Quadrilateral and Polygon

 

Quadrilateral and Polygon with Their Properties

 

QUADRILATERAL

It is a plane figure bounded by four straight lines. It has four sides and four internal angles. The sum of the internal angles of a quadrilateral is equal to 360°.

 

TYPES OF QUADRILATERAL

 

Trapezium

It is a quadrilateral where only one pair of opposite sides are parallel. ABCD is a trapezium as \[AB\parallel DC.\]

Area of trapezium = \[\frac{1}{2}(AB+CD)\times AE\]

Parallelogram

A quadrilateral in which the opposite sides are equal and parallel, is called a parallelogram. In a parallelogram,

·         The opposite sides are parallel and of equal lengths. AB = DC and AD = BC.

·         The sum of any two adjacent interior angles is equal to \[180{}^\circ .\]

\[\angle A+\angle B=\angle B+\angle C=\angle C+\angle D=\angle D+\angle A=180{}^\circ \]

·         The opposite angles are equal in magnitudes \[\angle A=\angle C\] and \[\angle B=\angle D.\]

·         The diagonals of a parallelogram are not equal in magnitudes, but they bisect each other. \[AC\ne BD\]but\[AO=OC\]and \[OB=OD.\]

 

Cyclic Quadrilateral

A quadrilateral whose vertices are on the circumference of a circle, is called a cyclic quadrilateral. The opposite angles of a cyclic quadrilateral are supplementary i.e.,

\[a+\beta =180{}^\circ .\]

 

Rhombus

A parallelogram in which all the sides are equal, is called a rhombus.

·         The opposite sides are parallel and all the sides are of equal lengths AB = BC = CD = DA.

·         The sum of any two adjacent interior angles is equal to

\[180{}^\circ \angle A\text{ }+\angle B=\angle B+\angle C=\angle C+\angle D=\angle D+\angle A=180{}^\circ \]

·         The opposite angles are equal in magnitudes, i.e.,

\[\angle A=\angle C\]and\[\angle A=\angle D.\]

·         The diagonals bisect each other at right angles and form four right angled triangles. They are not of equal magnitudes.

·         Area of the four right triangles,

\[\Delta AOB=\Delta BOC=\Delta COD=\Delta DOA\]and each equals \[\frac{1}{4}\]the area of the rhombus.

·         Sum of the squares of sides is equal to the sum of the squares of its diagonals.

Rectangle

A parallelogram in which the adjacent sides are perpendicular to each other, is called a rectangle.

·         The opposite sides are parallel and of equal lengths, i.e.,

AB = CD and AD = BC.

·         The adjacent sides are perpendicular i.e.,

\[\angle A=\angle B=\angle C=\angle D=90{}^\circ \]

·         The diagonals of a rectangle are of equal magnitudes and bisect each other i.e., AC = BD and OA = OB = OC = OD.

Square

A parallelogram in which all the sides are equal and perpendicular to each other, is called a square.                               

·         The opposite sides are parallel and all the sides are of equal lengths i.e.,

AB = BC = CD = DA.

·         The diagonals bisect each other at right angles and form four isosceles right angled triangles.

·         The adjacent sides are perpendicular i.e.,

\[\angle A=\angle B=\angle C=\angle D=90{}^\circ \]

·         The diagonals of a square are of equal magnitudes i.e., AC=BD

 

PROPERTIES OF QUADRILATERAL

·         The quadrilateral formed by joining the mid-points of the consecutive sides of a rectangle is rhombus.

Here, E, F, G, H are mid-points of AB, BC, CD, DA respectively. Then, EFGH is a rhombus.

·         The quadrilateral formed by joining the mid-point of the consecutive sides of a rhombus is a rectangle. Here, PQRS will be rectangle.

 

·         The quadrilateral formed by joining the mid-points of the sides of a square, is also a square. The figure formed by joining the mid-points of the pairs of consecutive sides of   quadrilateral is a parallelogram.

Here, ABCD is a quadrilateral while BFGH is a parallelogram.

 

·         Two parallelogram on the same base and between same parallel lines have equal areas.

·         One parallelogram and one rectangle on the same base and between same parallel lines have equal areas.

·         If a rectangle and parallelogram have same dimensions x and y, then area of rectangle > Area of parallelogram.

·         One rectangle/parallelogram one triangle on the same base and between same parallel lines are related as areas of rectangle \[=2\times \] Area of the triangle.

·         Two triangles on the same and between same parallel lines have equal areas.

 

POLYGON

Any figure in a plane bounded by three or more line segments is called a polygon. A regular polygon can be inscribed in a circle. They are named according to their number of sides as triangle, quadrilateral pentagon, hexagon, for 3, 4, 5, 6, sides, respectively.

 

TYPES OF POLYGON

There are all three types of polygon

(a) Convex Polygon       

(b) Concave Polygon

(c) Regular Polygon

Convex Polygon

A polygon in which none of its interior angles is more than

\[180{}^\circ \] is called convex polygon.

Concave Polygon

A polygon in which atleast one angle is more than 180° is called concave polygon.

Regular Polygon

A regular polygon has all its sides and angles equal.

 

Important Fact or Properties Polygon

(i)   In parallelogram, rectangle, rhombus and square, the diagonals bisect each other.

(ii) Sum of all the angles in a convex polygon is

\[(2n-4)\,90{}^\circ .\]

(iii) Exterior angle of a regular polygon is \[\frac{360{}^\circ }{n}.\]

(iv) Interior angle of a regular polygon is \[180{}^\circ -\frac{360{}^\circ }{n}.\]

(v) Number of diagonals of a convex polygon with sides is \[\frac{n\,(n-3)}{2}.\]

(vi) In a convex polygon on n sides, we have

a. sum of all interior angles \[=(2n-4)\times 90{}^\circ \]

b. sum of all exterior angles \[=360{}^\circ \]

c. number of diagonals of polygon on n side \[=\frac{n\,(n-3)}{2}\]