What is Pythagorean theorem: Definition and 46 Discussions

In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that 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 on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:










{\displaystyle a^{2}+b^{2}=c^{2},}
where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. The theorem, whose history is the subject of much debate, is named for the Greek thinker Pythagoras, born around 570 BC.
The theorem has been proven numerous times by many different methods—possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.
The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps, and cartoons abound.

View More On Wikipedia.org
  1. C

    B Question about Morin's time dilation explanation

    Hello everyone, I'm reading Morins book which I like, and I feel I kind of understand the part on time dilation, however I'm a little confused by the geometry of the Pythagorian theorem when applied to velocities. On the moving clock he shows the velocity of light on the diagonal it traces...
  2. Slipjoints

    Distance from a point on a circle to an arbitrary axis

    Hi all! In this assignment I have to formulate an equation for the shortest distance from a point on a circle perimeter to an arbitrary axis in a circle with angle theta. I included an image with the sketch. Anyone that can help?
  3. B

    I Dirac Matrices and the Pythagorean Theorem

    I understand that momentum, rest mass and energy can be put on the sides of a right triangle such that the Pythagorean Theorem suggests E^2=p^2+m^2. I understand that the Dirac equation says E=aypy+axpx+azpz+Bm and that when we square both sides the momentum and mass terms square while the cross...
  4. A

    MHB Pythagorean Theorem: Unexpected Finding

    Hello, today i was playing around with the pythagorean theorem and found out something that i can't really explaing or atleast explain it with probably a false answer. So i was putting every possible combination with the max digit of 10. For example 1^2+1^2=\sqrt{2}, 1^2+2^2=\sqrt{5}...
  5. M

    MHB Converse of Pythagorean Theorem

    Determine if the triangle with the given vertices is a right triangle. (7, -1), (-3, 5), (-12, -10) I must find the lengths of the sides using the distance formula for points on the xy-plane. The question then tells me to use the converse of the Pythagorean theorem. How do I use this...
  6. Kirito123

    Apply Area and Pythagorean Theorem to a prism

    Homework Statement Homework Equations Formula for Area of a retangle : A = L x W Pythagorean theorem: A2 + b2 = c2 The Attempt at a Solution So I am pretty sure I did it correct but I just want to be 100% certain I will get this right, By the way its a picture cause I found it easier to...
  7. Hatesmondays

    Cool ways to use the Pythagorean Theorem

    What are some cool things that people can do with the Pythagorean Theorem?
  8. caffeinemachine

    MHB A Fact I Observed While Looking at the Proof of the Pythagorean Theorem.

    Let $ABC$ be a right angled triangle, where the right angle is at $A$. Construct squares on $AC$, $AB$ and $BC$ as shown. Let $P$ be the point of intersection of $BK$ and $FC$ (Note that $P$ is not marked in the figure). Then I conjecture that $AP$ is parallel to $BD$.What I tried:By obsercing...
  9. C

    How accurate is pythagorean theorem?

    I'm just wondering: How accurate is the pythagaroen theorem? To what exact decimal point is the calculated length of the hypotenus in a triangle 100% corect when using pythagarean theorem? It must not be 100% accurate to an infinite descimal point since it's called a "Theorem"
  10. A

    Tension in landing cable on an aircraft carrier

    Homework Statement [/B] Keep in mind this is a Top Gun-themed homework assignment. Cougar comes in for a shaky landing. His 20422 kg airplane traveling at 85 m/s strikes the deck at 3.5 degrees below the horizontal. Cougar's plane snags the landing cable stretched across the deck. The landing...
  11. Ismael Nunez

    The Pythagorean Theorem

    Anyone want to take a crack at it? My class has been discussing it: Find the length of AB:http://t4.rbxcdn.com/84e25f3830d66e6bbaeaba48e35c0781
  12. L

    How to find angle after applying Pythagorean Theorem?

    Hello, so I have a question that states (these aren't the actual measurements but they are around about the same, I can't remember the exact numbers so I made these up, this way I could apply the same to the actual numbers) an object being 4km above me, 1.4km to the north of me, and 2km to the...
  13. P

    MHB Consequences of the Pythagorean Theorem

    Hello! I have a homework for tomorrow. The question is: What are the consequences of the Pythagorean theorem?? On Wikipedia, they mixed the consequences and the uses of the theorem in one paragraph: 5.1 Pythagorean triples 5.2 Incommensurable lengths 5.3 Complex numbers 5.4...
  14. T

    Pythagorean theorem based on cross product.

    I was developing a pythagorean theorem proof based on the cross product of two vectors..Below is my final solution...My problem is I had to get around using the distance/magnitude formula because that is using the pythagorean theorem to prove the pythagorean theorem. But after searching, it may...
  15. B

    Disproving Pythagorean Theorem

    Ever since I was in grade school I have been fascinated with the idea that the Pythagorean theorem, or any other universally respected theorem, could be wrong. When I was younger I found a little proof I made to disprove it, and I came across it in an old notebook of mine. Now after taking...
  16. S

    Proving the Pythagorean Theorem

    I am wondering what it means to "prove" the Pythagorean Theorem in modern mathematics. Most real analysis begins with the following definition of the n-dimensional Euclidean metric: ##d(x,y) = |y-x| = \sqrt{\sum\limits_{i=0}^{n} (y_i-x_i)^2}## This would seem to directly imply the...
  17. S

    Pendulum motion and pythagorean theorem

    Homework Statement Show that the relation between the horizontal and vertical components of the ball's position is given by the equation: y = L - [(L^2 - x^2)^1/2] http://www.flickr.com/photos/94066958@N08/8553595522/in/photostream/ Homework Equations y = L - [(L^2 - x^2)^1/2]...
  18. W

    Disproof to the Pythagorean Theorem?

    Ok I realize the Pythagorean Theorem is correct. I completely get that very basic concept this is just a question I have. On a right triangle a^2 + b^2 = c^2 with c being the hypotenuse. But if instead of the hypotenuse connecting the two legs you had a jagged line that went halfway up...
  19. M

    I found out new proof of Pythagorean theorem , how can I publish it ?

    Hello, I know that ,their is more than 97 proofs for Pythagorean theorem . but I think that I found new one ! which is very beautiful , also , this proof show us the relation between 2 branches of maths , and how can we look to one object by diffrent ways , also this proof shows us that we...
  20. Q

    Is this equivalent to the Pythagorean Theorem?

    So I was reading up on the Pythagoreans, and I came across this page: http://www.math.ufl.edu/~rcrew/texts/pythagoras.html [Broken]. I don't see the reasoning behind this statement. I tried some simple algebra on this statement and couldn't get Pythag to fall out of it. Can someone...
  21. C

    Solving for Line FC Using the Pythagorean Theorem

    Homework Statement Find line FC in the given problem. Homework Equations A^2 + b^2 = c^2 The Attempt at a Solution according to the diagram, 1/2ac = bc - since the the two right triangles share the same hypotenuse. bc = 12 so 1/2ac = 12, i got the answer as 16^2 + b^2 =...
  22. P

    Spherical Pythagorean theorem - finding length of longer side

    Picture of the problem: As seen by the diagram above, a2 < a1 But the spherical Pythagorean theorem states that cos c = (cos a)(cos b). The triangle can either have a1,b,c or a2,b,c as its sides, which means the above equation contradicts itself. Am I missing something? thanks.
  23. M

    Deriving Invariant Distance in Space Time: Pythagorean Theorem

    Why in deriving invariant distance in space time we use Pythagorean Theorem with a negative sign?
  24. E

    Proof of the Pythagorean theorem using Dimensional analysis

    hello, in the Wikipedia page for http://en.wikipedia.org/wiki/Dimensional_analysis#Proof_of_the_Pythagorean_theorem" how do we know that area = "largest edge2 • f (angle1, angle2)"? Isn't it just as reasonable to say something like area= (largest edge/40000)2 • f (angle1, angle2) ?
  25. G

    Solve for Building Height: Kinematics & Pythagorean Theorem

    Homework Statement A 0.21 kg rock is projected from the edge of the top of a building with an initial velocity of 7.82 m/s at an angle 56 above the horizontal. Due to gravity, the rock strikes the ground at a horizontal distance of 10.5 m from the base of the building. How tall is the...
  26. F

    Deriving the speed of light from the Pythagorean Theorem

    Has anyone heard of the speed of light being derived from the Pythagorean Theorem? Obviously I'm referring to using the time dilation effect of motion. Frank
  27. N

    Proof of Pythagorean Theorem

    I just learned the proof for the Pythagorean theorem.. my mind was blown! =D Simply amazing.
  28. C

    Solving Vector Components and Theta with Pythagorean Theorem

    Homework Statement Vector A= 5.00 m (See the attached image for the graphical representation) Vector B=17.0m Vector C=15.0 m Homework Equations Pythagorean Theorem, Basic trigonometry The Attempt at a Solution Finding the magnitude of the resultant vector is not...
  29. L

    Pythagorean Theorem: Relationships in Euclidean Space

    The Pythagorean theorem relates the length of a vector to its projection onto an orthonormal basis for Euclidean space. Does it also work in the same way for parallograms, and higher dimensional linear solids such as paralleopipeds? I take an n dimensional linear solid and project it onto an...
  30. L

    Displacement & Pythagorean Theorem: Triangle ABC

    Alright, we have triangle abc with hypotenuse c. So, if you add vector a and vector b, the answer is vector c. Now, according to the pythagorean theorem, this would not make sense. But the pythagorean theorem is DISTANCE. I am guessing that this phenomenon has something to do with using...
  31. Char. Limit

    Natural Numbers & the Pythagorean Theorem

    I was pondering square numbers today, and I noticed something interesting: every natural number contains information to construct a Pythagorean triple. Let me show what I mean for an odd natural number, also using the binomial theorem for square quadratic equations (equations of the form...
  32. K

    RMS and the Pythagorean Theorem

    Today I was thinking about the root mean square, and I figured out a definite relationship with the Pythagorean theorem. Specifically, the root mean square of the legs of a right triangle is equal to the "average leg," i.e. the leg of a square with the hypotenuse as it's diagonal. It appears to...
  33. D

    MSP430 Assembly Language Pythagorean Theorem HELP

    The problem statement given is this: Write a PUBLIC function in MSP430 assembly that implements the Pythagorean theorem A^2 + B^2 = C^2. Make A and B 16bit integers. What size should you make C? Your function must use the HW multiplier's multiply and accumulate feature. Also write a brief...
  34. P

    Photon Velocity and Pythagorean Theorem

    If two photons are traveling at right angles to each other with velocity vectors of c then what is the velocity vector of the hypotenuse of the right triangle and does this velocity triangle form an equilateral triangle with three angles of 90 degrees if c^2 + c^2 = c^2
  35. M

    Pythagorean Theorem to find the resultant

    A man walks 5 blocks east, then turns 10 blocks north, then walks 5 more block east. Total displacement____? is it... 10^2 + 10^2 = c^2 (Square root 200)=c^2 14.14 = c^2 is this right?
  36. R

    Pythagorean Theorem proof Vector Calc. Due tomorrow 9am. PLEASE HELP

    Pythagorean Theorem proof Vector Calc. Due tomorrow 9am. PLEASE HELP! Prove the Pythagorean theorem. That is, if a,b, and c are vectors in Rn such that a + b = c and a.b = 0, then ||a||2 + ||b||2 = ||c||2. Why is this called the Pythagorean theorem. Hint given: Given the hypotheses, you have a...
  37. U

    What happens to the pythagorean theorem in a non-euclidean space?

    Does the theorem still hold true, but triangle sides are curves?
  38. S

    1,f1x,f1y pythagorean theorem

    I have a question about those three forces. Let's say for example that f1x=0 and f1y=5 then f1=5 right? Or do I have to use the pathagorean therem to find f1 then it would be the square root of 5?
  39. S

    Pythagorean Theorem and Pendulums

    Homework Statement Calculate the maximum speed of the 100 g pendulum mass when it has a length of 100 cm and an amplitude of 50cm. sorry my computer won't access the other thread. i don't know any other laws of conservations of energy or trig very well. also, when the pendulum as an amplitude...
  40. E

    Pythagorean Theorem: Proving it with Maths

    I have used the pythagorean theorem quite frequently yet have never have it proved for me. I tried to do so myself but as i suck at maths i was unsuccessful. Any links proving it would be apppreciated.
  41. U

    Am I supposed to be thinking of the Pythagorean Theorem?

    I can't seem to understand this problem. Could someone give me a hint as to what I'm supposed to be thinking? Vectors A, B and C satisfy the vecotr equation A + B = C, and their magnitudes are related by the scalar equation A^2 + B^2 = C^2. How is vector A oriented with respect to vector B...
  42. L

    Learn the Pythagorean Theorem with Lou

    Hey everyone, I understand a2+b2=c2, but I have trouble figuring out which side is which in problems like these: http://img417.imageshack.us/img417/8238/py8gu.png [Broken] Can someone explain? Thanks, Lou
  43. J

    Pythagorean theorem and vectors

    The summit of a mountain, 2450 m above base camp, is measured on a map to be 4580 m horizonttally from the camp in a direction 32.4 degrees west of north. What are the components of the displacement vector from camp to summit? What is its magnitude? Choose the x-axis east, y-axis north and z...
  44. D

    3-D Pythagorean Theorem?

    Has anybody else tried this? a^3 + b^3 + c^3 = d^3 3^3 + 4^3 + 5^3 = 6^3 27 + 64 + 125 = 216 This seems to be a logical extension of the Pythagorean Theorem and it works if the values of 3, 4 and 5 are used for a, b and c. Has this already been discovered in mathematics or is this...
  45. O

    Searching for Perfect Squares using the Pythagorean Theorem

    perfect square? hello everyone! i have come across an on-going research onDETERMINING A PERFECT SQUARE GIVEN A DIFFERENCE . However, I have a feeling that this was not an original one. the researcher used the Pythagorean theorem to arrive at his so called "theorem". would anyone give...
  46. S

    How many ways are there to prove Pythagorean Theorem?

    This thread is intended just for fun. My favorite proof of the pythagorean theorem uses algebra, together with a very simple picture. A square is inscribed in another square, and then we use the fact that the whole area is equal to the sum of its parts. The resulting figure consists of an...