What are some Pythagorean Triples?
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Let's say at the outset that non-primitive triples (eg (6, 8, 10) and (9, 12, 15) are not particularly interesting. What are interesting are primitive ones eg (3, 4, 5) in which the 3 members of the triple have no common factors. After all, any fool can multiply a primitive triple by 2, 3, 4 etc, what we are after when we go triple-hunting are primitive ones. \n\nAnd what we really want is a method of generating triples, so that gaving found some, we can make more. Preferably a method that only generates primitive triples, so we don't have to exercise quality control and weed out non-primitive ones afterwards.\n\nAnd like the Spice Girls, what we really, really want is a method that generates triples from triples not from generating numbers that don't have any aparent connection with triples being plugged into a formula. Preferably a method that can generate all possible primitive triples just from (3, 4, 5), knowing only that one and no other. That would be so much more elegant and satisfying, especially if some findamental structure and relationships between triples were thereby revealed.\n\nThere are an infinite number of triples. So we want a method that generates the lot, not just a subset of them.\n\nAnd triples have many unusual and unexpected properties, making them rewarding to study, if you like finding patterns in nature. This can create a sense of awe and wonder and beauty, in addition to the aesthetic pleasure of elegance of solution.\n\n====\n\nAt the age of 5 I discovered two simple rules for generating primitive triples. As I recall, I only knew (3, 4, 5) and wondered if there were any more like that. Curiosity got me started. I knew no algebra at five.\n\n(1) take any odd number, square it, halve that, take the numbers either side.\n\nThus 3 => 9 => (3, 4, 5)\n5 => 25 => (5, 12, 13)\n7 => 49 => (7, 24, 25)\n9 => 81 => (9, 40, 41) \n11 => 121 => (11, 60, 61) etc\n\nAll such triples have their two largest sides differ by 1. Note how the other terms advance in an Arithmetic Progression, in steps of 8, 12, 16, 20 etc.\n\n(2) Take any even number divisible by 4, square it, divide that by 4 and take the numbers either side\n\nThus 4=> 16 => 4 => (4. 3. 5)\n8=> 64 => 16 => (8, 15, 17)\n12=> 144 => 36 => (12, 35, 37)\n16 => 256 => 64 => (16, 63. 65)\n20 => 400 => 100 => (20, 99, 101) etc\n\nNB Even numbers that are NOT divisible by 4 (6, 10, 14 etc) will generate triples but they are not primitive triples. And it doesn't work for 2.\n\nAll such primitive triples have their two largest sides differ by 2. Note how the other terms advance in an Arithmetic Progression, in steps of 12, 20, 28, 36 etc.\n\nHaving worked out these two rules for generating triples, in my naivete, I thought that was all there was to it. That these were the only triples in existence.\n\nI told parents and teachers, who didnt know of any other triples and weren't able to help me develop my theories.\n\nNobody told me about (20, 21, 29). If they had have done I would have realised that whilst there was a difference of 1 between two sides here, it was between the two shortest sides, not the two longest sides, so there was almost certainly a series of such triples to be discovered. and more than was dreamed of in my philosophy.\n\nNote that 20 x 20 = 400, divide by 16 = 25 and then take two numbers that are 8 apart either side => (20, 21, 29) so it has similar properties. \n\nIndeed there is such a series, the next triple in it is (119, 120, 169). \n\nNote that 120 x 120 = 14400, Divide by 100, and get 144, take two numbers that are 50 apart, either side and voila we get 119, and 169, so similar properties exist here, too.\n\nYou can generate each from the previous one in the series, and each new triple has legs that are about 6 times the size of the previous one. And the hypoteneise is about 1.41 times the lengths of the legs as these values converge o
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theorm of Pythagorus states that if one of the angle of triangle is right angle{90degrees} than the hypotenuese square{the side opposite to right angle} is egual to the sum of squares of the sides forming the right angle \ni.e. (Hyp)square =one side square + second side square \n a2=b2+c2
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the pythagorean theorem is "a squared + b squared = c squared"\n\nyou have a right triangle. the two short sides are "a" and "b" and the long side is "c." if you have two of those lengths, you can find the third.\n\nIn your example:\nCall the 34 meters "a" and 41 meters "b." You're trying to find "c."\n\nc squared = 34 squared plus 41 squared = 1156 + 1681 = 2837\n\nc squared = 2837\nc = approximately 53.26\n\nto find the amount saved, you add the two short sides together and subtract the long side: 34+41-53.26
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The "3, 4, 5" triangle is the most famous of the Pythagorean triples. It is the smallest right triangle with a right angle and integral side lengths.\n\nThe other four are: 5, 12, 13; 7, 24, 25; 8, 15, 17; and 20, 21, 29.
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sphere, cube, pyramid, cilinder
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pythagorean, soh cah toa (but that's easy), area, sector area\n\n\nbut you'll need to know the theorems. I need a theorem card....dang...\n\n\n*EDIT: I'd give up my formula card for a theorem card, but we can't use theorem cards. pfffft
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8r^3(8-π) \n64r^3-8πr^3\n(2r)^3(8-π)
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Here is a fun site about applications of the Pythagorean Theorem in relation to baseball:\n\nhttp://www.geom.uiuc.edu/~demo5337/Group3/bball.html\n\nThere are lots of other neat applications for the Pythagorean Theorem.  Such as how long of a ladder do I need to reach a window x feet above the ground if I can only get within y feet of the base of the house?  Will this new bed fit properly in this corner of the room?  the list goes on and on...\n\nTrigonometry is much the same, except now you don't have to have a perfect right triangle!!!  Great huh?\n\nIt can be used for building, navigation, engineering surveying, and so much more.\n\nBearings...hmm, I recall this from physics, but I definitely more mathy than sciency, so I can just give a few that I "think" apply.  Fishing reels, roller skates, etc.\n\nHope this helps!\n\nAngel, university mathematics instructor
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let \nc=hypotenuse\na,b = legs\n\ntherefore:\nc=25\na=2b-10\n\nform pythagorean theorem;\n\na^2+b^2 = c^2\n(2b - 10)^2 + b^2 = 25^2\n4b^2 - 40b + 100 + b^2 = 625\n5b^2 - 40b - 525 = 0\n\ndividing both sides by 5\nb^2 - 8b - 105 = 0\n\nfactoring the left side\n\n(b - 15)(b + 7) = 0\n\nb + 7 = 0\nb = -7 absurd (no side can be negative)\n\nb - 15 = 0\nb = 15\n\nsolve for a:\n\na = 2b - 10\na = 2*15 - 10\na = 20\n\ntherefor the two legs of the right triangle are\n20 cm and 15 cm\n\nchecking:\nc^2 = 20^2 + 15^2\nc^2 = 400 + 225\nc^2 = 625\nc = 25 cm.\n\n\nthat's it
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Martin Gardner, for many years the recreational maths columnist in Scientific American magazine,\n\nHe was a great populariser. A puzzle he set interested me so much that it prompted me to discover the trinary tree of Pythagorean Triples, some ten years ago, in order to solve the problem.\n\nThe problem was, there is a right-angled field with one side 48, there are ten different integer solutions, what are they?\n\nThe first 8 are fairly easy being multiples of \n\n(3, 4, 5) (yields two solutions: 12 x the 4 and 16 times the 3)  \n(5, 12, 13) x 4\n(7, 24, 25) x 2 \n(8, 15, 17) x 6\n(12, 35, 37) x 4\n(16, 63, 65) x 3\n(24, 143, 145) x 2\n\nThe ninth solution is further down that series and is a primitive triple\n(48, 575, 577)\n\nBut the tenth eluded me. I therefore started to research the topic. I reasoned there might be a clue in the intervals between the 3 numbers in a triple. I took the triple (5, 12, 13) and wrote it as\n\n(-12) ^2 + (-5) ^2 = 13^2 and wondered if there was another triple with intervals 7 18 and 25 that wouid be three positive numbers if I solved the quadratic equation created by putting x, x+7, and x+25 into Pythagoras' Theorem:\n\nx^2 + x^2 + 14x + 49 = x^2 + 50x + 625\nSimplifying\nx^2 - 36x - 576 = 0\n(x - 48) (x + 12) = 0\n\nSo one triple is (48, 55, 73) and the other is (-12, -5, 13) which is what we started from.\n\nThe triple (48, 55, 73) gave me the tenth solution to Gardner's problem but also a method of generating all possible primitive triples from (3, 4, 5).\n\nBy writing 5, 12, 13 as\n(-5)^2 + 12^2 = 13^2, a similar search for a triple with intervals 17, 1, 18 leads to seeking a triple (x, x+17, x+18) and the quadratic equation\n\nx^2 + x^2 + 34x + 289 = x^2 + 36x + 324\nx^2 - 2x - 35 = 0\n(x -7) (x + 5) = 0\nwith triples (7, 24, 25) and (-5, 12, 13) with which we started\n\nSimilarly \nBy writing 5, 12, 13 as\n(-12)^2 + 5^2 = 13^2, a similar search for a triple with intervals 17, 8, 25 leads to seeking a triple (x, x+17, x+25) and the quadratic equation\n\nx^2 + x^2 + 34x + 289 = x^2 + 50x + 625\nx^2 - 16x - 336 = 0\n(x - 28) (x + 12) = 0\nwith triples (28, 45, 53) and (-12, 5, 13) with which we \n\nThese three techniques generate three new triples, from any triple and can be expressed as matrices e.g. \n\n(2 1 2\n 1 2 2    X (5, 12, 13) = (48, 55, 73)\n 2 2 3)
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Don't you mean the Pythagorean Theorem?  That theorem is a basic tenent of Geometry that states: In any right triangle, the square of the length of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the other two sides.   A^2 + B^2 = C^2.
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http://www.freemathhelp.com/forum/viewforum.php?f=9\nhttp://www.purplemath.com/modules/
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Pythagorean Theorem\n \nLet's build up squares on the sides of a right triangle. Pythagoras' Theorem then claims that the sum of (the areas of) two small squares equals (the area of) the big one.\n\nIn algebraic terms, a2 + b2 = c2 where c is the hypotenuse while a and b are the legs of the triangle.\n\nThe theorem is of fundamental importance in the Euclidean Geometry where it serves as a basis for the definition of distance between two points. It's so basic and well known that, I believe, anyone who took geometry classes in high school couldn't fail to remember it long after other math notions got thoroughly forgotten.\n\nBelow is a collection of 69 approaches to proving the theorem. Some of the proofs are accompanied by Java illustrations, but most have been written in plain HTML.\n\nRemark\nThe statement of the Theorem was discovered on a Babylonian tablet circa 1900-1600 B.C. Whether Pythagoras (c.560-c.480 B.C.) or someone else from his School was the first to discover its proof can't be claimed with any degree of credibility. Euclid's (c 300 B.C.) Elements furnish the first and, later, the standard reference in Geometry. In fact Euclid supplied two very different proofs: the Proposition I.47 (First Book, Proposition 47) and VI.31. The Theorem is reversible which means that a triangle whose sides satisfy a2 + b2 = c2 is necessarily right angled. Euclid was the first (I.48) to mention and prove this fact.\n\nW. Dunham [Mathematical Universe] cites a book The Pythagorean Proposition by an early 20th century professor Elisha Scott Loomis. The book is a collection of 367 proofs of the Pythagorean Theorem and has been republished by NCTM in 1968.\n\nIn all likelihood, Loomis drew inspiration from a series of short articles in The American Mathematical Monthly published by B. F. Yanney and J. A. Calderhead in 1896-1899. Counting possible variations in calculations derived from the same geometric configurations, the potential number of proofs there grew into thousands. For example, the authors counted 45 proofs based on the diagram of proof #6 and virtually as many based on the diagram of #19 below. I'll give an example of their approach in proof #56. (In all, there were 100 "shorthand" proofs.)\n\nPythagorean Theorem generalizes to spaces of higher dimensions. Some of the generalizations are far from obvious.\n\nLarry Hoehn came up with a plane generalization which is related to the law of cosines but is shorter and looks nicer.\n\nThe Theorem whose formulation leads to the notion of Euclidean distance and Euclidean and Hilbert spaces, plays an important role in Mathematics as a whole. I began collecting math facts whose proof may be based on the Pythagorean Theorem.\n\nWherever all three sides of a right triangle are integers, their lengths form a Pythagorean triple (or Pythagorean numbers). There is a general formula for obtaining all such numbers.\n\nMy first math droodle was also related to the Pythagorean theorem. Unlike a proof without words, a droodle may suggest a statement, not just a proof.\n\nSeveral false proofs of the theorem have also been published. I shall collect some in a separate page. It is better to learn from mistakes of others than to commit one's own.\n\nIt is known that the Pythagorean Theorem is Equivalent to Parallel Postulate.\n\nThe Pythagorean configuration is known under many names, the Bride's Chair being probably the most popular. Besides the statement of the Pythagorean theorem, Bride's chair has many interesting properties, many quite elementary.\n\nProfessor Edsger W. Dijkstra found an absolutely stunning generalization of the Pythagorean theorem. If, in a triangle, angles a, b, g lie opposite the sides of length a, b, c, then\n\n(EWD) sign(a + b - g) = sign(a2 + b2 - c2),  \n\nwhere sign(t) is the signum function:\n\n  sign(t) = -1, for t < 0, \nsign(0) =  0, \nsign(t) =  1, for t > 0. \n \n\nThe theorem this page is devoted to is trea
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You can find a bunch of them in this website:\nhttp://www.mathpuzzle.com/
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umm none that im aware of, what kind of mathamatics leval are you looking at?
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Pythagorean Theorem: To find the hypotenuse:\nSquare each leg. Add the results. Take the square root of the result. \n\n7^2 + 9^2 = c^2
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The Pythagorean Theorem can only be applied to Right Triangles only.\n\nIn a right triangle, the side opposite the 90 degree angle is called the HYPOTENUSE. The other two sides are called LEGS.\n\nTHEOREM:\nThe square of the hypotenuse is equal to the sum of the squares of the legs. \n\nEQUATION: Supposing,\nc = hypotenuse\na = first leg\nb = second leg\n\nThen:\n\nc^2 = a^2 + b^2\n\nThis is the conventional way of assigning the letters to the sides of the right triangle. However, if other letters will be assigned, then determine which letter will represent the hypotenuse so that its square can be placed on the left side of the equation. The others will be placed on the right side.