What is a rhombus: definition, properties, signs

In this publication, we will consider the definition, properties and features (with drawings) of one of the main geometric shapes – a rhombus.

Diamond Definition

Rhombus is a figure on a plane; a type of parallelogram in which all four sides are equal and pairwise parallel. Usually a rhombus is denoted by the names of its vertices (for example, ABCD), and the length of its side – in a lowercase Latin letter (for example, a).

• AB = BC = CD = AD = a
• AB parallel CD, BC parallel AD.

Note: is a special case of a rhombus.

Rhombus Properties

Property 1

Opposite angles of a rhombus are equal to each other, and the sum of adjacent angles is 180°.

• ∠ABC = ∠ADC, ∠BAD = ∠BCD
• α + β = 180°

Property 2

The diagonals of the rhombus intersect at right angles and bisect at the point of intersection.

• diagonal BD (d₁) perpendicular diagonals AC (d₂)
• AE = EC
• BE = ED

As a result of the intersection of the diagonals, the rhombus is divided into 4 right triangles: ΔAEB, ΔBEC, ΔAED и ΔDEC.

Property 3

The diagonals of a rhombus are the bisectors of its angles.

Property 4

Side of a rhombus a can be found through its diagonals d₁ и d₂ (according to ).

• a – the hypotenuse of any of the 4 right triangles (for example, ΔBEC);
• half diagonals d₁ и d₂ – legs of triangles.

Property 5

Any rhombus can be inscribed with a circle whose center lies at the intersection of its diagonals.

Radius of a circle inscribed in a rhombus r calculated by the formula:

Signs of a rhombus

A parallelogram is a rhombus only if one of the following statements is true for it:

1. Its diagonals intersect at right angles.
2. If its diagonals are the bisectors of its angles.
3. Two adjacent sides are equal (hence all sides are equal).

Note: Any quadrilateral whose sides are equal is a rhombus.