Rectangle Properties

A rectangle is any quadrilateral with four right angles. Rectangle which consists of pair of two opposite sides along with the two diagonals. The diagonals intersect each other, and the lengths of the two diagonals are also equal. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles. (Source: Wikipedia)

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Properties of rectangle:

The rectangle has length and breadth. The horizontal sides of the rectangle are known as length of the rectangle and

Denoted by l.The vertical sides of the rectangle are known as width of the rectangle and denoted by the ‘w’ or ‘d’

A rectangle has 4 sides and 4 angles.

Four angles are equal right angles: 90

The opposite sides of a rectangle are equal in distance

Opposite sides of a rectangle are equal and all sides need not be equal.

If all sides of a rectangle are equal , it is called a square

The sum of four angles equal to 360 degree.

In a rectangle all angles are congruent

Axes of symmetry of a rectangle bisect opposite sides.

The diagonal of a rectangle are equal and intersect each other at 90 degree.

The area of a rectangle is measured with the help of the following formula:

Area of rectangle=length*width = l*d square unit

Perimeter of the rectangle= 2( length+ width)= 2(l+d) unit

This is a rectangle

Rectangle

And here AB and DC are the length of the rectangle and

AD and BC are the width of the rectangle

And AC and BD is the diagonal of the rectangle.

and the measure of all angles that are A,B,C,D is equal to 90 degree.

rect

Here length of the rectangle is 5 cm and width is equal to 11 cm

So the area of the rectangle =length*width =5*11=55 cm^2

And perimeter of the rectangle = 2(length+ width)=2(5+11)=2*16 =32cm

And the diagonal of the rectangle is equal to

=√(l^2+w^2)

=√(5^2+11^2)

=√(25+122)

=√147

=12.12cm

A square is always a rectangle because it always follows all the properties of a rectangle but a rectangle may be a square or not

Because a square have all sides equal but in rectangle it is not compulsory.

Proof of rectangle properties

Proof 1: If in a rectangle one angle is 90 degrees then all angles are 90 degrees.

Consider a rectangle ABCD, where it is given that Angle BCD = 90 degrees.

Since opposite angles of a rectangle are equal.

Hence, Angle DAB = Angle BCD ……………….(1)

Now, AB || DC,

Hence, By Sum of angles property in a parallel lines,

Angle DAB + Angle ACD = 180°.

Angle ADC = 180° – 90° =90° ………………………. (2)

Again by the property of parallelogram those opposite angles are equal.

Angle ABC =Angle ADC =90 ………………………… (3)

From Equations (1),(2) and (3)

Angle BCD = Angle DAB = Angle ADC =Angle ABC = 90

Hence all angle in a rectangle are equal to 90 degrees.

Proof 2: The opposite sides of a rectangle are equal.

Special type of rectangle is parallelogram whose all angles are 90 degrees. From the property the opposite sides of rectangle are hence equal.

Hence AB =DC ....................................... (4)

Proof 3: Diagonals of a rectangle are the same lenght and bisect each other.

Consider two Triangle ABD and ADC containing two diagonals BD and AC respectively.

Triangles ABD and ADC are Right Angled Triangles. In which the right angled at Angle (DAC) and Angle (ADC) respectively.

From Pythagorean theorem,

In Triangle ABD,

BD2 = (AB2 + AD2) ........................................... (5)

In Triangle ADC,

AC2 = AD2 + DC2 ............................................. (6)

From equations (4), (5) and (6)

BD2 = AC2 =(AD2 + DC2) =(AB2 + AD2)

BD2 = AC2

BD = AC

Hence Diagonals are of equal length in a rectangle.

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From property of parallelogram the diagonals are divided into two at their point of intersection. As diagonals are equal in a rectangle hence,

Diagonals of a rectangle are equal and bisect each other.

Example question on rectangle:

If the length of a rectangle is 4cm and width is 2 cm then find the area of the rectangle

The formula of the area of the rectangle is= length* width

=4*2

=8cm^2

And perimeter = 2(length+width)

=2(4+2)

12 cm

2:If the area of the rectangle is 20cm^2

And perimeter is 18cm then find the length and width of the rectangle?

Here l*w=20

And l+w  =9cm

So l(9-l)=20

From this equation l=5,4

So the length of the rectangle is 4cm or 5cm

And width is 5cm or 4cm

More example questions on rectangle

1: If the area of a rectangle is 20cm2 and width is 5cm . find the length of the rectangle?

ans: length=4cm

2: If the the area of a rectangle is 50cm2 and width is half of the length then find the length and width of the rectangle?

ans: width =5cm