From Wikipedia, the free encyclopedia Parallelogram
Parallelogram
Parallelogram Properties
• Opposite sides of a parallelogram are parallel (by
definition) and so 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 sides.
• Any line through the midpoint of a parallelogram
bisects the area.[3]
This parallelogram is a rhomboid as its angles are oblique. • Any non-degenerate affine transformation takes a
Type quadrilateral parallelogram to another parallelogram.
• A parallelogram has rotational symmetry of order 2
Edges and vertices 4
(through 180°). If it also has two lines of reflectional
Symmetry group C2, [2]+, (22) symmetry then it must be a rhombus or an oblong.
• The perimeter of a parallelogram is 2(a + b) where a
Area B × H;
and b are the lengths of adjacent sides.
ab sin θ
Properties convex
Types of parallelogram
In Euclidean geometry, a parallelogram is a convex • Rhomboid – A quadrilateral whose opposite sides are
quadrilateral with two pairs of parallel sides. The oppo- parallel and adjacent sides are unequal, and whose
site or facing sides of a parallelogram are of equal length angles are not right angles
and the opposite angles of a parallelogram are of equal • Rectangle – A parallelogram with four angles of
measure. The congruence of opposite sides and opposite equal size
angles is a direct consequence of the Euclidean Parallel • Rhombus – A parallelogram with four sides of equal
Postulate and neither condition can be proven without length.
appealing to the Euclidean Parallel Postulate or one of its • Square – A parallelogram with four sides of equal
equivalent formulations. The three-dimensional coun- length and four angles of equal size (right angles).
terpart of a parallelogram is a parallelepiped.
The etymology (in Greek παραλληλ-όγραμμον, a
shape "of parallel lines") reflects the definition.
Area formulas
Characterizations
A convex quadrilateral is a parallelogram if and only if
any one of the following statements is true:[1][2]
• Each diagonal divides the quadrilateral into two
congruent triangles with the same orientation.
• The opposite sides are equal in length.
• The diagonals bisect each other.
• The opposite angles are equal in measure.
• The sum of the squares of the sides equals the sum of
the squares of the diagonals. (This is the
The area of the parallelogram is the area of the blue region,
parallelogram law.) which is the interior of the parallelogram
• It possesses rotational symmetry.
• One pair of opposite sides are parallel and equal in • The area K of the parallelogram to the right (the blue
length. area) is the total area of the rectangle less the area of
• Adjacent angles are supplementary. the two orange triangles.
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From Wikipedia, the free encyclopedia Parallelogram
The area of the rectangle is
Proof that diagonals bisect
each other
and the area of a single orange triangle is
Therefore, the area of the parallelogram is
To prove that the diagonals of a parallelogram bisect
• Another area formula, for two sides B and C and each other, we will use congruent triangles:
angle θ, is (alternate interior angles
are equal in measure)
• The area of a parallelogram with sides B and C (B ≠ C) (alternate interior angles
and angle γ at the intersection of the diagonals is are equal in measure).
given by[4]
(since these are angles that a transversal makes with par-
allel lines AB and DC).
Also, side AB is equal in length to side DC, since oppo-
site sides of a parallelogram are equal in length.
The area on coordinate system Therefore triangles ABE and CDE are congruent (ASA
postulate, two corresponding angles and the included side).
Let vectors and let Therefore,
AE = CE
denote the matrix with BE = DE.
elements of a and b. Then the area of the parallelogram
Since the diagonals AC and BD divide each other into seg-
generated by a and b is equal to
ments of equal length, the diagonals bisect each other.
. Separately, since the diagonals AC and BD bisect each
Let vectors and let other at point E, point E is the midpoint of each diagonal.
See also
Then the
area of the parallelogram generated by a and b is equal to • Fundamental parallelogram
• Parallelogram law
. • Rhombus
Let points . Then the area of the par-
allelogram with vertices at a, b and c is equivalent to the References
absolute value of the determinant of a matrix built using [1] Owen Byer, Felix Lazebnik and Deirdre Smeltzer,
a, b and c as rows with the last column padded using ones Methods for Euclidean Geometry, Mathematical
as follows: Association of America, 2010, pp. 51-52.
[2] Zalman Usiskin and Jennifer Griffin, "The
Classification of Quadrilaterals. A Study of
Definition", Information Age Publishing, 2008, p.
22.
[3] Dunn, J.A., and J.E. Pretty, "Halving a triangle",
Mathematical Gazette 56, May 1972, p. 105.
[4] Mitchell, Douglas W., "The area of a quadrilateral",
Mathematical Gazette, July 2009.
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From Wikipedia, the free encyclopedia Parallelogram
External links • Equilateral Triangles On Sides of a Parallelogram at
cut-the-knot
• Parallelogram and Rhombus - Animated course • Definition and properties of a parallelogram with
(Construction, Circumference, Area) animated applet
• Weisstein, Eric W., "Parallelogram" from MathWorld. • Interactive applet showing parallelogram area
• Interactive Parallelogram --sides, angles and slope calculation interactive applet
• Area of Parallelogram at cut-the-knot
Retrieved from "http://en.wikipedia.org/w/index.php?title=Parallelogram&oldid=474112596"
Categories:
• Quadrilaterals
• Elementary shapes
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