PARALLELOGRAM

In geometry, a 'parallelogram' is a quadrilateral with two sets of parallel sides. The opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are congruent. The three-dimensional counterpart of a parallelogram is a parallelepiped.

Contents

Properties

Special cases

Vector spaces

Proof that diagonals bisect each other

Derivation of the area formula

Alternate method

See also

External links

Properties

★ The two parallel sides are of equal length.

★ The area, $A$, of a parallelogram is $A\; =\; BH$ 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 is also equal to the magnitude of the vector cross product of two adjacent sides.

★ The diagonals of a parallelogram bisect each other.

★ It is possible to create a tessellation with any parallelogram.

Special cases

★ The rhombus is a special case of a parallelogram where all four sides are of equal length.

★ The rectangle is a special case of a parallelogram where the two sets of parallel sides are perpendicular to each other.

★ The square is a special case of a parallelogram where all four sides are of equal length ''and'' the two sets of parallel sides are perpendicular to each other.

★ The parallelogram is itself a special case of a trapezoid.

Vector spaces

In a vector space, addition of vectors is usually defined using the parallelogram law. The parallelogram law distinguishes Hilbert spaces from other Banach spaces.

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:
:
:
Since they are angles that a transversal makes with parallel lines $AB$ and $DC$.
Also, since they are a pair of vertical angles.
Therefore, $riangle\; ABE\; sim\; riangle\; CDE$ since they have the same angles.
From this similarity, we have the ratios
:$\{AB\; over\; CD\}\; =\; \{AE\; over\; CE\}\; =\; \{BE\; over\; DE\}$
Since $AB\; =\; DC$, we have
:$\{AB\; over\; CD\}\; =\; 1$.
Therefore,
:$AE\; =\; CE$
:$BE\; =\; DE$
$E$ bisects the diagonals $AC$ and $BD$.

Derivation of the area formula

The area formula,
:$A\_\{parallelogram\}\; =\; B\; imes\; H$,
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
:$A\_\{rect\}\; =\; (B+A)\; imes\; H$
and the area of a single orange triangle is
:
Therefore, the area of the parallelogram is
:$A\_\{parallelogram\}\; =$
A_{rect} - 2 imes A_{tri} =
left( (B+A) imes H
ight) - left( A imes H
ight) =
B imes H

Alternate method

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

See also

★ Fundamental parallelogram

★ Parallelogram of force

★ Rhombus

★ Synthetic geometry

★ Gnomon (figure)

External links

★ Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)

★

★ Interactive Parallelogram --sides, angles and slope

★ Area of Parallelogram at cut-the-knot

★ National Parallogram Dealers WebsiteNational Parallelogram Dealers Association

★ 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