** Parallelogram vs Rectangle **

Parallelogram and rectangle 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*

**Rectangle**

A quadrilateral with four right angles is known as a rectangle. It is a special case of the parallelogram where the angles between any two adjacent sides are right angles.

In addition to all the properties of a parallelogram, additional characteristics can be recognized when considering the geometry of the rectangle.

*• Every angle at the vertices is a right angle.*

*• The diagonals are equal in length, and they bisect each other. Therefore, the bisected sections are also equal in length.*

*• The length of the diagonals can be computed using Pythagoras` theorem:*

*PQ ^{2 }+ PS^{2 }= SQ^{2}*

*• The area formula reduces to the product of length and width.*

*Area of rectangle = length × width*

*• Many symmetric properties are found on a rectangle, such as;*

*– A rectangle is cyclic, where all the vertices can be placed on the perimeter of a circle.*

*– It’s equiangular, where all the angles are equal.*

*– It is isogonal, where all corners lie within the same symmetry orbit.*

*– It has both reflectional symmetry and rotational symmetry.*

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

• Parallelogram and rectangle are quadrilaterals. Rectangle 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 rectangle are equal in length and bisect each other; bisected sections are equal in length. The diagonals bisect the rectangle into two congruent right triangles.

• Considering the internal angles;

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

– All four internal angles of the rectangle are right angles.

• 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)

– In rectangles, the sum of the squares of the two adjacent sides is equal to the square of the diagonal at the ends. (Pythagoras` Rule)

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