** Parallelogram vs Rhombus **

Parallelogram and rhombus are quadrilaterals. The geometry of these figures were known to man for thousands of years. The subject is explicitly treated in the book “Elements” written by Greek mathematician Euclid.

**Parallelogram**

Parallelogram can be defined as the geometric figure with four sides, with opposite sides parallel to each other. More precisely it is a quadrilateral with two pairs of parallel sides. This parallel nature gives many geometric characteristics to the parallelograms.

A quadrilateral is a parallelogram if following geometric characteristics are found.

*• Two pairs of opposing sides are equal in length. (AB = DC, AD = BC)*

*• Two pairs of opposing angles are equal in size. ([latex]D\hat{A}B = B\hat{C}D, A\hat{D}C = A\hat{B}C[/latex])*

*• If the adjacent angles are supplementary [latex]D\hat{A}B + A\hat{D}C = A\hat{D}C + B\hat{C}D = B\hat{C}D + A\hat{B}C = A\hat{B}C + D\hat{A}B = 180^{\circ} = \pi rad[/latex]*

*• A pair of sides, which are opposing each other, is parallel and equal in length. ( AB = DC & AB∥DC)*

*• The diagonals bisect each other (AO = OC, BO = OD)*

*• Each diagonal divides the quadrilateral into two congruent triangles. (∆ADB ≡ ∆BCD, ∆ABC ≡ ∆ADC)*

Further, the sum of the squares of the sides is equal to the sum of the squares of diagonals. This is sometimes referred to as the * parallelogram law* and has widespread applications in physics and engineering. (AB

^{2 }+ BC

^{2 }+ CD

^{2 }+ DA

^{2 }= AC

^{2 }+ BD

^{2})

Each of the above characteristics can be used as properties, once it’s established that the quadrilateral is a parallelogram.

Area of the parallelogram can be calculated by the product of the length of one side and the height to the opposite side. Therefore, area of the parallelogram can be stated as

*Area of parallelogram = base × height = AB×h*

The area of the parallelogram is independent of the shape of individual parallelogram. It is dependent only on the length of base and the perpendicular height.

If the sides of a parallelogram can be represented by two vectors, the area can be obtained by the magnitude of the vector product (cross product) of the two adjacent vectors.

If sides AB and AD are represented by the vectors ([latex]\overrightarrow{AB}[/latex]) and ([latex]\overrightarrow{AD}[/latex]) respectively, the area of the parallelogram is given by [latex]\left | \overrightarrow{AB}\times \overrightarrow{AD} \right | = AB\cdot AD \sin \alpha [/latex], where α is the angle between [latex]\overrightarrow{AB}[/latex] and [latex]\overrightarrow{AD}[/latex].

Following are some advanced properties of the parallelogram;

*• The area of a parallelogram is twice the area of a triangle created by any of its diagonals.*

*• The area of the parallelogram is divided in half by any line passing through the midpoint.*

*• Any non-degenerate affine transformation takes a parallelogram to another parallelogram*

*• A parallelogram has rotational symmetry of order 2*

*• The sum of the distances from any interior point of a parallelogram to the sides is independent of the location of the point*

**Rhombus**

A quadrilateral with all sides are equal in length is known as a rhombus. It is also named as an * equilateral quadrilateral*. It is considered to have a diamond shape, similar to the one in the playing cards.

Rhombus is also a special case of the parallelogram. It can be considered as a parallelogram with all four sides equal. And it has following special properties, in addition to the properties of a parallelogram.

*• The diagonals of the rhombus bisect each other at right angles; diagonals are perpendicular.*

*• The diagonals bisect the two opposite internal angles.*

*• At least two of the adjacent sides are equal in length.*

The area of the rhombus can be calculated in the same method as the parallelogram.

**What is the difference between Parallelogram and Rhombus?**

• Parallelogram and rhombus are quadrilaterals. Rhombus is a special case of the parallelograms.

• Area of any can be calculated using the formula base ×height.

• Considering the diagonals;

– The diagonals of the parallelogram bisect each other, and bisect the parallelogram to form two congruent triangles.

– The diagonals of the rhombus bisect each other at right angles, and the triangles formed are equilateral.

• Considering the internal angles;

– Opposing internal angles of the parallelogram are equal in size. Two adjacent internal angles are supplementary.

– The internal angles of the rhombus are bisected by the diagonals.

• Considering the sides;

– In a parallelogram, the sum of squares of the sides is equal to the sum of the squares of the diagonal (Parallelogram law).

– As all four sides are equal in a rhombus, four times the square of a side is equal to the sum of the squares of the diagonal.

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