Parallelogram

A parallelogram or rhomboidal ( diamond- like ) is a planar convex quadrilateral, wherein opposite sides are parallel.

Parallelograms are special harnesses and two-dimensional parallelepipeds. Rectangle, rhombus (diamond ) and square are special cases of the parallelogram.

Properties

A nondegenerate quadrilateral is a parallelogram if and only if one of the following conditions is met:

  • Opposite sides are equal and no two opposite sides intersect (no on proposed square, called Antiparallelogramm ).
  • Opposite angles are equal.
  • Two adjacent angles totaling 180 degrees.
  • The diagonals bisect each other.
  • It is point-symmetric ( two-numbered rotational symmetry ).

For each parallelogram applies:

  • Each diagonal divides it into two ( equal -oriented ) congruent triangles.
  • The center of symmetry is the intersection of the diagonals.

Use in the art

Parallelograms are often found in mechanics. By means of four joints, a portable parallel storage faithful are made, the so-called parallelogram. Examples:

Parallel wiper

Platform

Pantograph

Formulary

About transformation into a rectangle with the determinant:

Generalizations

A generalization to dimensions is the parallelotope explained as the amount and their parallel shifts. Here are the linearly independent vectors.

Proof of the area formula for a parallelogram

The area of ​​the adjacent black parallelogram can get one by subtracting from the area of the large rectangle, the six small areas with colored edges. Because of the symmetry and the commutativity of multiplication, you can also deduct from the large rectangle twice the three small areas below the parallelogram. It is therefore:

Convex set Windscreen wiper Internal and external angle Translation (geometry)
633126
de