Parallelogram

A parallelogram.

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. The opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are congruent. In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i The three-dimensional counterpart of a parallelogram is a parallelepiped. Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism

1 Properties
2 Computing the area of a parallelogram
3 Proof that diagonals bisect each other
4 Derivation of the area formula
- 4.1 Alternate method
5 See also
6 External links

Properties

The area, $A$ , of a parallelogram is $A = B H$ , where $B$ is the base of the parallelogram and $H$ is its height.
The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which Adjacent is an adjective meaning contiguous, adjoining or abutting.
The diagonals of a parallelogram bisect each other. A diagonal can refer to a line joining two nonconsecutive vertices of a Polygon or Polyhedron, or in contexts any upward or downward sloping line In Geometry, bisection is the division of something into two equal or Congruent parts usually by a line, which is then called a bisector
It is possible to create a tessellation of a plane with any parallelogram. A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps
The parallelogram is a special case of the trapezoid. A trapezoid (in North America or a trapezium (in Britain and elsewhere is a Quadrilateral (a closed plane shape with four linear sides that has at least one
The rectangle is a special case of the parallelogram. In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as
The rhombus is a special case of the parallelogram. In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses

Computing the area of a parallelogram

Let $a,b\in\R^2$ and let $V=[a\ b]\in\R^{2\times2}$ denote the matrix with columns $a$ and $b$ . Then the area of the parallelogram generated by $a$ and $b$ is equal to $| det(V) |$

Let $a,b\in\R^n$ and let $V=[a\ b]\in\R^{n\times2}$ . Then the area of the parallelogram generated by $a$ and $b$ is equal to $\sqrt{\det(V^T V)}$

Let $a,b,c\in\R^2$ , and let $V=[a\ b\ c]\in\R^{2\times2}$ . Then the area of the parallelogram is equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows:

$V = \left| \det \begin{bmatrix} a_1 & a_2 & 1 \\ b_1 & b_2 & 1 \\ c_1 & c_2 & 1 \end{bmatrix} \right|$

Proof that diagonals bisect each other

To prove that the diagonals of a parallelogram bisect each other, first note a few pairs of equivalent angles:

$\angle ABE \cong \angle CDE$

$\angle BAE \cong \angle DCE$

Since they are angles that a transversal makes with parallel lines $A B$ and $D C$ .

Also, $\angle AEB \cong \angle CED$ since they are a pair of vertical angles. pair of Angles is said to be vertical (US English or opposite (British English if the angles share the same vertex and are bounded by the same pair of

Therefore, $\triangle ABE \sim \triangle CDE$ since they have the same angles.

From this similarity, we have the ratios

${AB \over CD} = {AE \over CE} = {BE \over DE}$

Since $A B = D C$ , we have

${AB \over CD} = 1$ . Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other

Therefore,

A E = C E

B E = D E

$E$ bisects the diagonals $A C$ and $B D$ . In Geometry, bisection is the division of something into two equal or Congruent parts usually by a line, which is then called a bisector

Derivation of the area formula

Area of the parallelogram is in blue

The area formula,

$A_\text{parallelogram} = B \times H,\,$

can be derived as follows:

The area of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is

$A_\text{rect} = (B+A) \times H\,$

and the area of a single orange triangle is

$A_\text{tri} = \frac{1}{2} A \times H\,$

Therefore, the area of the parallelogram is

$A_\text{parallelogram} = A_\text{rect} - 2 \times A_\text{tri} = \left( (B+A) \times H \right) - \left( A \times H \right) = B \times H\,$

Alternate method

Step one: ends of parallelogram are chopped off

Step two: pieces are rearranged

An alternative, less mathematically sophisticated method, to show the area is by rearrangement of the area. First, take the two ends of the parallelogram and chop them off to form two more triangles. Each of these two new triangles are equal in every way with the orange triangles. This first step is shown to the right.

The second step is merely swap the left orange triangle with the right blue triangle. Clearly, the two blue triangles plus the blue rectangle have an area equivalent to $B H$ .

To further demonstrate this, the first image on the right could be printed off and cut up along the lines:

Cut along the lines between the orange triangles and the blue parallelogram
Cut along the vertical lines on the end to form the two blue triangles and the blue rectangle
Rearrange all five pieces as shown in the second image

External links

Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)
Eric W. Weisstein, Parallelogram at MathWorld. In Mathematics, a fundamental pair of periods is an Ordered pair of Complex numbers that define a lattice in the Complex plane. Italic text See also Vector addition The Parallelogram of forces is a method for solving (or visualizing the results of applying several different In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses Synthetic geometry is the branch of Geometry which makes use of Theorems and synthetic observations to draw conclusions as opposed to Analytic geometry In Geometry, a gnomon is a plane figure formed by removing a similar Parallelogram from a corner of a larger parallelogram Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Interactive Parallelogram --sides, angles and slope
Area of Parallelogram at cut-the-knot
Equilateral Triangles On Sides of a Parallelogram at cut-the-knot
Varignon and Wittenbauer Parallelograms by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
Van Aubel's theorem Quadrilateral with four squares by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
Parallelogram Quiz
Definition and properties of a parallelogram with animated applet
Interactive applet showing parallelogram area calculation interactive applet

Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics. Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics.

Dictionary

-noun

(geometry) A convex quadrilateral in which each pair of opposite edges are parallel and of equal length.

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