large outer square. The outer square has fixed area equal to the area of the blue hypoteneuse square plus the areas of all four orange triangles. As the orange triangles move about in the animation, the outer square is preserved and consequently, the area of the outer square does not change.

19. NTNU JAVA :: View Topic - Theorem Of Pythagoras A*a + B*b = C*c How ancient Chinese people discovers the same theorem. (much earlier than Pythagoras).You can change the interval delta T (in second, default value = 2 http://www.phy.ntnu.edu.tw/~hwang/abc/Pythagoras.html

NTNU JAVA Virtual Physics Laboratory Fu-Kwun Hwang FAQ Search Memberlist Usergroups Register Profile Log in to check your private messages Log in Theorem of Pythagoras a*a + b*b = c*c NTNU JAVA Physics simulations Misc Registered user will be able to get files related to java applets for offline use View previous topic View next topic hwang Site Admin Joined: 28 Jan 2004 Posts: 257 Posted: Thu Jan 29, 2004 4:11 pm Post subject: Theorem of Pythagoras a*a + b*b = c*c This java applet shows you (automatically - step by step) How ancient Chinese people discovers the same theorem. You can change the interval delta T (in second, default value = 2 second). Click mouse button for manual control mode : Click right mouse button : show the following step Click left mouse button : show the previous step When you reach the last step, Press reset button to restart Registed user can get files related to this applet for offline access. Back to top Display posts from previous: All Posts 1 Day 7 Days 2 Weeks 1 Month 3 Months 6 Months 1 Year Oldest First Newest First NTNU JAVA Physics simulations Misc All times are GMT Page of Jump to: Select a forum Physics simulations kinematics Dynamics Wave Thermodynamics Electromagnetics Optics Misc Comment about NTNUJAVA Mirror sites Molecular Workbench Easy java simulations Information and Download Optics Kinematics Dynamics Wave Thermodynamics Modern Physics misc You cannot post new topics in this forum You cannot reply to topics in this forum You cannot edit your posts in this forum You cannot delete your posts in this forum

20. Keymath.com : Discovering Geometry : JavaSketchpad™ Activities : The Theorem Of The theorem of pythagoras. The Pythagorean Theorem relates the lengths of thethree sides of a right triangle. In the investigation on this web page, http://www.keymath.com/DG/dynamic/pythagorean_theorem.html

Home Discovering Geometry Dynamic Geometry Explorations : The Theorem of Pythagoras Dynamic Geometry Exploration The Theorem of Pythagoras The Pythagorean Theorem relates the lengths of the three sides of a right triangle. In the investigation on this web page, you will learn more about the Pythagorean Theorem and see whether or not it works for triangles that are not right triangles. This investigation will help you understand Lesson 9.1 on pages 462-464 of Discovering Geometry: An Investigative Approach. Sketch The sketch below shows a right triangle with squares constructed on its three sides. You can drag vertex A to change the size and shape, but it will remain a right triangle. Sorry, this page requires a Java-compatible web browser. Investigate Click "Construct Center of Square" to show center O of the square on the longer leg. The center of a square is located at the intersection of the diagonals. Click "Construct j and k" to show two lines through O : line j perpendicular to the hypotenuse and line k perpendicular to line j . Lines j and k divide the square on the longer leg into four quadrilaterals. Click "Construct Quadrilaterals" to show the quadrilaterals and the square on the shorter leg.

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