Parallelogram Definition and 126 Threads

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.
By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English.
The three-dimensional counterpart of a parallelogram is a parallelepiped.
The etymology (in Greek παραλληλ-όγραμμον, parallēl-ógrammon, a shape "of parallel lines") reflects the definition.

View More On Wikipedia.org
Recent contents Top users
  1. M

    How to calculate the fourth vertex of a parallelogram

    given 3 vectors : a=(1,3,4) b=(2,5,3) c=(1,0,0)
  2. M

    Does It Matter Which Vector You Start With When Using the Parallelogram Rule?

    Homework Statement Determine resultanat force of vectors P (40N) and Q (60N). P is 20 degrees from reference plane and Q is 45 degrees from reference plane. Homework Equations When using the parallelogram rule, should you always start with the lowest angle vector? (like in the bottom...
  3. W

    Geometry Area of sections of a parallelogram

    [b]1. The area of a parallelogram is 60 square units. A segment is drawn from one vertex to the midpoint of an opposite side. The diagonal is drawn between the other two other vertices. Find the area of the four regions formed. Homework Equations [b]3. I found that two of the...
  4. D

    Parallelogram Vector Proof: Using Vector Addition and Triangle Relationships

    Homework Statement Incl. diagrams: http://i.imgur.com/w6Exj.png Homework Equations Just vector addition (?)The Attempt at a Solution a.) ED = x(BD) ED = x(a+b) b) EQ = y(AQ) = y(b-0.5a) From the smaller triangle: ED = EQ + QD ED = [y(b-0.5a)] + (0.5a) I am not sure if this is what part b is...
  5. L

    Find the Area of the Parallelogram

    Homework Statement Find the area of the paralellogram with vertices P1, P2, P3, and P4. *Not all pairs of vertices will give rise to a side. P1 = (1, 2, -1) P2 = (4, 2, -3) P3 = (6, -5, 2) P4 = (9, -5, 0) Homework Equations ||U X V || is the area of the parallelogram having U≠0 and...
  6. B

    Evaluating a Surface Integral: A Parallelogram

    Homework Statement Evlute the surface integral Homework Equations f(x,y,z)=x+y+z where sigma is the parallelogram with parametric equations x=u+v, y=u-v and z=1+2u+v where 0 <=u<=2 and 0<=v<=1. The Attempt at a Solution I have no idea how to tackle this. Any suggestions?
  7. N

    Proving the Parallelogram Theorem: Sum of Side Squares = Diagonal Squares

    Homework Statement Apply the formula for the distance between two points to prove the well-known theorem: In a parallelogram the sum of the squares of the sides is equal to the sum of the squares of the diagonals. Homework Equations It gave a hint saying to put one of the...
  8. E

    What is the area of this parallelogram?

    1. I'll illustrate the question, I've been having a lot of troubles with it, the image is in the attachements. 2. I've tried it many times, with different methods but keep getting different answers. Really need help with it. Most recently tried by splitting it into two triangles then working...
  9. K

    Understanding the Parallelogram Area Formula: A Quest for Proof

    Hi there! My question involves the area of a parallelogram. Now, I know how to prove the commonly used formula (b*h) very easily, however, there is a formula given on Wikipedia as an alternative that states...Given two sides B and C with angle (theta), B*C*sin(theta)=Area of a parallelogram. Now...
  10. A

    Parallelogram law calculation is this an error in the text?

    In Kolmogorov and Fomin's Real Analysis book, pg. 161, they make the following claim: For any vectors f,g,h in a real Hilbert space, we have \|f + g + h\|^2 + \|f - h - g\|^2 = 2\|f - h\|^2 + 2\|g\|^2. They attempt to justify this using the parallelogram law: \|x + y\|^2 + \|x -...
  11. L

    Exploring Parallelogram Diagonals: Vector Algebra

    Homework Statement P is the point where the diagonals of the parallelogram abcd intersect one another let \alpha = AB and \beta = AD and let s and t be scalars such that AP = sAC and BP = tBD use vector algebra to show that s(\alpha + \beta) = AP = \alpha + t(\beta - \alpha)The Attempt at...
  12. H

    Parallelogram Rule for combining forces (moments)

    (2 dimensions.) Given 2 forces acting on an object (not modeled as particle), you can project their lines so that you can find a point of intersection - X. On this point of intersection exists no moment caused by the 2 forces since the line action makes 0 degrees with the forces. It follows that...
  13. T

    Understanding the Diagonals of a Parallelogram in the Plane

    Consider the parallelogram with adjacent sides OP; OQ where P is the point (x1; x2); Q is the point (y1; y2) and O is the origin.What does this ||x + y||^{2}+ ||x - y||^{2} = 2(||x||^{2} + ||y||^2)say about a parallelogram in the plane? I know ||x + y|| & ||x -y|| represent the diagonals...
  14. Jadaav

    Parallelogram method to calculate the resultant force

    In school, we've been using the parallelogram method to calculate the resultant force of 2 forces. I've noticed the teacher using his calculator to give the answer, so I think there must be a formula to calculate Resultant force. Can someone tell me the formula please ?
  15. S

    Measuring how similar to a parallelogram

    This is a (fairly basic) lemma without proof I saw in a research paper. Wasn't sure how to classify it exactly, but decided it's closest to vector (and linear) algebra. It goes like this, consider a quadrilateral in the plane with vertices A, B, C, D in clockwise order. It is given that...
  16. Z

    Possible Fourth Corners of a Parallelogram Given Three Non-Collinear Points

    Homework Statement If three corners of a parallelogram are (1,1), (4,2), and (1,3), what are all the possible fourth corners? Homework Equations The Attempt at a Solution The only possible fourth corner is (4,4) if the other 3 points are set. The solution says (4,0) and...
  17. C

    Vectors (prove a quadrilateral is a parallelogram)

    Homework Statement The diagonals of quadrilateral ABCD bisect each other. Use vectors to prove that ABCD is a parallelogram. The Attempt at a Solution let O = point of intersection AO = AD + DO DO = 1/2DB AO = AD + 1/2 DB AO = AB + BO BO = -1/2DB AO = AB - 1/2DB 2AO = AD...
  18. S

    Sketching a parallelogram question

    Homework Statement Sketch the parallelogram spanned by <2,1,0> and <-3,1,0> in the xy-plane, and compute its area. Homework Equations The Attempt at a Solution I have already computed the area, so I don't need help with that part. I do need help sketching the parallelogram...
  19. G

    Prove that ABCD is a parallelogram

    Homework Statement Let z_1, z_2, z_3 and z_4 be te position vectors of the vertices of quadrilateral AMCD. Prove that ABCD is a parallelogram if and only if z_1-z_2-z_3+z_4=0. Homework Equations The Attempt at a Solution The solution obviously uses the fact that collinear vectors of equal...
  20. L

    Multivariable Calculus - Parallelogram

    Homework Statement Suppose u=(-2,-10) and v=(-2,-2) are two vectors that form the sides of a parallelogram. Then the lengths of the two diagonals of the parallelogram are... Homework Equations The Attempt at a Solution I tried using pythagrean theorem and some trig to find the...
  21. A

    Linear Algebra Help, Area of a Parallelogram using vectors

    Homework Statement Find the area of the parallelogram defined by the vectors v = {1 1 3 1} w = {-2 -1 2 2} Homework Equations Area = v dot w * sin(theta) theta = cos^-1(v dot w / |v|*|w|) The Attempt at a Solution Solved General Solution: Area of a parallelogram for non-R^3 vectors = v dot...
  22. M

    Mass Moment of Inertia: slanted solid parallelogram: Well past wit's end

    Perhaps Tensor Calculus holds the answer; but I just can't justify the time for studying that as I know nothing of it. The end objective is to calculate the mass moment of inertia of the yellow solid parallelepiped about rectangular axes through its centre of mass as in the diagram here...
  23. C

    Which Vectors Correctly Determine the Interior Angle at B in a 3D Parallelogram?

    Homework Statement Parallelogram in 3d with vectors. Points: A(2,-1,4); B(1,0,-1); C(1,2,3); D(2,1,8) I need the interior angle at B in degrees. Homework Equations cos(theta) = (Vector1 dot product vector2) / (magnitude of v1 * magnitude of v2) The Attempt at a Solution I...
  24. P

    Find length of two line segments in parallelogram.

    Homework Statement ABCD is a parallelogram with <BAD=60. Lines AM and BM bisect Angles BAD and ABC respectively. Perimeter of ABCD is 6. Find lengths of the sides of triangle ABM. [PLAIN]http://img709.imageshack.us/img709/2440/stumped.jpg The Attempt at a Solution I'm stumped...
  25. S

    Area of a Parallelogram from 3D points

    Homework Statement Find the area of the parallelogram with vertices: P(0,0,0), Q(-3,0,-1), R(-3,1,0), S(-6,1,-1) Homework Equations A=BH The Attempt at a Solution I think I know why this is incorrect, but i don't know what else to try. I found vector PQ and called it a...
  26. A

    Calculating the Area of a Parallelogram Using Diagonals: A Vector Approach

    Homework Statement Find the area of the parallelogram with diagonals a = 3i + j − 2k and b = i − 3j + 4k The attempt at a solution I know that |x| X |y| will give the area, but will it hold for diagonals? Or do I have to find x and y vectors?
  27. J

    Functions not satisfying parallelogram identity with supremum norm

    Homework Statement Find two functions f, g \in C[0,1] (i.e. continuous functions on [0,1]) which do not satisfy 2 ||f||^2_{sup} + 2 ||g||^2_{sup} = ||f+g||^2_{sup} + ||f-g||^2_{sup} (where || \cdot ||_{sup} is the supremum or infinity norm) Homework Equations Parallelogram identity...
  28. V

    What is a Suitable Transformation for a Double Integral on a Parallelogram?

    Homework Statement Let Ω ⊂ R^2 be the parallelogram with vertices at (1,0), (3,-1), (4,0) and (2,1). Evaluate ∫∫_Ω e^x dxdy. Hint: It may be helpful to transform the integral by a suitable (affine) linear change of variables.Homework Equations The Attempt at a Solution Ok here is what I have...
  29. C

    Area of triangle inside parallelogram

    Homework Statement http://folk.uio.no/robinbj/gg/ggstart.pdf" I am supposed to find the area of the triangle PQD. The numbers given are the areas of the other triangles.Homework Equations A= \frac{1}{2} a b \sin{C} As well as Heron's formula, possibly? A= \sqrt{s(s-a)(s-b)(s-c)} where s =...
  30. 1

    How to Prove that AB=2BC without using any of the given angles?

    Homework Statement ABCD is a paralleelogram.angle DAE=angleEAB and angle CBE=angleEBA .Prove that AB=2BC Homework Equations none The Attempt at a Solution I just got that angle AEB =90 degrees (But it isn't of any help) Please help !
  31. C

    Showing the parallelogram law with a simplistic knowledge of vectors

    Homework Statement Show that the sum of the squared side lengths of a parallelogram is equal to the sum of its squared diagonals. 2. Somewhat relevant thoughts I've decided to try to show the parallelogram law with vectors, since I already managed to find an Elements-inspired proof of it...
  32. Z

    Parallelogram is formed by joining midpoints of a quadrilateral

    Homework Statement P1(x1,y1), P2(x2,y2), P3(x3,y3), P4(x4,y4) are the vertices of a quadrilateral. Show that the quadrilateral formed by joining the midpoints of adjacent sides is a parallelogram. Homework Equations Midpoint: M = ((x0+x1)/2, (y0+y1)/2) The Attempt at a Solution...
  33. S

    Vector proof of generalized parallelogram law for quadrilaterals

    Homework Statement The following theorem in geometry suggests a vector identity involving three vectors A, B, and C. Guess the identity and prove that it holds for vectors in Vn. This provides a proof of the theorem by vector methods. "The sum of the squares of the sides of any...
  34. jbunniii

    Norm Satisfying the Parallelogram Law

    Let V be a vector space over the complex field. If V has an inner product <\cdot,\cdot>, and ||\cdot|| is the induced norm, then it's easy to show that the norm must satisfy the parallelogram law, to wit: ||x+y||^2 + ||x-y||^2 = 2||x||^2 + 2||y||^2 Much more interestingly, given an arbitrary...
  35. G

    Area of a parallelogram with vectors

    Homework Statement Determine the area of the parallelogram spanned by the vectors < 0, 9, 6 > and < −10, −6, −4 > Homework Equations Area = A X B The cross product of < 0, 9, 6 > and < −10, −6, −4 > = 0i - 60j + 90k The Attempt at a Solution I know the area is the cross...
  36. F

    Law of cosines/ Parallelogram rule

    I am not sure if this question should be in here but it does pertain more to these problems as compared to a general math question. So, could someone explain why when using the parallelogram rule for obtaining the sum of 2 forces by the means of the Law of Cosines that the controller -2bc is...
  37. D

    Proving that Angles OBC and CDO are Equal in a Parallelogram

    Homework Statement A parallellogram ABCD has an interior point O sucht that \alpha + \beta = 180^o http://img413.imageshack.us/img413/5636/post102741235319763.png Prove that: \angle{OBC}=\angle{CDO} Homework Equations Definitions of a parallellogram. The Attempt...
  38. M

    Resolving forces on Hydraulic Parallelogram frame

    Hi guys, I want to check that I am going about this problem in the right way: The attached image shows a parallelogram type frame with a hydraulic ram located from corner to corner as shown (E to C). There is a load applied to a platform in the far bottom right hand side of the drawing...
  39. C

    Area of a parallelogram using determinants

    Homework Statement let v = (1,0,1) and u = (0,2,1) Find the area of the parallelogram {sv + tu : 0 <= s, t <=1) Homework Equations The Attempt at a Solution I know the area of a parallelogram is the determinant of a 2x2 matrix, but they gave v and u in R^3. Would I just...
  40. F

    Calculating the Area of a Parallelogram with Two Sets of Parallel Lines

    Homework Statement This isn't really homework, but a not assigned problem out of my mathbook which is kinda confusing me.. #74. We have two pairs of parallel lines in R^2 defined by the linear equations below: a1x + b1y = r1 a1x + b1y = s1 a2x + b2y = r2 a2x + b2y = s2 We assume...
  41. G

    Multiple Integrals with parallelogram

    Use the given transformation to evaluate the given integral, where R is the parallelogram with vertices (-2, 2), (2, -2), (4, 0), and (0, 4). ∫∫(2x+8y)dA; x=1/2(u+v) y=1/2(v-u) I found the bounds of the parallelogram of -4≤u≤4 and 4≤v≤0 so i set the equation to be...
  42. C

    Area of a Parallelogram with out cross products

    Homework Statement given the vertices: (0,0) (3,1) (2,3) (5,4) Homework Equations solving without cross products and with mathematica if available The Attempt at a Solution Te1=[3] [1] Te2=[2] [3] [det[Ay]] ________ [det[A]] get you the solution...
  43. S

    Calculating the Area of a Parallelogram with Given Vertices

    i'm posting this problem in the calculus forum because i got this question in a calculus class. It seems like a straightforward area problem, but i don't think that's the case and i can't figure out another way to do the problem using vectors. "Find the area of the parallelogram with vertices...
  44. I

    Finding the Area of a Parallelogram with Two Vectors in R^4

    I need to find the area of a parallelogram with two vectors in R^4 my book has nothing on this and I don't know how to do it.
  45. M

    Parallelogram Law of Vector Addition

    Homework Statement I'm having trouble with a word problem involving the parallelogram law of vector addition. The problem reads as follows: Two forces with magnitudes 8N and 11N act on a large object. The angle between the forces is 30 degrees. a) Draw a diagram to represent the combined...
  46. D

    What are the coordinates of C in a parallelogram given vertices A, B, and D?

    Homework Statement Given a parallelogram ABCD has vertices A(-1,2,-1), B(2,-1,3) and D(-3,1-3). Find the coordinates of C. Homework Equations The Attempt at a Solution I'm extremely confused here. I do not know how to tell which coordinate is for which vertex on the parallelogram. I...
  47. Saladsamurai

    Anybody know anything about parallelogram mounts

    My girlfriend bought me some sweet astro-binoculars from Orion for Christmas (20x80s) and I would love to use them but they are heavy as hell (and shaky). I was looking around for one of those parallelogram mounts. I found a couple like...
  48. P

    Parallelogram with coordinates

    Homework Statement Parallelogram ABCD is given, with A(2,4), B(5,7), C(12,8). Find the coordinates of D(x,y). Homework Equations The Attempt at a Solution Sorry but, I don't have any idea.
  49. V

    Circumference of a parallelogram (diagnoals given only)

    Homework Statement Given is a parallelogram which has diagonals of the length 7 (e) and 9 units (f). How big is its circumference? The sides are a,b,c,d; a being the bottom side, rest is anti-clockwise... alpha is the angle of a etc... Homework Equations no are given, i guess...
  50. quasar987

    Normed spaces and the parallelogram identity

    Homework Statement My professor stated the theorem "If (X,<,>) is an an inner product space and || || is the norm generated by <,>, then we have ||x+y||² + ||x-y||² = 2(||x||² + ||y||²)." But then she also said that the converse was true. I suppose this means that "Given (X, || ||) a normed...
Back
Top