Parallelogram
A parallelogram is a polygon with four sides (a quadrilateral). It has two pairs of parallel sides (line segments which never meet if the lines were allowed to extend beyond their end points). The opposite sides of a parallelogram have the same length (they are equally long). The word "parallelogram" comes from the Greek word "parallelogrammon" (bounded by parallel lines).[1] Rectangles, rhombuses, and squares are all parallelograms.
| Parallelogram | |
|---|---|
| File:Parallelogram.svg This parallelogram is a rhomboid as it has no right angles and unequal sides. | |
| Type | quadrilateral, trapezium |
| Edges and vertices | 4 |
| Symmetry group | C2, [2]+, |
| Area | b × h (base × height); ab sin θ (product of adjacent sides and sine of the vertex angle determined by them) |
| Properties | convex |
As shown in the picture on the right, because triangles ABE and CDE are congruent (have the same shape and size),
- [math]\displaystyle{ AE = CE }[/math]
- [math]\displaystyle{ BE = DE. }[/math]
In all Parallelogram's opposite angles are equal to each other. Angles which are not opposite in the Parallelogram will add up to 180 degrees.
Characterizations
A simple (non self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:[2][3]
- Two pairs of opposite sides are equal in length
- Two pairs of opposite angles are equal in measure
- The diagonals bisect each other
- One pair of opposite sides are parallel and equal in length
- Adjacent angles are supplementary
- Each diagonal divides the quadrilateral into two congruent triangles
- The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law)
- It has rotational symmetry of order 2
- It has two lines of symmetry
Properties
- Opposite sides of parallelogram are parallel.
- Any line through the midpoint of a parallelogram bisects the area.
- Parallelograms are quadrilaterals.
Area formula
A parallelogram can be cut into a trapezoid and a right triangle, and be rearranged to make a rectangle. This mean the area of a parallelogram is the same with a rectangle of the same height and base as that parallelogram:
[math]\displaystyle{ K=bh. }[/math]
Parallelogram Media
- ParallelogramArea.svg
A parallelogram can be rearranged into a rectangle with the same area.
- Parallelogram area animated.gif
Animation for the area formula K = bh.
- Parallelogram area.svg
The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram
- Parallelogram1.svg
A parallelogram with the diagonals drawn and relevant points labeled
- Lattice of squares.svg
- 000000* * B5E61D * FFF200
- Lattice of rectangles.svg
- 000000* * B5E61D * FFF200
- Lattice of rhombuses.svg
- 000000* * B5E61D * FFF200
- Lattice of rhomboids.svg
- 000000* * B5E61D * FFF200
- Varignon parallelogram.svg
Proof without words of Varignon's theorem
References
- ↑ "Online Etymology Dictionary". etymonline.com. Retrieved 10 January 2011.
- ↑ Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, pp. 51-52.
- ↑ Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 22.
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