Rhombus

 Rhombus

Rhombus
Rhombus.svg

Two rhombi

VideoProperties of a Rhombus

Type
quadrilateral, parallelogram, kite
Edges
and
vertices
4
Schläfli symbol
{ } + { }
Coxeter diagram
CDel node 1.pngCDel sum.pngCDel node 1.png
Symmetry group
Dihedral (D2), [2], (*22), order 4
Area
  (half the product of the diagonals)
Dual polygon
rectangle
Properties
convex, isotoxal

The rhombus has a square as a special case, and is a special case of a kite and parallelogram.

In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.

Every rhombus is a parallelogram and a kite. A rhombus with right angles is a square.

Characterizations

A simple (non-self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:

  • a parallelogram in which a diagonal bisects an interior angle
  • a parallelogram in which at least two consecutive sides are equal in length
  • a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram)
  • a quadrilateral with four sides of equal length (by definition)
  • a quadrilateral in which the diagonals are perpendicular and bisect each other
  • a quadrilateral in which each diagonal bisects two opposite interior angles
  • a quadrilateral ABCD possessing a point P in its plane such that the four triangles ABP, BCP, CDP, and DAP are all congruent
  • a quadrilateral ABCD in which the incircles in triangles ABC, BCD, CDA and DAB have a common point

Basic properties

Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties:

  • Opposite angles of a rhombus have equal measure.
  • The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral.
  • Its diagonals bisect opposite angles.

The first property implies that every rhombus is a parallelogram. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel; adjacent angles are supplementary; the two diagonals bisect one another; any line through the midpoint bisects the area; and the sum of the squares of the sides equals the sum of the squares of the diagonals (the parallelogram law). Thus denoting the common side as a and the diagonals as p and q, in every rhombus

Not every parallelogram is a rhombus, though any ......

 

 

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