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Parallelogram
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a parallelogram is a quadrilateral
Quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
 with two sets of parallel
Parallel

From Greek language: pa???????? Parallel may refer to:...
 sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size. The three-dimensional counterpart of a parallelogram is a parallelepiped
Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
.

Types of parallelograms

>(alternate)

Since they are angles that a transversal makes with parallel lines
Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more line s or plane , or a combination of these....
  and .

Also, since they are a pair of vertical angles
Vertical (angles)

A pair of angles is said to be vertical or opposite if the angles share the same vertex and are bounded by the same pair of Line but are opposite to each other....
.

Therefore, since they have the same angles.

From this similarity
Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape. Equivalently and more precisely, one is congruence to the result of a uniform Scaling of the other....
, one has the ratios

Since , we have .

Therefore,

bisects
Bisection

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line , which is then called a bisector. The most often considered types of bisectors are segment bisectors and angle bisectors....
 the diagonals and .

It can also be proved that the diagonals bisect each other, by placing the parallelogram on a coordinate grid, and assigning variables to the vertices, it can be shown that the diagonals have the same midpoint.

There is yet another way to prove that the diagonals of a parallelogram bisect each other.

It is known that AB = CD, because opposite sides of a parallelogram are equal.






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Parallelogram
In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a parallelogram is a quadrilateral
Quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
 with two sets of parallel
Parallel

From Greek language: pa???????? Parallel may refer to:...
 sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size. The three-dimensional counterpart of a parallelogram is a parallelepiped
Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
.

Properties


  • Opposite sides of a parallelogram are equal in length.
  • Opposite angles of a parallelogram are equal in measure.
  • The area, , of a parallelogram is , where is the base of the parallelogram and is its height.
  • Opposite sides of a parallelogram will never intersect.
  • 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
    Adjacent

    Adjacent is an adjective meaning contiguous, adjoining or abutting.In geometry, adjacent is when sides meet to make an angle....
     sides.
  • The diagonal
    Diagonal

    A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
    s of a parallelogram bisect
    Bisection

    In geometry, bisection is the division of something into two equal or congruent parts, usually by a line , which is then called a bisector. The most often considered types of bisectors are segment bisectors and angle bisectors....
     each other.
  • Any non-degenerate affine transformation
    Affine transformation

    In geometry, an affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation :...
     takes a parallelogram to another parallelogram. There are infinite affine transformations which take any given parallelogram to a square
    Square

    Square may mean:...
    .
  • Opposite sides of a parallelogram are parallel (by definition).


Types of parallelograms

  • Parallelogram - A quadrilateral whose opposite sides are parallel.
  • Rectangle - A quadrilateral with four angles of equal size (right angles).
  • Rhombus - A quadrilateral with four sides of equal length.
  • Square - A quadrilateral with four sides of equal length and four angles of equal size (right angles).


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:

(alternate) (alternate)

Since they are angles that a transversal makes with parallel lines
Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more line s or plane , or a combination of these....
  and .

Also, since they are a pair of vertical angles
Vertical (angles)

A pair of angles is said to be vertical or opposite if the angles share the same vertex and are bounded by the same pair of Line but are opposite to each other....
.

Therefore, since they have the same angles.

From this similarity
Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape. Equivalently and more precisely, one is congruence to the result of a uniform Scaling of the other....
, one has the ratios

Since , we have .

Therefore,

bisects
Bisection

In geometry, bisection is the division of something into two equal or congruent parts, usually by a line , which is then called a bisector. The most often considered types of bisectors are segment bisectors and angle bisectors....
 the diagonals and .

It can also be proved that the diagonals bisect each other, by placing the parallelogram on a coordinate grid, and assigning variables to the vertices, it can be shown that the diagonals have the same midpoint.

There is yet another way to prove that the diagonals of a parallelogram bisect each other.

It is known that AB = CD, because opposite sides of a parallelogram are equal. It is also known that since segment AB is parallel to segment CD (definition of parallelogram), then AIA are congruent. ASA postulate proves that these two triangles are congruent. Therefore, segment AE is equal to segment CE (corresponding parts of congruent triangles are equal), and therefore, point E bisects segment AC.

Derivation of the area formula


The area formula,

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

and the area of a single orange triangle is

or

Therefore, the area of the parallelogram is

Computing the area of a parallelogram


Let and let denote the matrix with columns and . Then the area of the parallelogram generated by and is equal to

Let and let . Then the area of the parallelogram generated by and is equal to

Let . 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:

See also


  • Fundamental parallelogram


External links

  • at cut-the-knot
    Cut-the-knot

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

    Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics....
  • by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
  • Quadrilateral with four squares by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
  • with animated applet
  • interactive applet