What is Parallelogram: Definition and 126 Discussions

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.

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  1. J

    Finding the center of mass of a piecewise function

    I understand that I can divide this shape into a few parallelograms and a triangle and calculate the center of mass of each, but am confused as to what I should do after that. My physics teacher also wants us to use integrals, but I'm assuming I can calculate the COM of each parallelogram and...
  2. karush

    MHB Tikz for parallelogram problem

    https://mathhelpboards.com/threads/3-coordinates-of-parallelogram-stuv.6195/ this problem had over 6000 views so I was interest in posting it to Linkedin hoping it would recruit some new members however it had a graph in the OP which has been deleted but another was posted using some code...
  3. N

    Proof Involving the Diagonals of Parallelogram

    Chapter 1, Section 1.1. Look at the picture. Question 57. Let me see. To show this prove, I must find the midpoint of the diagonals. The midpoint of (b, c) and (a, 0) must be the same as the midpoint of (0, 0) and (a + b, c). You say?
  4. S

    MHB Need help with parallelogram proof

    Hello, we are learning about similar triangles and this was a problem. So I know that opposite sides of a parallelogram are congruent as are opposite angles, so I can establish similarity with triangles WYS and STW, but I don't understand how that proves SX x YW = SV x WT because the proportions...
  5. A

    MHB Calculating Perimeter & Area of a Parallelogram & Triangle

    Find the perimeter and area of CD, if ABCE is a parallelogram and ADE is an equilateral triangle.
  6. anemone

    MHB Parallelogram ABCD: Finding AC from PB & PD

    In parallelogram $ABCD$, $\angle B$ and $\angle D$ are acute while $\angle A$ and $\angle C$ are obtuse. The perpendicular from $C$ to $AB$ and the perpendicular from $A$ to $BC$ intersect at $P$ inside the parallelogram. If $PB=700$ and $PD=821$, find $AC$.
  7. M

    I PDEs: Laplace's Equation over a Parallelogram

    Hi, I have been learning about Laplace's equation recently, and have been wondering: how would we approach the problem if the region was a parallelogram (or some other shape that isn't a standard rectangle or circle)? Is this something that could feasibly be solved by hand, or would it require...
  8. C

    Finding angles of a Parallelogram

    Hi, I work as a math/science tutor, and a student had this question. He said you're supposed to find the values of x, y, and z and the only information given is that it is a parallelogram. It looks like it might be under-determined to me, every equation I write for it ends up reducing to 15x...
  9. Robin04

    Probability - Camera in a parallelogram shaped room

    I tried to mess around with some equations in Mathematica with more or less success. Let's call ##a,b## the two sides and ##\delta## the angle between them. I contructed vectors that point to the corners of the room. ##\vec{a} = (0, 0)## ##\vec{b} =(a, 0)## ##\vec{c}=(a+b \cos{\delta}, b...
  10. Akash47

    Finding the area of a parallelogram inside another

    Through symmetry of parallelogram,I have come to this: Here 1,2,3,4 denotes the area of the particular regions.Then I am stuck.Please help what to do next or whether there is any other way.
  11. S

    MHB Parallelogram with diagonals. Need to find the area (S).

    diagonal 1=20cm. diagonal 2=37cm. AB=25.5cm S (AMC)= 306cm. S (ABCD)=?
  12. T

    B Trig Problem Altitudes of Parallelogram

    Basically the altitudes of a parallelogram have two lengths, 8 and 10 and they intersect at an angle whose sine is 1/4. I'm attempting to find the area of the parallelogram. I'm having trouble creating a good picture of it. Seeing if I could receive some help, thanks!
  13. DLeuPel

    B Understanding the Parallelogram Method for Solving Forces

    How does the parallelogram method work to solve one of two forces with different angles? Or better said, how do you derive the method so that I can get a better understanding of it ?
  14. Z

    MHB How to Calculate the Diagonal Length of a Parallelogram?

    The diagonal divides the parallelogram into two triangles, each triangle circumference is 6.21 m. The circumference of the parallelogram is 7.12 m. Calculate the diagonal length. I need tips how to solve this problem.
  15. Monoxdifly

    MHB [ASK] Parallelogram in Cuboid

    In an ABCD.EFGH cuboid with AB = 4 cm, BC = 3 cm, and CG = 5 cm there is a parallelogram OBFPH with O is located at the center of ABCD and P is located at the center of EFGH. The distance between the lines HO and PB is ... A. 5\sqrt3 cm B. 5\sqrt2 cm C. \sqrt5 cm D. \frac{5}{2}\sqrt2 cm E...
  16. Richie Smash

    ZPQM is a parallelogram, express Vector OZ in terms of u and v

    Homework Statement u and v are two vectors in the same plane. Vector OM = u+2v Vector OP = 6u+v Vector OQ = 5u +2v Vector OR =2(VecOM) +v Given that ZPQM is a parallelogram, express Vector OZ in terms of u and v.Homework EquationsThe Attempt at a Solution First they wanted me to find Vectors...
  17. Z

    Area of Parallelogram ABCD: Find the Correct Solution | 4*6=24? [Picture]

    Homework Statement Find the area of Parallelogram ABCD where AD=4 and CD=6 and angle ABC=125 degrees, tell whether the area is greater than 24 or not? See the attached picture Homework Equations Area of Parallelogram = length * breadth The Attempt at a Solution Sol: 4 * 6 = 24 but answer is...
  18. S

    I Solving Confusion About Parallelograms in Curved Spacetime

    One way to see that spacetime is curved is to try and draw a "rectangle" in spacetime (see the figure in the Feynman lectures, ch 42.7): If I wait for 100 seconds and then move upwards on earth, I end up at a different point in spacetime than when I first move upwards and then wait for 100...
  19. A

    Parallelogram area (coordinates)

    Homework Statement The coordinates of the parallelogram ABCD are: A (-2; 1) B (5; 2) C (6; 5) D (-1; 4) We also know that the diagonals intercept in the middle of each other (so if the diagonals are AC and BD, and the intercept in point M, then AM = MC, and BM = MD). Not sure if this...
  20. N

    MHB Find value of x and area of parallelogram: a,b,c,P

    Hello, could someone please help me with this question? I don't even know where to begin. Given vectors a = (2, x, 0), b = (1, 0, −1) and c = (5, −9, 3), and let P(2, 1, −1) be a point. Find the value of x in a such that the angle between a and b is π/4, then find the area of parallelogram...
  21. I

    MHB Parallelogram on graph problem

    Is there a way to do this other than guess and check?
  22. karush

    MHB S6.793.12.4.27area of a parallelogram

    $\tiny{s6.793.12.4.27}$ $\textsf{area of a parallelogram with adjacent sides}\\$ $\textsf{$\vec{P}{Q}$ and $\vec{P}{R}$ is the length of this cross product:}$ \begin{align} |PQ \times PR|&=\sqrt{()^2+()^2}= \end{align} $\textit{new on vectors... not sure what goes into $()^2$}$
  23. M

    MHB Prove Equal Area Triangle & Parallelogram, $\angle ADC=90^{\circ}$

    Workings $\triangle ADE \cong \triangle CFE \left(AAS\right)$ $\angle AED = \angle CEF $( vertically opposite angles ) $\angle CFE= \angle EDA $( alternate angles ) $AE=EC $( E midpoint ) $ii.$ADCF is a parallelogram because diagonals bisect each other. Where is help needed How should...
  24. Matejxx1

    Vector algebra (proving you have a parallelogram by using vectors)

    Homework Statement 23. In a ABCD quadrilateral let P,Q,R,S be midpoints of sides AB,BC,CD and DA. Let X be the intersection of BR and DQ, and let Y be the intersection of BS and DP. If ##\vec{BX}=\vec{YD} ## show that ABCD is a parallelogram . Homework Equations ## (\vec{a}\cdot\vec{b})=0##...
  25. M

    MHB Proof of Parallelogram ABCD: Midpoint X & Y Show Area $\frac{1}{4}$

    ABCD is a parallelogram . X is the midpoint of AD & Y is the midpoint of BC. Show that the area of $\triangle {ABX}$ is $\frac{1}{4}$ the area of ABCD Can you help me with this proof ? were should i start ? I think It should be by proving $\triangle{DBC} \cong \triangle{DBA} $ using SAS as...
  26. M

    MHB Problem on Parallelogram proof

    Can anyone solve the following question Please try to make you answer detail as possible :)
  27. parshyaa

    I Parallelogram law of vector addition

    Acording to this diagram vector P = vector BC and vector Q = vector OB(their magnitudes are also respectively equal.) Therefore acoording to the congruency of triangles angle alpha = (theta)/2. But this is not right( what's wrong ) {Tan(alpha) = Qsin(theta)/(P + Qcos(thets)) ] Why resultant...
  28. J

    Given: parallelogram and right angle; Prove: parallelogram

    I'm curious if it looks like I defined the reasons correctly to prove that FBCD is indeed a parallelogram. Specifically, I'm unsure if I'm using #4 "definition of coincident" and #5,6 "reflexive property" correctly in their terminology. I don't have anyone else, or any teacher to ping this...
  29. J

    Proving a parallelogram is a rhombus, given an Isosceles triangle

    I believe I could work this problem fine had the instructor not placed statements #3,4, and 6, as well as the reasons # 5 and 7. I can't seem to understand where he's using the transitive property in #5. If these steps weren't here I would assume, I'm only using substitution to prove that "RS"...
  30. S

    MHB Solve Parallelogram: Lengths X & Y, Angle Z, A=15in, B=20.5in

    A parallelogram exists within a rectangle which measures 15 inches tall by 20.5 inches wide. A=15 inches B=20.5 inches C=1.5 inches (C makes a 90° angle with X) Solve for lengths X and Y and angle Z of the parallelogram and please tell me how you did it!
  31. Julian102

    What is the relative area of a rectangular field?

    Homework Statement If the area of a rectangular field has length of 110 m and 80 m.If a spaceship is traveling with 0.9c velocity along the diagonal of the field.Then what is the area of the field seen by the astronaut in the spaceship? Homework Equations L=L(initial) Root over (1- (v/c)^2)...
  32. Naton

    Resolving angle of parallelogram in 3D space

    Hi there, I'm doing a bit of amateur scale model making as a hobby, building shapes out of flat-cut pieces. Trigonometry is a huge help with this, but I've hit a snag where I'm trying to calculate the angle of a particular parallelogram so it fits with the rest of the geometry. 1...
  33. T

    Finding angles of a parallelogram.

    Homework Statement This question has popped up recently and I was completely stumped. How to find the angles of a parallelogram given only area and the length of the diagonals? I'm trying to find a generic solution or formula that works for a non-rhombus. Homework EquationsThe Attempt at a...
  34. A

    Given two vectors, find vector of the parallelogram height

    Homework Statement Find the coordinates of the vector of the height of the parallelogram formed by vectors a={1, 2, 1} and b={2, -1, 0} Homework Equations A=|axb|, A=|a|*h The Attempt at a Solution I can find the intensity of the vector h i.e the length of the height, but not its vector. I...
  35. B

    MHB Sketch of the Reflection Transformation of a Parallelogram

    $\textbf{Problem:}$ Let $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation that reflects each point through the $x_2$ axis. Make two sketches that illustrate properties of linear transformation. $\textbf{Solution:}$ Let $T(\textbf{x}) = \begin{bmatrix} -1 & 0 \\ 0 & 1...
  36. B

    MHB Transformation of a Parallelogram

    $\textbf{Problem}$ Let $\textbf{u}$ and $\textbf{v}$ be vectors in $\mathbb{R}^n$. It can be shown that the set $P$ of all points in the parallelogram determined by $\textbf{u}$ and $\textbf{v}$ has the form $a\textbf{u} + b\textbf{v}$, for $0 \le a \le 1, 0 \le b \le 1$. Let $T: \mathbb{R}^n...
  37. R

    Draw a Parallelogram: Step-by-Step Guide

    I'd really need help with this. Thanks. 1. Homework Statement Draw a parallelogram: a = 4cm e + f (diagonals) = 11cm angle between a and e = 22° Homework Equations 2 and 2 sides are parallel both diagonals halve each other (diagonal e goes from A to C) The Attempt at a Solution (attachment)
  38. F

    Starting the Sander's Parallelogram Proof

    Can anyone help me set this problem up? I am trying to figure out how to Prove the Sander's Parallelogram. See it here: http://www.tigel.nl/fun/files/opticals/Ill...ith_line-25.htm basically it is proving that the bisectors are of equal length The question is: what would be needed...
  39. J

    MHB What is the area of a parallelogram without knowing the height?

    I am working on a task right now. I am currently trying to find the area of a parallelogram. I do not have the height. I only have the dimensions. I have tried suggestions like dividing the parallelogram into triangles and doing 1/2bh. The dimensions I have are the 1/2,1/2,\sqrt{2}/4...
  40. Z

    Find area of parallelogram given vertices

    Homework Statement Find the are of the parallelogram ABCD where A is (1,2,-3), B is (-1,3,-4) and D is (1,5,-2)Homework Equations Area=\left|AxB\right| where A and B are the vectors AD, and AB respectively.The Attempt at a Solution I have calculated AD to be= (0,-3,-1) and AB=(2,-1,1) ∴ to...
  41. L

    MHB Is the area of a parallelogram equal to a/b times sine alpha?

    demonstrate or show that the figure area is = a/b sen alpha triangle a = triangle B maybe this is the main premise
  42. Z

    Why is the parallelogram rule for the addition of forces as it is?

    Why is the parallelogram rule for the addition of forces as it is? I feel it must have some deep origin and pointing to something fundamental. Though I know this problem may have no answer: God design it as such. But I wonder how the first person came up with this rule, where does his/her...
  43. T

    4th Vertex in a 3D Parallelogram

    Homework Statement Let P, Q, and R be the vertices of a parallelogram with adjacent sides PQ and PR. Find the other vector S. P (2, 0 ,-1), Q (-2, 4, 1), R (3, -1, 0) Homework Equations PR = QS PQ = RS The Attempt at a Solution I took the two equations and solved both of them...
  44. karush

    MHB *3 coordinates of parallelogram STUV

    (a) $\vec{ST} = \pmatrix{9 \\ 9}$ so $V=(5,15)-(9,9)=(-4,6)$ (b) $UV = \pmatrix{-4,6}-\lambda \pmatrix{9,9}$ (c) eq of line $UV$ is $y=x+10$ so from position vector $\pmatrix{1 \\11}$ we have $11=1+10$ didn't know how to find the value of $\lambda$ (d) ?
  45. U

    Finding Angle θ in Parallelogram ABCD

    Homework Statement In the parallelogram ABCD the internal bisectors of the consecutive angles B and C intersect at P. Use vector method to find angle BPC. Homework Equations The Attempt at a Solution I assume angle BPC to be θ and the point of intersection of internal bisectors to be P...
  46. Albert1

    MHB Prove Parallelogram Point P's Angles are Equal

    Point P is an iner point of a parallelogram ABCD given $\angle PAB=\angle PCB$ please prove :$\angle PBA=\angle PDA$
  47. S

    MHB Area of the parallelogram when diagonal vectors are given.

    I can find the area of the parallelogram when two adjacent side vectors are given. But how to find the area of the parallelogram when diagonals of the parallelogram are given as \alpha = 2i+6j-k and \beta= 6i-8j+6k
  48. W

    Calculating area of a parallelogram defined by 2 vectors

    Homework Statement "Find the area of a parallelogram defined by the two vectors P=(4,-10,3) and Q=(2,1,0)" Homework Equations The area of the parallelogram is equal to the magnitude of the cross product of the two vectors? i.e. Area = |PXQ| The Attempt at a Solution PXQ =...