Understanding Quadrilaterals - 8th math

The word polygon is derived from two Greek words "Polus" which means "many" and "gonia" which means "corner or angle". Thus, the word "polygon" means figure with many corners or angles.

In other words a polygon has many sides or angles. For example: a triangle is a polygon.

In other words "A simple closed curve made up of only line segments is called a polygon."

Types of Polygons

Polygons can be classified on the basis of their sides or angles.

(a) Triangle : A triangle has three angles and three sides. Thus, a triangle is a polygon. In other words a polygon with three sides or angles is called A TRIANGLE.

The word triangle is the combination of two words "tri + angle" which means of three angles.

(b) Quadrilateral : A polygon with four sides or four angles is known as QUADRILATERAL.

The word Quadrilateral is the combination of two words "Quadri + Lateral" which means having four sides.

(c) Petagon: A polygon with five sides or five angles is called A PENTAGON

Here "penta" means five and "gon" means angle.

(d) Hexagon: A polygon with six angles or sides is known as HEXAGON.

Here "hexa" means six and "gon" means angles or sides.

(e) Heptagon: A polygon with seven angles or sides is known as HEPTAGON. Here "hepta" means seven.

(f) Octagon: A polygon with eight angles or sides is known as OCTAGON. Here "octa" means eight similar in the word "Octopus" which means a creature with eight feet.

Diagonals

A line segment that connects two non-consecutive vertices of a polygon is known as a diagonal.

Example:

(a) In the figure of given rectangle ABCD, AC and DB are diagonals.

(b) In the figure of given pentagon EFGHL, EG, EH, LF and LG are diagonals.

(c) In the figure of given figure of MONP, PO and MN are diagonals but, MN is external diagonal.

Regular and Irregular Polygons

Polygons in which all the angles and sides are equal are called REGULAR POLYGONS. This means in regular polygon is both "EQUIANGULAR & EQUILATERAL".

And those polygons in which all angles and sides are not equal are called IRREGULAR POLYGONS.

Example:

(a) An equilateral triangle has all sides and angles equal; and thus an equilateral triangle is a regular polygon. Similarly a square is also a regular polygon.

(b) A rectangle is not a regular polygon as a rectangle has only all angles are equal and opposite sides equal, i.e. all sides are not equal in a rectangle.

Types of Quadrilateral

Parallelogram

A parallelogram is a quadrilateral in which opposite sides parallel and equal.

Properties of Parallelogram

In the given parallelogram ABCD

Side AB || to side DC and Side AD || side BC

Angle A = Angle C and Angle B = Angle C

Side AB = Side DC and side AD = side BC

OA = OC and OB = OD

Angle A + Angle D = 180⁰

And Angle A + Angle B = 180⁰

And Angle B + Angle C = 180⁰

And Angle C + Angle D = 180⁰

(a) Opposite sides of a parallelogram are parallel

(b) Opposite sides of a parallelogram are equal in length.

(c) The opposite angles of a parallelogram are of equal measure, i.e. opposite angles of a parallelogram are equal.

(d) The adjacent angles in a parallelogram are supplementary. This means the sum of adjacent angles in a parallelogram is equal to 180⁰

(e) The diagonals of a parallelogram bisect each other.

(f) Area of a parallelogram = 1/2 × Base × Height

(g) Area of a parallelogram = 1/2 × Diagonal1 × Diagonal2

(g) Perimeter of a parallelogram = 2(Base + Side length)

Square

A rectangle with all sides equal and all angles equal is called A Square. The length of both of the diagonals of a square is also equal.

Properties of square

In the given square

Side AB = BC = DC = AD

Diagonal AC = DB

Diagonal

And, AO = OC = DO = OB

And,

And,

(a) All the four sides are of equal length.

(b) All the four angles are equal.

(c) Length of both of the diagonals is equal.

(d) Both of the diagonals are perpendicular to each other.

(e) Since opposite side of a square are parallel thus a square is a parallelogram also.

(f) In a square diagonals bisect one another, and hence a square is a parallelogram also>

(g) In a square the diagonals are of equal length, so a square is a rectangle

(h) Area of a Square = Side²

(i) Area of a square = 1/2 × diagonal²

(j) Perimeter of a square = 4 × side

(k) Diagonal of a square = side×

Rectangle

A quadrilateral with opposite sides equal and all angles equal is called a rectangle.

In other words a rectangle is a parallelogram with equal angles.

In other words a rectangle is a parallelogram in which every angle is a right angle.

In the given figure of rectangle

(i) Side AB = Side DC

(ii) And Side AD = side BC

(iii) Angle A = angle B = angle C = Angle D = 90⁰

(iv) Diagonal AC = Diagonal DB

(v) And, OA = OC = OD = OB

(vi) Angle AOB = angle DOC and Angle AOD = angle COB

Properties of a Rectangle

(a) The opposite sides are equal in length in a rectangle.

(b) The opposite sides are parallel in a rectangle.

(c) Angle made with adjacent sides of a rectangle is equal to 90⁰.

(d) The diagonals of a rectangle are equal in length.

(e) The diagonals of a rectangle bisects one another.

(f) Opposite angles at the meeting point of diagonals are equal in measure.

(g) Area of a rectangle = Length × Height

(h) Area of a rectangle = 1/2 × square of diagonal

(i) Perimeter of a rectangle = 2(Length + Breadth)

Kite

A type of quadrilateral in which exactly two pair of consecutive sides is equal in length is called a kite.

In the given figure of kite

(i) Side AB = side BC

(ii) And, side AD = side DC

(iii) Angle A = Angle C

(iv) But, angle B ≠ angle D

(v) Diagonal AC ⊥ BD

Properties of a kite

(a) Exactly two pair of consecutive sides is equal in length a kite.

(b) One of the diagonal bisects other in a kite.

(c) Diagonals are perpendicular to one another in a kite.

(d) Angles are equal where two pair of sides meets in a kite.

(e) Area of a kite = 1/2 × product of diagonals.

(f) Perimeter of a kite = 2(side1 + side2)

Rhombus

A parallelogram with all the four sides are equal in length is called a rhombus.

In other words, a quadrilateral in which opposite sides parallel, all sides are of equal length and opposite angles are equal in measure is called a rhombus.

In other words rhombus is parallelogram in which all sides are equal and opposite angles are equal.

In the given figure of Rhombus

(i) AB || DC and AD || BC

(ii) And AB = BC = CD = AD

(iii) Angle A = angle C and Angle D = angle B

(iv) AO = OC and OD = OB

Properties of Rhombus

(a) Opposite sides are parallel in a rhombus

(b) All sides are equal in a rhombus

(c) Diagonals are perpendicular bisector in a rhombus

(d) Adjacent angles are supplementary in a rhombus. This means sum of adjacent angles are equal to 180⁰ in a rhombus

(e) Since adjacent sides of a rhombus are equal in length, thus a rhombus is a kite.

(e) Area of a rhombus = side²

(f) Area of a = 1/2 × diagonal²

(g) Perimetre of a rhombus = 4 × side

Trapezium

Image of a Trapezium

Isosceles trapezium

A quadrilateral in which a pair of sides is parallel is called trapezium.

In the given figure of trapezoid

(i) Side AB || DC

(ii) DE is the height of the trapezium

(iii) When two sides of a trapezium are equal, it is called Isosceles trapezium.

(iv) The red arrow shows the parallel sides.

Trapezium and Trapezoid

In UK a quadrilateral with a pair of parallel sides is called a trapezium. And a quadrilateral with no parallel side is called a trapezoid.

While in USA a quadrilateral with a pair of parallel sides is called a trapezioid. And a quadrilateral with no parallel side is called a trapezium.

Properties of Trapezium

(a) A pair of sides is parallel in a trapezium.

(b) In a trapezium when two sides are of equal length then it is known as isosceles trapezium

(c) Area of a trapezium = 1/2 (sum of parallel sides)