From Wikipedia, the free encyclopedia - View original article
Parallelogram | |
---|---|
This parallelogram is a rhomboid as it has no right angles and unequal sides. | |
Type | quadrilateral |
Edges and vertices | 4 |
Symmetry group | C_{2}, [2]^{+}, (22) |
Area | b × h (base × height); ab sin θ (product of adjacent sides and sine of any vertex angle) |
Properties | convex |
Parallelogram | |
---|---|
This parallelogram is a rhomboid as it has no right angles and unequal sides. | |
Type | quadrilateral |
Edges and vertices | 4 |
Symmetry group | C_{2}, [2]^{+}, (22) |
Area | b × h (base × height); ab sin θ (product of adjacent sides and sine of any vertex angle) |
Properties | convex |
In Euclidean geometry, a parallelogram is a (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. The three-dimensional counterpart of a parallelogram is a parallelepiped.
The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") reflects the definition.
A simple (non self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:^{[2]}^{[3]}
Let vectors and let denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to .
Let vectors and let . Then the area of the parallelogram generated by a and b is equal to .
Let points . Then the area of the parallelogram with vertices at a, b and c 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:
To prove that the diagonals of a parallelogram bisect each other, we will use congruent triangles:
(since these are angles that a transversal makes with parallel lines AB and DC).
Also, side AB is equal in length to side DC, since opposite sides of a parallelogram are equal in length.
Therefore triangles ABE and CDE are congruent (ASA postulate, two corresponding angles and the included side).
Therefore,
Since the diagonals AC and BD divide each other into segments of equal length, the diagonals bisect each other.
Separately, since the diagonals AC and BD bisect each other at point E, point E is the midpoint of each diagonal.
An automedian triangle is one whose medians are in the same proportions as its sides (though in a different order). If ABC is an automedian triangle in which vertex A stands opposite the side a, G is the centroid (where the three medians of ABC intersect), and AL is one of the extended medians of ABC with L lying on the circumcircle of ABC, then BGCL is a parallelogram.
The midpoints of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its Varignon parallelogram. If the quadrilateral is convex or concave (that is, not self-intersecting), then the area of the Varignon parallelogram is half the area of the quadrilateral.
For an ellipse, two diameters are said to be conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram, sometimes called a bounding parallelogram, formed by the tangent lines to the ellipse at the four endpoints of the conjugate diameters. All tangent parallelograms for a given ellipse have the same area.
It is possible to reconstruct an ellipse from any pair of conjugate diameters, or from any tangent parallelogram.
A parallelepiped is a three-dimensional figure whose six faces are parallelograms.
Wikimedia Commons has media related to Parallelograms. |