Parallelogram

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Parallelogram
Parallelogram.svg
This parallelogram is a rhomboid as it has no right angles and unequal sides.
Typequadrilateral
Edges and vertices4
Symmetry groupC2, [2]+, (22)
Areab × h (base × height);
ab sin θ
Propertiesconvex
 
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Parallelogram
Parallelogram.svg
This parallelogram is a rhomboid as it has no right angles and unequal sides.
Typequadrilateral
Edges and vertices4
Symmetry groupC2, [2]+, (22)
Areab × h (base × height);
ab sin θ
Propertiesconvex

In Euclidean geometry, a parallelogram is a simple (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.

Special cases[edit]

Characterizations[edit]

A simple (non self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:[2][3]

Properties[edit]

Area formula[edit]

A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle
A parallelogram can be rearranged into a rectangle with the same area.

A = bh

The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram
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
K = A_\text{rect} - 2 \times A_\text{tri} = ( (B+A) \times H) - ( A \times H) = B \times H
K = B \cdot C \cdot \sin \theta.\,
K = \frac{|\tan \gamma|}{2} \cdot \left| B^2 - C^2 \right|.
K=2\sqrt{S(S-B)(S-C)(S-D_1)}
where S=(B+C+D_1)/2 and the leading factor 2 comes from the fact that the number of congruent triangles that the chosen diagonal divides the parallelogram into is two.

The area on coordinate system[edit]

Let vectors \mathbf{a},\mathbf{b}\in\R^2 and let V = \begin{bmatrix} a_1 & a_2 \\ b_1 & b_2 \end{bmatrix} \in\R^{2 \times 2} denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to |\det(V)| = |a_1b_2 - a_2b_1|\,.

Let vectors \mathbf{a},\mathbf{b}\in\R^n and let V = \begin{bmatrix} a_1 & a_2 & \dots & a_n \\ b_1 & b_2 & \dots & b_n \end{bmatrix} \in\R^{2 \times n} Then the area of the parallelogram generated by a and b is equal to \sqrt{\det(V V^\mathrm{T})}.

Let points a,b,c\in\R^2. 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:

K = \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[edit]

Parallelogram ABCD

To prove that the diagonals of a parallelogram bisect each other, we will use congruent triangles:

\angle ABE \cong \angle CDE (alternate interior angles are equal in measure)
\angle BAE \cong \angle DCE (alternate interior angles are equal in measure).

(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,

AE = CE
BE = DE.

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.

See also[edit]

References[edit]

  1. ^ http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf
  2. ^ Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, pp. 51-52.
  3. ^ Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 22.
  4. ^ Dunn, J.A., and J.E. Pretty, "Halving a triangle", Mathematical Gazette 56, May 1972, p. 105.
  5. ^ Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", The College Mathematics Journal 37(5), 2006, pp. 390–391.
  6. ^ Mitchell, Douglas W., "The area of a quadrilateral", Mathematical Gazette, July 2009.

External links[edit]